Lagrangian discretization of incompressibility, congestion and linear diffusion

Transcription

1 Lagrangian discretization of incompressibility, congestion and linear diffusion Quentin Mérigot Joint works with (1) Thomas Gallouët (2) Filippo Santambrogio Federico Stra (3) Hugo Leclerc Seminaire du LJLL 19 Octobre 2018

2 Outline 1. General formalism 2. Incompressible Euler equations 3. Crowd motion 4. Linear diffusion

3 Lagrangian formulation of (some) PDEs We track the evolution of a population of particles (= probability distribution) ρ 0 P 2 (R d ), whose displacement field s : [0, T ] L 2 (ρ 0, R d ) satisfies ṡ E(s) (gradient flow) or s E(s) (hamiltonian flow) The distribution of particles at time t is then ρ(t) = s(t) # ρ 0. Main assumption: The energy/entropy E only depends on the distribution of particles, i.e. E(s) = E(s # ρ 0 ), with E : P 2 (R d ) R {+ }. { ρ log ρ if ρ P ac 2 (Ω) E ent (ρ) = + if not { 0 if ρ Leb Ω E cong (ρ) = + if not E inc (ρ) = E pot (ρ) = V d ρ { 0 if ρ = Leb Ω + if not NB: E is typically non-convex, with values in R {+ }.

4 Lagrangian-Eulerian dictionnary ṡ E(s) s E(s) E inc incompressible Euler equation E cong + E pot crowd motion pressureless Euler equation E ent + E pot linear Fokker-Planck equation isentropic Euler equation Wasserstein gradient-flows: [Otto 99], [Jordan-Kinderlehrer-Otto 98], [Ambrosio-Gigli-Savaré 08], etc. [Evans-Gangbo-Savin 08] incompressible Euler as geodesics: [Arnold 66], [Ebin Marsden 70], [Brenier], Numerical advantages of using a Lagrangian formulation: 1) handling many phases 2) tracking of individual particles / interfaces 3) allowing singular solutions (e.g. Dirac masses) Moreau-Yosida regularization: E ε (s) := inf s ṡ ε = E ε (s ε ) or s ε = E ε (s ε ) 1 2ε s s 2 + E(s )

5 2. Incompressible Euler equations Joint work with Thomas Gallouët

6 Geodesics on a submanifold of R d K = submanifold of R d, d K = distance to K, Π K = projection on K { { E(s) = 0 if s K Nor s K if s K E(s) = + if not if not E ε (s) = 1 2ε d2 K(m) E ε (s) = s Π K(s) ε a.e. Equation of geodesics: s E(s) ( s Nor s K = (T s K) and s K) approximated by s ε = E ε (s ε ) s ε + s ε Π K (s ε ) ε = 0 Nor K (Π K s ε ) Example: K = R {0} R 2, s(0) = 0, s ε (0) = (0, h ε ), ṡ(0) = ṡ ε (0) = (1, 0) h ε with s ε = (x ε, y ε ) we have { ẍ ε = 0 ÿ ε + 1 ε y ε = 0 i.e. { x ε (t) = t y ε (t) = h ε cos(t/ ε)

7 Incompressible fluids as geodesics Setting: Ω R d bounded connected, Ω = 1, ρ 0 = Leb Ω [Arnold 1966] Measure-preserving maps: K = {s L 2 (ρ 0, R d ) s # ρ 0 = ρ 0 }, E = i K s E(s) ( s Nor K (s) and s K) s 0 (x) s t (x) ( s = p s and s K) [ ] Lemma: a) Norid K = (T id K) { p p H 1 (Ω)} b) Nor s K = {w s w Nor id K} With u t := s t s 1 t (= velocity in Eulerian coordinates), one recovers Euler s eqs: t u + div(u u) = p in Ω divu = 0 in Ω u n = 0 on Ω Can this formulation be used for numerical computations (Brenier)? Minimizing geodesics (with JM Mirebeau, 15) / Cauchy pb (with T. Gallouet, 16).

