On numerical simulations of nonlinear convection-aggregationequations by evolving diffeomorphisms
J. Carrillo and M.T. Wolfram
BIRS Workshop on ’Entropy Methods, PDEs, Functional Inequalities, and ApplicationsBanff, June 2014
Special thanks to
Jose
Jean David Benamou
Michael Neilan
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Outline
1 IntroductionGradient flowsEvolving diffeomorphisms
2 Numerical schemesImplicit-in-time discretizationFinite element methods for the Monge Ampere equation
3 Numerical results
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Nonlinear continuity equations
Let us consider a time dependent unknown probability density ρ(⋅, t) on a domainΩ ⊂ Rd , which satisfies the nonlinear continuity equation
∂tρ = −∇ ⋅ (ρu) ∶= ∇ ⋅ (ρ∇ [U ′(ρ) +V +W ∗ ρ]) .
U ∶ R+ → R denotes the internal energy.
V ∶ Rd → R is the confining potential.
W ∶ Rd → R corresponds to an interaction potential.
Nonlinear velocity is given by u = −∇ δFδρ
, where F denotes the free energy or entropy
functional
F(ρ) = ∫Rd
U (ρ)dx + ∫Rd
V (x)ρ(x)dx + 1
2∫Rd×Rd
W (x − y)ρ(x)ρ(y)dxdy.
Free energy is decreasing along trajectories
d
dtF(ρ)(t) = −∫
Rdρ(x , t)∣u(x , t)∣2dx .
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Nonlinear continuity equations
Included models:
U (s) = s log s, V = 0, W = 0 heat equation.
U (s) = 1m−1
sm , V = 0, W = 0 porous medium (m > 1) or fast diffusion (m < 1)equation.
U (s) = s log s, V given, W = 0 Fokker Planck equations or Patlak-Keller-Segelmodel.
U = 0, V = 0, W = log(−∣x ∣) or W = 12∣x ∣2 − 1
4∣x ∣4 correspond to
attraction-(repulsion) potentials in swarming, herding and aggregation models.
(a) Dictyostelium discoideum (b) Fish school
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Gradient flow formalism1
Solutions ρ can be constructed by the following variational scheme:
ρn+1∆t ∈ arg inf
ρ∈K 1
2∆td2W (ρn∆t , ρ) +F(ρ),
with K = ρ ∈ L1+(Rd) ∶ ∫Rd ρ(x)dx =M , ∣x ∣2ρ ∈ L1(Rd).
Quadratic Euclidean Wasserstein distance dW between two probability measuresµ and ν,
d2W (µ, ν) ∶= inf
T ∶ν=T#µ∫Rd
∣x −T(x)∣2dµ(x).
Variational scheme corresponds to the time discretization of an abstract gradientflow in the space of probability measures.
Solutions can be constructed by this variational scheme; naturally preservepositivity and the free-energy decreasing property.
1Jordan, Kinderlehrer and Otto (1999); Otto (1996, 2001); Ambrosio, Gigli and Savare (2005); Villani(2003).....
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Gradient flow formalism2
Let Ω and Ω be smooth, open, bounded and connected subsets of Rd . We denote
by Φ ∈ D the set of diffeomorphisms from Ω to ¯Ω (mapping ∂Ω onto ∂Ω).
Classic L2-gradient flow:
Φn+1∆t ∈ arg inf
Φ∈D 1
2∆t∥Φn
∆t −Φ∥2L2(Ω)
+ I(Φ)
converges to solutions of the PDE
∂Φ
∂t∶= u(t) ⋆Φ
= ∇ ⋅ [Ψ′(detDΦ)(cof DΦ)T ] −∇V Φ − ∫Ω∇W (Φ(x) −Φ(y))dy,
and
I(Φ) = ∫Ω
Ψ(detdΦ)dx + ∫ΩV (Φ(x))dx + 1
2∫
Ω∫
ΩW (Φ(x) −Φ(y))dxdy.
2Evans, Savin and Gangbo, Diffeomorphisms and nonlinear heat flows, 2004
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Gradient flow formalism
For a given diffeomorphism Φ ∈ D the corresponding density ρ ∈ K is given by
ρ = Φ#LN Ω which is equivalent to ρ(Φ(x))det(DΦ) = 1 on Ω
for sufficiently smooth functions.
PDE for the evolving diffeomorphisms Φ is the Lagrangian coordinaterepresentation of the original Eulerian formulation for ρ.
Idea can be generalized for different transportation costs, e.g. relativistic costsinstead of the Euclidean.
