Statistical Approach Continuity and perturbation results Temporal approximation Stationary Statistical Properties of Dissipative Systems Wang, Xiaoming [email protected]Florida State University Stochastic and Probabilistic Methods in Ocean-Atmosphere Dynamics, Victoria, July 2008 Wang, Xiaoming [email protected]Stationary Statistical Properties of Dissipative Systems
Transcript
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistical Properties of DissipativeSystems
Wang Xiaomingwxmmathfsuedu
Florida State University
Stochastic and Probabilistic Methods in Ocean-AtmosphereDynamics Victoria July 2008
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Outline
1 Statistical Approach
2 Continuity and perturbation results
3 Temporal approximation
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Logistic map
T (x) = 4x(1minus x) x isin [0 1]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Lorenz 96 modelEdward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 0 1 middot middot middot J J = 5 F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
L96_demoavi
Media File (videoavi)
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Outline
1 Statistical Approach
2 Continuity and perturbation results
3 Temporal approximation
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Logistic map
T (x) = 4x(1minus x) x isin [0 1]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Lorenz 96 modelEdward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 0 1 middot middot middot J J = 5 F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
L96_demoavi
Media File (videoavi)
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Logistic map
T (x) = 4x(1minus x) x isin [0 1]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Lorenz 96 modelEdward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 0 1 middot middot middot J J = 5 F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
L96_demoavi
Media File (videoavi)
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Lorenz 96 modelEdward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 0 1 middot middot middot J J = 5 F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
L96_demoavi
Media File (videoavi)
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Lorenz 96 modelEdward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 0 1 middot middot middot J J = 5 F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
L96_demoavi
Media File (videoavi)
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistical approach
dudt
= F(u) u isin H
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Spatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutions
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v) vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure Joseph Liouville1809-1882
Figure Eberhard Hopf1902-1983
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Liouville and Hopf equations
Liouville type equation
ddt
intH
Φ(v) dmicrot(v) =
intH
lt Φprime(v) F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((v v1) middot middot middot (v vN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Hopfrsquos equation (special case of Liouville type)
ddt
intH
ei(vg) dmicrot(v) =
intH
i lt F(v) g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Stationary Statistics Solutions (IM)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintH
lt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Stationary statistical solutions IM are not necessarily supportedon steady state solutions or periodic orbitsUncertainty in both initial condition and parameter(s) (modelerror)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intH
ϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Maximum entropy for conservative case
Maximum entropy principle most probable pdf maximize theShannon entropy S = minus
intp ln p
Entropy is conserved in the deterministic case with Liouvilleproperty (nabla middot F = 0)
d ~Xdt
= ~F (~X ) + εd ~Wdt
Fokker-Planck equation (Kolmogorov Smoluchowski)
partppartt
+ ~F middot nabla~X p minus ε2
2∆~X p = 0
Equation for the density of Shannon entropy Q = minusp ln p
partQpartt
+nabla~X middot (~FQ)minus ε2
2∆~X Q =
ε2
2p|nablap|2
Monotonicity of Shannon entropy (noise increases uncertainty)ddtS(p(t)) ge 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
More on statistical theories for complex dynamical systems can befound
