Stochastic Differential Equations

SSES, Spring 2015

Radu Hubei

⚫️

⚫️

⚫️

Today

Most of the material comes from Sigrist et al . ( 2015)

I will not"

present" the

paper ,rather use it for some results .

Overview :

Briefintro to Fourier analysis

PDES and Fourier functions

SPDES and Fourier functions

Conclusions

Fourier transformationei 't ask ) +, isinlx )

f :1R→1R f EL ,( IR)

{ ( w ) = fµfln) tin " da → the Fourier transform of f fh=F( f)

f In ) = zt ) { ( w ) eiwrdw → inverse Fourier transform f = F-'

(F)112

Basic properties

Hfhlko £ IHH, (f*g)H=ffH-y)g1y)dy

Ilw) is uniformly continuous on - a

Fourier series

A periodic function f C ) flx )=f( xtzt )

admits the series representation

flx )= Azt ¥,

aw as ( wx ) + bwsiwlwx )

{ cos ( nx ),

sin ( nx ) } form a complete Ortho normal system

Discrete Fourier Transform

given a sequence { fo , f , , . . . , fn , } its DFT is the sequence { To , ... ,fn . , }

I u=m⇐ofi . exp ( - Ziti F- . k ) k=0 , I , . . . , ntThe inverse DFT fm=nt⇐o§a

- exp ( Zti htm ) m= 0,1 , . . . , ni

Notation wh=2t . F → frequency

{ 0 , 1 , 2 , . . . , nt ) ' 2¥ → set of frequenciesOr

10,

1

, , . . . ,±z - i ,- he , . . . , -1 ] . 2¥ ( same sin and cos )

🔴

Real Fourier functions

replace expfiwx ) with [ cos ( wx ) , sin ( wx ) ]

In =m±}of... exp f- Ziti nfuik)becomes

I =n¥of,i asfztmnh ) I=n¥of,isinf2Tm⇒lk 24The inverse is

ht

fm=tnI,

I,u

. cos ( zit . mw± ) + In . sin ( 2T .m÷ )

compare with fmth "£o§a - exp ( Zti In .m )

Example real Fourier functions aslwix ) and sin ( wix )Wo W , Wz Wz

§

3is

N

Say flx ) = x 't sinhox ) observed at x⇐ho , Nt , In , . . . ,'¥ }

.. .

The DFT is found using the fft function ( most languages )

n ^

f ,µ fzn

Ww Ww

⚫️

⚫️

⚫️

⚫️

ZD Fourier representations

everything extends naturally to functions f=f( say )

Given a ZD sequence ( Amw ) 0£ m< M 0 en < N

the ZD DFT is defined as the sequence (Ak ,e ) OE k

Example teal as( kjs )=us( kijxtkwy ) and sin(kB)=sin( kjxtkuy )

n

:a

⚫️

⚫️

⚫️

⚫️ 😊

PDES and Fourier representations

¥÷= cheat b ¥ f=f( t , x ) + initial condition

For a fixed t felt ,x ) = function of x

= ÷€ A ,j . cos ( W ; x ) + Azj sin ( wjx )

J L

make these depend on t

Model : flt ,x)=¥o9iH ) wslwix) + autttsinlwix )

Solving the PDE above is now easy

It is !

Ft = Z an'

Has( win ) + au.

'

H sin ( wix )

feta = Z a,i It ) wit sin ( wix )) tazilt ) wi as ( wi D

Tah = E a ,i H ) w ,? f- us ( wi xD + an. It ) within lwix ))

an !It ) = - to we qi It ) + cwiazitt 'au

! It ) = . bw,

? au . HI - cwi a ,iH

😊

" an " "

Hateful,:HHFha 'iH= A ayt ) - aiH=exp( At ) ailo )

(matrix exp)

this can be solved

easilyutility

.mu#H:llY.itYLesson learned : we solved a PDE by solving a " much simpler " ODE .See Cressie & Wikle

,Sec

.7.26 for more discussion on this topic .