8 Distance to measure-preserving maps I Measure-preserving maps: K = {s L 2 (ρ 0, R d ) s # ρ 0 = ρ 0 }, ρ 0 = Leb Ω Wasserstein distance: W 2 2(ρ, µ) = min{ x T (x) 2 d ρ(x) T # ρ = µ} Polar factorization (Brenier): (i) d 2 K(s) = W 2 2(ρ 0, s # ρ 0 ) (ii) If µ := s # ρ 0 is absolutely continuous,! optimal transport map S between µ and ρ 0 and then Π K (s) = {S s}. Moreover, S = φ where φ : Ω R is a convex function. [Brenier 92] With E = i K, E ε (s) = 1 2ε d2 K(s) E ε (s) = W 2 2(s # ρ 0, ρ 0 ) E ε (s) = s Π K(s) ε E ε (s) = id S ε s S = OT between s # ρ and ρ s ε = p ε s ε p ε = 1 ε ( 1 2 x 2 φ ε ) Formally, s ε = E ε (s ε ) [assuming µ ε Leb Ω ] µ ε = s ε# ρ 0 φ ε# µ ε = ρ 0, φ ε convex

9 Distance to measure-preserving maps II Definition: Given y 1,..., y N R d, let D(y 1,..., y N ) := N W 2 2(ρ 0, 1 N i δ y i ). Example: N = 900, ρ 0 = Leb [0,1] 2. y 1,..., y N 1 N D 0, 031 (y ( yi D) 1 i N i yi D) 1 i N

10 Distance to measure-preserving maps II Definition: Given y 1,..., y N R d, let D(y 1,..., y N ) := N W 2 2(ρ 0, 1 N i δ y i ). Example: N = 740, ρ 0 = Leb [0,1] 2. y 1,..., y N 1 N D 0, 14 ( y i D) 1 i N (y i yi D) 1 i N yi D(y 1,..., y N ) = B i (y 1,..., y N ) where a) T = is the optimal transport map from ρ 0 to 1 N b) V i := T 1 ({y i }) i δ y i c) B i (y 1,..., y N ) = N V i x d ρ 0 (x) = barycenter of V i

11 Space discretization of Euler s equations [Brenier, CMP 2000] [Gallouët M., 2016] Setting: let M N := {s = i y i1 ωi y 1,..., y N R d } L 2 (ρ 0, R d ) where ρ 0 (ω i ) = 1 N, ρ 0(ω i ω j ) = 0 and diam(ω i ) N 1 d. Given s = i y i1 ωi M N, (1) s + E ε MN (s) = 0 (2) ÿ i + 1 ε N yi D(y 1,..., y N ) = 0 (3) ÿ i + 1 ε N (y i B i (y 1,..., y N )) = 0 y i (t) is attached by a spring to the time-dependent barycenter B i (y 1 (t),..., y N (t)). Theorem: Let (u, p) be a regular (e.g. C 1,1 ) solution to Euler s equations and let s satisfy (1) with initial condition (s(0), ṡ(0)) = (Π MN (id), Π MN (u 0 )). Then, t [0, T ], ṡ t u t s t 2 L 2 (ρ 0,R d ) h2 N ε + ε + h N w. h N = N 1/d Proof: Gronwall on modulated energy E u (t) = 1 2 ṁ t u t m t ε 2 d 2 S(m t )

12 Numerical computation of D Computation of the Wasserstein distance: D(y 1,..., y N ) := N W 2 2(ρ 0, 1 N i δ y i ) We seek φ convex s.t. φ # ρ = 1 N i δ y i = φ = max i y i ψ i for some ψ R N. = φ # ρ = i ρ(v i(ψ))δ yi, where V i (ψ) = {x R d j, x y i ψ i x y j ψ j }. The partition (V i (ψ)) 1 i N can be computed in near-linear time in 2D/3D Corollary: Computing D(y 1,..., y N ) Finding φ convex such that φ # ρ = ν, Finding ψ R N s.t. i, ρ(v i (ψ)) = 1 N. Numerics: [Cullen Purser 84], [Oliker Prussner 88], [Aurenhammer Aronov Hoffman 98] [M. 11], [de Goes et al 12], [Lévy 15], [Kitagawa M. Thibert 16] [Lévy 15] + [KMT 16] handles points in R 3, within 20min on a laptop.