Majda AJ and Wang X Nonlinear Dynamics and StatisticalTheories for Basic Geophysical Flows Cambridge UniversityPress 2006Majda AJ Abramov R and Grote M Information theory andstochastics for multiscale nonlinear systems CRM monographseries American Mathematical Society 2005Foias C Manley O Rosa R Temam R Navier-StokesEquations and Turbulence Cambridge University PressCambridge UK 2001
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Dissipative system
Definition
A dynamical system S(t) t ge 0 on a phase space H is calleddissipative if there exists a global attractor A such that
A is invariant under S(t)A is compactA attracts any bounded set B in H ie
limtrarrinfin
dist(S(t)BA) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and attractors
Theorem (IM and the global attractors W 08)1 IM is a convex compact set (with respect to the weak topology)2 suppmicro sub Aforallmicro isin IM
Singular nature of invariant measureEarlier work under existence of a compact absorbing set (CiprianFoias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
IM and time averages
Theorem (IM and time averages W 08)
Assume H reflexive S dissipativeforallu0 isin HforallLIM rArr existmicro isin IM such that
LIMTrarrinfin1T
int T
0ϕ(S(t)u0) dt =
intH
ϕ(u) dmicro(u)forallϕ isin C(H)
Earlier work under existence of a compact absorbing set or smallerclass of weakly continuous functionals (Foias et al)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Ergodicity and extremal points
Theorem (Ergodicity and extremal points W 08)
Let IM be the set of all invariant probability measures of adissipative dynamical system S(t) t ge 0 Then an invariantmeasure micro is ergodic if micro is an extreme point of IM Moreover if thedynamical system is injective on the global attractor A then everyergodic invariant measure must be an extremal point of IM
Other versions with group (ODE) assumption is well-known
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Regular perturbation
Theorem (Conv of IM regular version W 08)
Assume for S(t ε)1 (uniformly dissipativity) pre-compactness of K =
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Two time scale set-up
Two-time-scale problem X1 X2 Hilbert spaces
ε(du1
dt+ g(u1 u2)) = f1(u1 u2) u1(0) = u10
du2
dt= f2(u1 u2) u2(0) = u20
Limit problem ( ε = 0)
0 = f1(u01 u0
2)
du02
dt= f2(u0
1 u02) u0
2(0) = u20
y = f1(u1 u2) hArr u1 = F1(u2 y)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Conv of stat prop 2 time scale case
Theorem (Conv of IM singular version W 08)
Assume1 (uniform dissipativity) pre-compactness of K =
⋃0ltεltε0
Aε
2 (dissipativity of the limit system) A0 in X23 (conv of the slow variable)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Conv of stat prop 2 time scale case
Theorem (Conv of IM singular version W 08)
Assume1 (uniform dissipativity) pre-compactness of K =
⋃0ltεltε0
Aε
2 (dissipativity of the limit system) A0 in X23 (conv of the slow variable)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Conv of stat prop 2 time scale case
Theorem (Conv of IM singular version W 08)
Assume1 (uniform dissipativity) pre-compactness of K =
⋃0ltεltε0
Aε
2 (dissipativity of the limit system) A0 in X23 (conv of the slow variable)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Conv of stat prop 2 time scale case
Theorem (Conv of IM singular version W 08)
Assume1 (uniform dissipativity) pre-compactness of K =
⋃0ltεltε0
Aε
2 (dissipativity of the limit system) A0 in X23 (conv of the slow variable)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Conv of stat prop 2 time scale case
Theorem (Conv of IM singular version W 08)
Assume1 (uniform dissipativity) pre-compactness of K =
⋃0ltεltε0
Aε
2 (dissipativity of the limit system) A0 in X23 (conv of the slow variable)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Conv of stat prop 2 time scale case
Theorem (Conv of IM singular version W 08)
Assume1 (uniform dissipativity) pre-compactness of K =
⋃0ltεltε0
Aε
2 (dissipativity of the limit system) A0 in X23 (conv of the slow variable)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Conv of stat prop 2 time scale case
Theorem (Conv of IM singular version W 08)
Assume1 (uniform dissipativity) pre-compactness of K =
⋃0ltεltε0
Aε
2 (dissipativity of the limit system) A0 in X23 (conv of the slow variable)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Rayleigh and Beacutenard