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Examples and first simulations3
Heat equation
∂ρ
∂t= ρxx ⇒ ∂Φ
∂t= − ∂
∂x( 1
Φx) = Φxx
(Φx )2,
Porous medium equation
∂ρ
∂t= ∂xx (ρm) ⇒ ∂Φ
∂t= − ∂
∂x( 1
(Φx )m) =m
Φxx
(Φx )m+1.
1D simulation of the PME for m = 2:
(c) Density ρ (d) Diffeomorphism Φ
3Carrillo and Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuityequations by evolving diffeomorphism (2009)
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Examples and first simulations3
Heat equation
∂ρ
∂t= ρxx ⇒ ∂Φ
∂t= − ∂
∂x( 1
Φx) = Φxx
(Φx )2,
Porous medium equation
∂ρ
∂t= ∂xx (ρm) ⇒ ∂Φ
∂t= − ∂
∂x( 1
(Φx )m) =m
Φxx
(Φx )m+1.
Why do we make our life so much harder ?
Automatic mesh adaptation in regions of high density.
Delta Dirac correspond to a degeneration of the transport map - numericallymore tractable than blow up maps.
Energy dissipation.
3Carrillo and Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuityequations by evolving diffeomorphism (2009)
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Boundary conditions
We consider with no flux boundary conditions, i.e.
∇ρ ⋅ n = 0 on ∂Ω,
The corresponding natural boundary conditions for the equation in Lagrangianformulation are
nT (cof DΦ)T ∂Φ
∂t= (cof DΦ)n ⋅ ∂Φ
∂t= 0.
On the square Ω = [−1,1]2 these boundary conditions translate to
∂Φ1
∂t
∂Φ2
∂x1− ∂Φ2
∂t
∂Φ1
∂x1= 0 for x1 = −1, x1 = 1
−∂Φ1
∂t
∂Φ1
∂x2+ ∂Φ2
∂t
∂Φ1
∂2= 0 for x2 = −1, x2 = 1.
Consider only diffeomorphisms which map each edge of ∂Ω to the correspondingone of ∂Ω without rotation.
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Implicit Euler scheme
Implicit in time scheme:
Φn+1 −Φn
∆t= [∇V Φn+1 + ∫ ∇W (Φn+1(x) −Φn+1(y))dy] .
Conforming finite element discretization
F(Φn+1, ϕ) = 1
∆t∫
Ω(Φn+1 −Φn)ϕ(x)dx − ∫ ∇V (Φn+1)ϕ(x)dx
− ∫Ω[∫
Ω∇W (Φn+1(x) −Φn+1(y))dy]ϕ(x)dx .
We use lowest order H 1 conforming finite elements, i.e.
Φ(x1, x2) =∑k
(Φ1k
Φ2k
)ϕk (x1, x2).
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Calculating the initial diffeomorphism
Rectangular mesh: diffeomorphism can be constructed by subsequently solving twoone-dimensional Monge Kantorovic problems in x1 and x2 direction.
Triangular mesh: No natural ordering !
Monge Ampere equation gives the optimal transportation planT = T(x) ∶ Ω→ Ω, which maps a given probability density ρX on Ω to anotherprobability density ρY on Ω with respect to the quadratic cost
∫Ω∥x −T(x)∥2ρX (x)dx .
Unique minimizing map is the gradient of a convex function u, i.e. T = ∇u,which satisfies the MA equation
det(D2u(x)) = ρX (x)ρY (∇u(x))
Initial diffeomorphism Φ0 maps the constant density one to ρI , hence
det(D2u(x)) = 1
ρI (∇u(x)).
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Vanishing viscosity approach for the MA equation
Vanishing viscosity approach: approximate a given fully nonlinear second orderPDE by a sequence of well chosen higher order quasi-linear PDEs.
Quasi-linear fourth order problem:
−ε∆2uε + det(D2uε) = 1
ρ0(∇uε)0 < ε≪ 1,
Abstract discrete formulation:
εAhuεh +Bh(uεh) = Ch(uεh),
where B(u) = det(D2u) and C(u) = 1ρi (∇u)
.
Corresponding Newton iteration creates a sequence uεk satisfying
εAhuεk+1 + (DBh [uεk ] −DCh [uε]k ])(uεk+1 − uεk ) = −Bh(uεk ) +Ch(uεk ).
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Vanishing viscosity approach for the MA equation
Spatial discretization: Hybrid C 0 conforming finite elements.Discrete fourth order operator: find u ∈ V = H 1 ×H (div) such that
⟨Ahu, v⟩ = ∫Ω
∆u∆vdx − ∑e∈Ei
∫e(∆u [ ∂v
∂n] +∆v [∂u
∂n] + α
he[∂u∂n
] [ ∂v∂n
])ds,
for all test functions v ∈ V , where [ ∂u∂n
] ∶= ∂u∂n
− ∂nu and α ∈ R.
