Convex discretization of functionals involving the Monge-Amp` ere - PowerPoint PPT Presentation

Convex discretization of functionals involving the Monge-Amp` ere operator Quentin M´ erigot CNRS / Universit´ e Paris-Dauphine Joint work with J.D. Benamou, G. Carlier and ´ E. Oudet Workshop on Optimal Transport in the Applied Sciences December 8-12, 2014 — RICAM, Linz 1

1. Motivation: Gradient flows in Wasserstein space 2

Background: Optimal transport R d p 2 � x � 2 d µ ( x ) < + ∞} P 2 ( R d ) = { µ ∈ P ( R d ); � ν P ac 2 ( R d ) = P 2 ( R d ) ∩ L 1 ( R d ) π ◮ Wasserstein distance between µ, ν ∈ P 2 ( R d ) , p 1 Γ( µ, ν ) := { π ∈ P ( R d × R d ); p 1# π = µ, p 2# π = ν } R d µ � x − y � 2 d π ( x, y ) . Definition: W 2 � 2 ( µ, ν ) := min π ∈ Γ( µ,ν ) 3

Background: Optimal transport R d p 2 � x � 2 d µ ( x ) < + ∞} P 2 ( R d ) = { µ ∈ P ( R d ); � ν P ac 2 ( R d ) = P 2 ( R d ) ∩ L 1 ( R d ) π ◮ Wasserstein distance between µ, ν ∈ P 2 ( R d ) , p 1 Γ( µ, ν ) := { π ∈ P ( R d × R d ); p 1# π = µ, p 2# π = ν } R d µ � x − y � 2 d π ( x, y ) . Definition: W 2 � 2 ( µ, ν ) := min π ∈ Γ( µ,ν ) Def: K := finite convex functions on R d ◮ Relation to convex functions: 3

Background: Optimal transport R d p 2 � x � 2 d µ ( x ) < + ∞} P 2 ( R d ) = { µ ∈ P ( R d ); � ν P ac 2 ( R d ) = P 2 ( R d ) ∩ L 1 ( R d ) π ◮ Wasserstein distance between µ, ν ∈ P 2 ( R d ) , p 1 Γ( µ, ν ) := { π ∈ P ( R d × R d ); p 1# π = µ, p 2# π = ν } R d µ � x − y � 2 d π ( x, y ) . Definition: W 2 � 2 ( µ, ν ) := min π ∈ Γ( µ,ν ) Def: K := finite convex functions on R d ◮ Relation to convex functions: Theorem (Brenier): Given µ ∈ P ac 2 ( R d ) the map φ ∈ K �→ ∇ φ # µ ∈ P 2 ( R d ) is surjective and moreover, R d � x − ∇ φ ( x ) � 2 d µ ( x ) W 2 � 2 ( µ, ∇ φ # µ ) = [Brenier ’91] 3

Background: Optimal transport R d p 2 � x � 2 d µ ( x ) < + ∞} P 2 ( R d ) = { µ ∈ P ( R d ); � ν P ac 2 ( R d ) = P 2 ( R d ) ∩ L 1 ( R d ) π ◮ Wasserstein distance between µ, ν ∈ P 2 ( R d ) , p 1 Γ( µ, ν ) := { π ∈ P ( R d × R d ); p 1# π = µ, p 2# π = ν } R d µ � x − y � 2 d π ( x, y ) . Definition: W 2 � 2 ( µ, ν ) := min π ∈ Γ( µ,ν ) Def: K := finite convex functions on R d ◮ Relation to convex functions: Theorem (Brenier): Given µ ∈ P ac 2 ( R d ) the map φ ∈ K �→ ∇ φ # µ ∈ P 2 ( R d ) is surjective and moreover, R d � x − ∇ φ ( x ) � 2 d µ ( x ) W 2 � 2 ( µ, ∇ φ # µ ) = [Brenier ’91] Given any µ ∈ P ac 2 ( R d ) , we get a ”parameterization” of P 2 ( R d ) , as ”seen” from µ . 3

Motivation 1: Crowd Motion Under Congestion ◮ JKO scheme for crowd motion with hard congestion: [Maury-Roudneff-Chupin-Santambrogio 10] 2 τ W 2 1 ρ τ 2 ( ρ τ k , σ ) + E ( σ ) + U ( σ ) ( ∗ ) k +1 = min σ ∈P 2 ( X ) where X ⊆ R d is convex and bounded, and � 0 if d ν = f d H d , f ≤ 1 � E ( ν ) := R d V ( x ) d ν ( x ) U ( ν ) := + ∞ if not congestion potential energy 4