Figure Lord Rayleigh (JohnWilliam Strutt) 1842-1919
Figure Henri Beacutenard (left)1874-1939
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to RBC W CPAM 07
Boussinesq system for Rayleigh-Beacutenard convection
1Pr
(partupartt
+ (u middot nabla)u) +nablap = ∆u + Ra kθ nabla middot u = 0 u|z=01 = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to RBC W CPAM 07
Boussinesq system for Rayleigh-Beacutenard convection
1Pr
(partupartt
+ (u middot nabla)u) +nablap = ∆u + Ra kθ nabla middot u = 0 u|z=01 = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to RBC W CPAM 07
Boussinesq system for Rayleigh-Beacutenard convection
1Pr
(partupartt
+ (u middot nabla)u) +nablap = ∆u + Ra kθ nabla middot u = 0 u|z=01 = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to RBC W CPAM 07
Boussinesq system for Rayleigh-Beacutenard convection
1Pr
(partupartt
+ (u middot nabla)u) +nablap = ∆u + Ra kθ nabla middot u = 0 u|z=01 = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
RBC set-up and numerics
Figure RBC set-up Figure Numerical simulation
(infin Pr simulation)
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
m1mpg
Media File (videompeg)
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Possible deficiency of classical schemes
dudt
= F(u) u isin H
classical scheme of order m
u(n∆t)minus un le C(n∆t)∆tm
Dependence on TC(T ) = exp(αT )
Error in approximation of long time averages
| lim supNrarrinfin
1N
Nsumn=1
(Φ(u(n∆t))minus Φ(un))|
le c lim supNrarrinfin
∆tm exp((N + 1)α∆t)minus exp(α∆t)exp(α∆t)minus 1
= infin
Classical schemes may not be able to capture the climatealthough they may work very well for weather
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Difficulty with large Rayleigh number
Infinite Pr number model
partθ0
partt+ Ra Aminus1(kθ0) middot nablaθ0 minus Ra Aminus1(kθ0)3 = ∆θ0
A Stokes operatorAlternative form with s = Ra t
partθ0
parts+ Aminus1(kθ0) middot nablaθ0 minus Aminus1(kθ0)3 =
1Ra
∆θ0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Theorem (General result W 08)
Assume1 (Uniform dissipativity) K =
⋃0ltklek0
Ak
2 (Finite time uniform convergence)supuisinAk nkisin[t0T ] Sn
k uminus S(nk)u rarr 0 as k rarr 0
3 (Uniform continuity of the continuous system)limtrarrT supuisinK S(t)uminus S(T )u = 0
Then1 (Conv of stationary stat prop)
microk micro microk isin IMk micro isin IM
2 (Conv of attractors)
limkrarr0
dist(Ak A) = 0
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Application to infin Pr model
infin Prandtl number model (alternative form)
partθ
partt+ Ra Aminus1(kθ) middot nablaθ minus Ra Aminus1(kθ)3 = ∆θ θ|z=01 = 0
Semi-implicit scheme
θn+1 minus θn
k+ Ra Aminus1(kθn) middot nablaθn+1 + Ra Aminus1(kθn)3 = ∆θn+1
Equivalent form (T = θ + 1minus z)
T n+1 minus T n
k+ Ra Aminus1(kT n) middot nablaT n+1 = ∆T n+1
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Nusselt number (recast)
For the infinite Prandtl number model
Nu = supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
|nablaT (x s)|2 dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kT (x s))3T (x s) dxds
= 1 + Ra supθ0isinL2
lim suptrarrinfin
1tLxLy
int t
0
intΩ
Aminus1(kθ(x s))3θ(x s) dxds
For the scheme
Nuk = 1 + Ra supθ0isinL2
lim supNrarrinfin
1NLxLy
Nsumn=1
intΩ
Aminus1(kθn(x))3θn(x) dx
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Summary
Invariant measures of appropriate numerical approximationsconverge to invariant measures of the continuous in timedynamical system under approximationSpecific stationary statistical properties (such as Nusseltnumber) also convergeSame idea can be applied to many dissipative dynamicalsystemsUniform dissipativity is crucialLong way to go to reach our goal
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems
Statistical Approach
Continuity and perturbation results
Temporal approximation
Statistical ApproachContinuity and perturbation results
Temporal approximation
Questions
Continuity instead of upper semi-continuity Linear responseStatistical hysteresisUniqueness of physical invariant measure Noise effectBalancing mixing rate and errorConvergence rate At least for certain good statisticalquantitiesHigh order schemesExplicit time stepping (Perhaps with posterior approach)Fully discrete schemesOther schemesPartiallyweakly dissipative systems
Wang Xiaoming wxmmathfsuedu Stationary Statistical Properties of Dissipative Systems