13 Numerical computation of D Computation of the Wasserstein distance: D(y 1,..., y N ) := N W 2 2(ρ 0, 1 N We seek φ convex s.t. φ # ρ = 1 N V i (ψ) = {x R d j, x y i ψ i x y j ψ j }. i δ y i = φ = max i y i ψ i for some ψ R N. = φ # ρ = i ρ(v i(ψ))δ yi, where The partition (V i (ψ)) 1 i N can be computed in near-linear time in 2D/3D i δ y i ) Corollary: Computing D(y 1,..., y N ) Finding φ convex such that φ # ρ = ν, Finding ψ R N s.t. i, ρ(v i (ψ)) = 1 N. For such a ψ R N, the Wasserstein distance and its gradient are given by D(y 1,..., y N ) = N i V i (ψ) x y i 2 d ρ 0 (x) 1 2 y i D(y 1,..., y N ) = y i bary(v i (ψ)) =: B i (y 1,..., y N )

14 Numerical result: Stationary flow on [0, 1] 2 Stationary flow on [0, 1] 2 : speed: u(x) = (cos(πx 1 ) sin(πx 2 ), sin(πx 1 ) cos(πx 2 )) pressure: p(x) = 1 4 (sin2 (πx 1 ) + sin 2 (πx 2 )) X = [0, 1] 2

15 Numerical result: Instabilities Objectives: Larger computations, with more complex behaviour. Preservation of the Hamiltonian by the discrete scheme. A. Discontinuous initial velocity u 0 = (.5, 0) u 0 = (1, 0) Hamiltonian preservation X = [0, 2] [.5,.5]/(x = 0 x = 2) 200k particles, 2000 timesteps, t max = 8

16 Numerical result: Instabilities Objectives: Larger computations, with more complex behaviour. Preservation of the Hamiltonian by the discrete scheme. B. Rayleigh-Taylor instability (Inhomogeneous fluid) ρ = 3 gravity velocity Verlet symplectic Euler ρ = 1 Hamiltonian preservation X = [ 1, 1] [ 3, 3] 50k particles, 2000 timesteps, t max = 2

17 3. Crowd motion Joint work with F. Santambrogio and F. Stra

18 Constrained gradient flows in R d K = compact convex subset of R d V C 1 (R d ), V 0 in all cases, y(0) = y 0 R d Constrained gradient flow: ẏ ( V (y) + i K (y)) ẏ = V (y) w y K w N y K ( ) ε-constrained gradient flow: ẏ ε = (V + 1 2ε d2 K)(y ε ) ẏ ε = V (y ε ) 1 ε (yε Π K (y ε )) Nor ΠK (y ε )K As ε 0, y ε converges locally uniformly, up to subsequence, to a weak solution of ( ). K Π K (y ε ) y ε

19 Crowd motion model [Maury, Roudneff-Chupin, Santambrogio 09] Ω R d compact, Ω > 1, V C 1 (R d ), V 0 K = {s L 2 (Ω, R d ) s # ρ 0 Leb Ω } where ρ 0 P ac (Ω) E pot (s) = V s d ρ 0 E cong (s) = i K (s) Lagrangian formulation: ṡ (E pot + E cong )(s) If s K and ρ = s # ρ 0 is the density of people, one expects T s K = {v s div(v) 0 on {ρ = 1} + no flux b.c.} i.e. E cong (s) = N s K = { p s p 0, p(1 ρ) = 0}. ṡ = V s p s ρ = s # ρ 0 Leb Ω p 0, p(1 ρ) = 0 Eulerian formulation: t ρ + div(ρv) = 0, v = V p p 0, p(1 ρ) = 0 ρ Leb Ω Gradient flow in (P(Ω), W 2 ) of the energy F(ρ) := V d ρ + i LebΩ (ρ)

20 Moreau-Yosida regularisation of E cong Admissible displacements: K = {s L 2 (ρ 0, R d ) s # ρ 0 Leb Ω }, E cong = i K E cong,ε (s) = 1 2ε d2 K(s), Formally, ṡ ε = ( E pot + E cong,ε )(s ε ) E cong,ε (s) = s Π K(s) ε ṡ ε = V (s ε ) p ε Π K (s ε ) σ ε = Π K (s ε ) # ρ 0 p ε 0, p ε (1 σ ε ) = 0 The computation of E cong,ε is justified using optimal transport techniques. Proposition: (i) d 2 K(s) = min σ P(Ω), σ LebΩ W 2 2(σ, s # ρ 0 ) (ii)!σ realizing the minimum (iii) If µ := s # ρ 0 is a.c. then Π K (s) = {S s} where S = OT between µ and σ Moreover, S 1 = id p with p 1-concave, p 0, p(1 σ) = 0. [M, RC, S 09] i.e. ε E cong,ε (s) = s Π K (s) = s S s = (S 1 id) S s = p Π K (s)