Motivation 1: Crowd Motion Under Congestion ◮ JKO scheme for crowd motion with hard congestion: [Maury-Roudneff-Chupin-Santambrogio 10] 2 τ W 2 1 ρ τ 2 ( ρ τ k , σ ) + E ( σ ) + U ( σ ) ( ∗ ) k +1 = min σ ∈P 2 ( X ) where X ⊆ R d is convex and bounded, and � 0 if d ν = f d H d , f ≤ 1 � E ( ν ) := R d V ( x ) d ν ( x ) U ( ν ) := + ∞ if not congestion potential energy ◮ Assuming σ = ∇ φ # ρ τ k with φ convex, the Wasserstein term becomes explicit: 1 R d � x − ∇ φ ( x ) � 2 ρ τ k ( x ) d x + E ( ∇ φ # ρ τ k ) + U ( ∇ φ # ρ τ � ( ∗ ) ⇐ ⇒ min φ k ) 2 τ On the other hand, the constraint becomes strongly nonlinear: U ( ∇ φ # ρ τ ⇒ det D 2 φ ( x ) ≥ ρ k ( x ) k ) < + ∞ ⇐ 4

Motivation 2: Nonlinear Diffusion ∂ρ ρ (0 , . ) = ρ 0 ∂t = div [ ρ ∇ ( U ′ ( ρ ) + V + W ∗ ρ )] ( ∗ ) ρ ( t, . ) ∈ P ac ( R d ) 5

Motivation 2: Nonlinear Diffusion ∂ρ ρ (0 , . ) = ρ 0 ∂t = div [ ρ ∇ ( U ′ ( ρ ) + V + W ∗ ρ )] ( ∗ ) ρ ( t, . ) ∈ P ac ( R d ) ◮ Formally, ( ∗ ) can be seen as the W 2 -gradient flow of U + E , with �  R d U ( f ( x )) d x if d ν = f d H d  U ( ν ) := ⇒ entropy internal energy, ex: U ( r ) = r log r = + ∞ if not  � � E ( ν ) := R d V ( x ) d ν ( x ) + R d W ( x − y ) d[ ν ⊗ ν ]( x, y ) potential energy interaction energy 5

Motivation 2: Nonlinear Diffusion ∂ρ ρ (0 , . ) = ρ 0 ∂t = div [ ρ ∇ ( U ′ ( ρ ) + V + W ∗ ρ )] ( ∗ ) ρ ( t, . ) ∈ P ac ( R d ) ◮ Formally, ( ∗ ) can be seen as the W 2 -gradient flow of U + E , with �  R d U ( f ( x )) d x if d ν = f d H d  U ( ν ) := ⇒ entropy internal energy, ex: U ( r ) = r log r = + ∞ if not  � � E ( ν ) := R d V ( x ) d ν ( x ) + R d W ( x − y ) d[ ν ⊗ ν ]( x, y ) potential energy interaction energy ◮ JKO time discrete scheme: for τ > 0 , [Jordan-Kinderlehrer-Otto ’98] k , σ ) 2 + U ( σ ) + E ( σ ) 1 ρ τ 2 τ W 2 ( ρ τ k +1 = arg min σ ∈P ( R d ) 5

Motivation 2: Nonlinear Diffusion ∂ρ ρ (0 , . ) = ρ 0 ∂t = div [ ρ ∇ ( U ′ ( ρ ) + V + W ∗ ρ )] ( ∗ ) ρ ( t, . ) ∈ P ac ( R d ) ◮ Formally, ( ∗ ) can be seen as the W 2 -gradient flow of U + E , with �  R d U ( f ( x )) d x if d ν = f d H d  U ( ν ) := ⇒ entropy internal energy, ex: U ( r ) = r log r = + ∞ if not  � � E ( ν ) := R d V ( x ) d ν ( x ) + R d W ( x − y ) d[ ν ⊗ ν ]( x, y ) potential energy interaction energy ◮ JKO time discrete scheme: for τ > 0 , [Jordan-Kinderlehrer-Otto ’98] k , σ ) 2 + U ( σ ) + E ( σ ) 1 ρ τ 2 τ W 2 ( ρ τ k +1 = arg min σ ∈P ( R d ) − → Many applications: porous medium equation, cell movement via chemotaxis, Cournot-Nash equilibra, etc. 5

Displacement Convex Setting For X convex bounded and µ ∈ P ac ( X ) , 2 τ W 2 1 2 ( µ, ν ) + E ( ν ) + U ( ν ) min ν ∈P ( X ) spt( ν ) ⊆ X 6