Example0¥=c2£+b It b= 0.007 c= 0.03

f1÷

tt ,x)=

Two dimensional example

z÷5H,s)=-µt95H,s

) +

TEDSH,s) -

TH,s) stay )

µ=( µ , ,µ)T = velocity vector 5>0 ="

sink"

term ( damping )

[ = ( EI,

EY,) = diffusion matrix

% = - µ , £ - µz9y+£( E" £+524 )+Fy( Eu Z +En5y ) - 59

Model :JH ,s)=⇐&g.lt) exp (ikjts) K ;= ( th ; , kz ;)= spatial freq .

or esths ) = ¥1,4 ,tHws( Kots)+%Hsinks)

Easily seen that the coefficients xjtt ) satisfy : ( complex version )

gift ) =L ; - L ;H) he;= -iµK; -KYEK;- E

µt4exp(IKB) =ink;explikots)9 . ETexplikots) = -KFEK; exp (ikjts)thusftp.T.TT- 5) [{44explikjs) = ¥dh;gHexp(iksts)= II

,diHuplil#

.

Thus

xjtt )=h ; . X ; It )

The solution is a ;H= exp ( hjt ) g :( o)

Real Fourier functions esths ) =÷},4,tHcos(KB)+%.Hsin( KB)Find derivatives : JH , D=hIz9sHcoslkBtastHsinCkIDFe5HD-EE9.HFfsinCkIsDth.HkY@slkY5DsyFesH.s) = . . .

It follows that dijtt ) and dutt ) must satisfy :

gilt )=[k¥kj.E) x , ;H ) - (MTK; )x↳Hdzstt )= ( - k¥k; .f) %.Htfnk;)gottajH)= [ L , ;H) &zjH[Xjtt)=H ; a ;H ) -k¥kj . E -MTK ;As '= µtk; - k¥k;Z:t)=exp( Hjt )XoH)

( matrix exp )

⚫️

⚫️

⚫️

What did we learn

Using Fourier functions PDE simplifies to ODE ( this case with exact sol )

ODES are easier to discretize ( no need for CFL condition)

Previous examp

: ( tto ) = exp ( H ;D ) a ; It )or

...a

; ( tto ) = ( It H;D) djtt )

Example z÷5H,s)=-µTHH,s)+TEDSH,s) - SJH,s)E

: := . leg( 99)

,

µ=(0.2

,0.2 ) ,

E=diag ( 0.001 , 0.001 )

Stochastic PDES

÷ 5ft ,s)= . MtHH ,s ) + TEDSH,s) - SJH,s) + EH ,s )

EH ,S) = Gaussian process temporally white and spatially colored .

Sigrist et al . consider the Matern spatial covariance structure

Proposition ( Sigrist et al . ) If - Gaussian while noise

5( o ,s)=I€ djlo ) exp ( i KYD Ett ,s) = ,÷} EH exp ( i KYDthe "

§(t,s) =÷E, djtt) exp ( i KYS) with

&jH= exp ( hit )g( o) + ftexp ( h ; H - D) Flu) du

⚫️

⚫️

What does that mean ?

Easy to simulate ! Just propagate the coefficients d ; ( i ) via

Xjttto ) = exp ( H; b) gtt ) + N ( 0 , § )§ = diag ( f ( k ;)

1 - exPt2o( KEK +5 ))2k¥Emig ]

Ready for Bayesian inference .

Simulations of SPDE

while noise EHS)

Matern EH ,s )

⚫️

⚫️

⚫️

⚫️

Discussion

Fourier representations provide several advantages

( elegant solutions , computational efficiency )

Method works nicely when there is a"

model grid" (data on a grid)

( If not , use date augmentation , an incidence matrix , etc.)

Suitable for Bayesian inference . Easy to incorporate Meas . error .

Extensions ?