21 Discretization of the crowd motion model Eulerian schemes: [Maury, Roudneff-Chupin, Santambrogio 10] [Benamou, Carlier, Laborde 15] catching up ALG2-JKO Lagrangian setting: Let M N := {s = i y i1 ωi y 1,..., y N R d } L 2 (ρ 0, R d ) [with s = 1 N i δ y i ] We let µ N (t) = 1 N where ρ 0 (ω i ) = 1 N and ρ 0(ω i ω j ) = 0, { ṡ = ( E pot MN + E cong,εn MN )(s) s(0) = Π MN (id) { y i (t) = V (y i (t)) 1 ε N yi D (y 1 (t),..., y N (t)) y i (0) = N ω i x d ρ 0 where D (y 1,..., y N ) := N 2 inf σ 1 W 2 2(σ, 1 N i δ y i ) i δ y i (t) P(Ω) the corresponding distribution of particles.

22 Interpretation of the discretization Given y 1,..., y N R Nd : (example: y 1,..., y N iid in [0,.8] 2, Ω = [0, 2] 2 ) y 1,..., y N σ 1)!σ arg min ρ 1 W 2 2(ρ, 1 N i δ y i ) [Maury, Roudneff-Chupin, Santambrogio] 2)!T OT map from σ to 1 N [Brenier] i δ y i T 1 ({y i }) B i (y 1,..., y N ) 3) With V i := T 1 ({y i }) spt(σ), D (y 1,..., y N ) = N i V i x y i 2 d x 1 2 y i D (y 1,..., y N ) = N V i y i x d x =: y i B i (y 1,..., y N ) y i = V (y i ) 1 ε N (B i (y 1,..., y N ) y i )

23 Convergence theorem Let M N := {s = i y i1 ωi y 1,..., y N R d } L 2 (ρ 0, R d ), { ṡ N = ( E pot MN + E cong,εn MN )(s) s N (0) = Π MN (id) We denote µ N (t) = 1 N i δ y i (t) P(Ω) the corresponding distribution of particles. Theorem: If W 2 2(ρ 0, µ N (0)) ε N and ε N 0, then µ N ρ C 0 ([0, T ], P(Ω)), where ρ satisfies in the sense of distribution, for some p L 2 ([0, T ], H 1 (Ω)), t ρ + div(ρv) = 0, ρ K, v = V p, p 0, p(1 ρ) = 0, [M., Stra, Santambrogio 18]

24 Numerical examples 1) Square + gravity 1) Nonconvex domain + gravity 3) Gravity + aggregation 4) Two populations Fast solver for computing E cong,ε by Hugo Leclerc (LMO):.1s for N = 10 5!

25 4. Linear diffusion

26 Linear diffusion Setting: E ent (s) = E ent (s # ρ 0 ) with E ent (ρ) = { ρ log ρ if ρ P ac 2 (R d ) + if not Moreau-Yosida regularization in L 2 M-Y regularization in (P(R d ), W 2 ): i.e. E ent,ε (s) = E ent,ε (s # ρ 0 ) where E ent,ε (s) = min s E ent,ε (µ) = min σ P ac (R d ) 1 2ε s s 2 + E ent (s) 1 2ε W2 2(σ, µ) + E ent (σ) Letting D ent,ε (y 1,..., y N ) = E ent,ε ( 1 N i δ y i ), we can compute the ε-entropy of a point cloud in near-linear time in 2D/3D. Particle discretization of the heat equation: ẏ i = yi D ent,εn (y 1,..., y N ) yields a time dependent measure µ N = 1 N i δ y i. { t ρ + div(ρv) = 0 Under assumptions, µ N converges to a weak solution of v = log ρ

27 Summary Optimal transport can be used to construct particle discretization of (some) evolution PDEs with congestion/incompressibility constraints or diffusion terms Some perspectives: more general models: non-linear diffusion, reaction terms (e.g. tumor growth), multiple population, surface tension, interaction forces (e.g. Keller-Segel)? convergence towards Lagrangian solutions (à la [Evans-Gangbo-Savin 05])? getting rid of ε for 2nd order models (cf [de Goes et al, SIGGRAPH 2015])? Thank you for your attention!