Displacement Convex Setting For X convex bounded and µ ∈ P ac ( X ) , K X := { φ convex ; ∇ φ ∈ X } 2 τ W 2 1 2 ( µ, ν ) + E ( ν ) + U ( ν ) min ν ∈P ( X ) 2 τ W 2 1 ⇐ ⇒ 2 ( µ, ∇ φ # µ ) + U ( ∇ φ # µ ) + E ( ∇ φ # µ ) min φ ∈K X ( ∗ X ) 6

Displacement Convex Setting For X convex bounded and µ ∈ P ac ( X ) , K X := { φ convex ; ∇ φ ∈ X } 2 τ W 2 1 2 ( µ, ν ) + E ( ν ) + U ( ν ) min ν ∈P ( X ) 2 τ W 2 1 ⇐ ⇒ 2 ( µ, ∇ φ # µ ) + U ( ∇ φ # µ ) + E ( ∇ φ # µ ) min φ ∈K X ( ∗ X ) � d ν When is the minimization problem ( ∗ X ) convex ? � � U ( ν ) := R d U d x d H d � E ( ν ) := R d ( V + W ∗ ν ) d ν 6

Displacement Convex Setting For X convex bounded and µ ∈ P ac ( X ) , K X := { φ convex ; ∇ φ ∈ X } 2 τ W 2 1 2 ( µ, ν ) + E ( ν ) + U ( ν ) min ν ∈P ( X ) 2 τ W 2 1 ⇐ ⇒ 2 ( µ, ∇ φ # µ ) + U ( ∇ φ # µ ) + E ( ∇ φ # µ ) min φ ∈K X ( ∗ X ) � d ν When is the minimization problem ( ∗ X ) convex ? � � U ( ν ) := R d U d x d H d � E ( ν ) := R d ( V + W ∗ ν ) d ν Theorem: ( ∗ X ) is convex if (H1) V, W : R d → R are convex functions, (H2) r d U ( r − d ) is convex non-increasing, U (0) = 0 . [McCann ’94] � � ρ ( x ) � MA[ φ ]( x ) := det(D 2 φ ( x )) NB: U ( ∇ φ # ρ ) = U MA[ φ ]( x ) d x MA[ φ ]( x ) 6

Displacement Convex Setting For X convex bounded and µ ∈ P ac ( X ) , K X := { φ convex ; ∇ φ ∈ X } 2 τ W 2 1 2 ( µ, ν ) + E ( ν ) + U ( ν ) min ν ∈P ( X ) 2 τ W 2 1 ⇐ ⇒ 2 ( µ, ∇ φ # µ ) + U ( ∇ φ # µ ) + E ( ∇ φ # µ ) min φ ∈K X ( ∗ X ) � d ν When is the minimization problem ( ∗ X ) convex ? � � U ( ν ) := R d U d x d H d � E ( ν ) := R d ( V + W ∗ ν ) d ν Theorem: ( ∗ X ) is convex if (H1) V, W : R d → R are convex functions, (H2) r d U ( r − d ) is convex non-increasing, U (0) = 0 . [McCann ’94] � � ρ ( x ) � MA[ φ ]( x ) := det(D 2 φ ( x )) NB: U ( ∇ φ # ρ ) = U MA[ φ ]( x ) d x MA[ φ ]( x ) Goal: convergent and convex spatial discretization of ( ∗ X ) & numerical applications. 6

Numerical applications of the JKO scheme ◮ Numerical applications of the (variational) JKO scheme are still limited: 7

Numerical applications of the JKO scheme ◮ Numerical applications of the (variational) JKO scheme are still limited: 1D : monotone rearrangement e.g. [Kinderleherer-Walkington 99] [Blanchet-Calvez-Carrillo 08] [Agueh-Bowles 09] 7

Numerical applications of the JKO scheme ◮ Numerical applications of the (variational) JKO scheme are still limited: 1D : monotone rearrangement e.g. [Kinderleherer-Walkington 99] [Blanchet-Calvez-Carrillo 08] [Agueh-Bowles 09] 2D: optimal transport plans → diffeomorphisms [Carrillo-Moll 09] U = hard congestion term [Maury-Roudneff-Chupin-Santambrogio 10] 7

Numerical applications of the JKO scheme ◮ Numerical applications of the (variational) JKO scheme are still limited: 1D : monotone rearrangement e.g. [Kinderleherer-Walkington 99] [Blanchet-Calvez-Carrillo 08] [Agueh-Bowles 09] 2D: optimal transport plans → diffeomorphisms [Carrillo-Moll 09] U = hard congestion term [Maury-Roudneff-Chupin-Santambrogio 10] ◮ Our goal is to approximate a JKO step numerically, in dimension ≥ 2 : For X convex bounded and µ ∈ P ac ( X ) , 2 τ W 2 1 2 ( µ, ν ) + E ( ν ) + U ( ν ) ( ∗ X ) min ν ∈P ( X ) 7