Nonlinear discrete operator Bh is calculated from the discrete linearizationLh ∶= DBh , which is consistent with L given by
L(w) ∶= limt→0
B(u + tw) − B(u)t
= − cof(D2u) ∶ D2(w) = −∇ ⋅ (cof(D2u)∇w).
Hence we obtain
⟨Lhw , v⟩ = ∫Ω
cof(D2u)∇w∇vdx + βhe ∑e∈Ei
∫e[∂w∂n
] [ ∂v∂n
]ds.
with β ∈ R.
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Numerical simulations
Numerical solver:
1 Given an initial density ρ0 = ρ0(x) calculate the corresponding initialdiffeomorphism by determining the solution uε = uε(x) the Monge Ampereequation for a sequence of decreasing ε.
2 Map the optimal transportation plan to the initial diffeomorphism Φ0(x) = ∇uε.
3 Solve the implicit-in-time discretization for Φ = Φ(x , t) via Newton’s method inevery time step.
4 Calculate the corresponding density ρ = ρ(x , t) via
ρ(Φ(x))det(DΦ(x)) = 1.
Simulation parameters:
Computational domain Ω = [−1,1]2 discretized into 5732 triangles.
Monge Ampere solver: H 1 conforming finite elements of order 3.Vanishing viscosity approach: ε = 1,0.1,0.01,0.001.Max error for the Newton solver: 10−6.
Diffeomorphism solver: Time steps ∆t = 0.2.Max error for the Newton solver: 10−4.
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Attraction: W (x) = 12 ∣x ∣
2
(e) t = 0.2 (f) t = 0.8 (g) Entropy
(h) t = 0.2 (i) t = 0.8
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Attraction-repulsion:W (x) = 12 ∣x ∣
2−
14 ∣x ∣
4
(j) t = 0.8 (k) t = 1.4 (l) Entropy
(m) t = 0.8 (n) t = 1.4
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Attraction-repulsion: W (x) = 12 ∣x ∣
2− ln(∣x ∣),
V (x) = −14 ln(∣x ∣)
(o) t = 0.4 (p) t = 1 (q) Entropy
(r) t = 0.4 (s) t = 1
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Conclusion and future work
Conclusions:
Presented an numerical algorithm for solving nonlinear convection-aggregationequations, which automatically adapts the mesh in case of mass aggregation.
It is based on the Lagrangian formulation and preserves energy dissipation.
Can be used for general geometries due to triangular mesh.
Future work:
Include nonlinear diffusion.
Speed up simulations (operator splitting, assembling of the convolution matrix).
Higher order basis functions for the diffeomorphism solver.
Boundary conditions for more general geometries; better ’treatment’ in thenumerical solver.
Thank you very much for your attention !
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Conclusion and future work
Conclusions:
Presented an numerical algorithm for solving nonlinear convection-aggregationequations, which automatically adapts the mesh in case of mass aggregation.
It is based on the Lagrangian formulation and preserves energy dissipation.
Can be used for general geometries due to triangular mesh.
Future work:
Include nonlinear diffusion.
Speed up simulations (operator splitting, assembling of the convolution matrix).
Higher order basis functions for the diffeomorphism solver.
Boundary conditions for more general geometries; better ’treatment’ in thenumerical solver.
Thank you very much for your attention !
19 / 20
Bibliography
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Numerical simulation of diffusive and aggregation phenomena in nonlinear continuityequations by evolving diffeomorphisms, SIAM J. Sci. Comput., 31 (2009/10).
S. Brenner, T. Gudi, M. Neilan, and L.-Y. Sung
C0 penalty method for the fully nonlinear Monge-Ampere equation, Mathematics ofComputation, 80 (2011).
L. Evans, O. Savin, and W. Gangbo,
Diffeomorphisms and nonlinear heat flows, SIAM Journal on Mathematical Analysis, 37(2005).
X. Feng and M. Neilan,
Analysis of Galerkin methods for the fully nonlinear Monge-Ampere equation, Journal ofScientific Computing, 47 (2011).
M. D’Orsogna, Y.L. Chuang, A.L. Bertozzi, L.S., Chayes,
Self-propelled particles with soft-core interactions: patterns, stability, and collapse, Physicalreview letters, 96 (2006)
Y. Huang and A. L. Bertozzi,
Asymptotics of blowup solutions for the aggregation equation, Discrete Contin. Dyn. Syst.,Ser. B, 17 (2012).
D. Balague, J.A. Carrillo, T. Laurent, and G. Raoul,
Nonlocal interactions by repulsive–attractive potentials: radial ins/stability, Physica D:Nonlinear Phenomena, 260 (2013).
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