Convex discretization of functionals involving the Monge-Ampère operator

Convex discretization of functionals involving the Monge-Ampère operator Quentin Mérigot CNRS / Université Paris-Dauphine Joint work with J.D. Benamou, G. Carlier and É. Oudet GeMeCod Conference October 26-30, 2015, Marne-la-Vallée 1

2 1. Discretization of the space of convex functions

Motivation: variational problems over convex functions Principal-agent problem in economy min φ K,φ 0 [1,2] 2 1 2 ( φ(x) x 2 φ(x)) d x K := {φ convex } [Rochet-Choné 98] φ # Leb [1,2] 2 3

Motivation: variational problems over convex functions Principal-agent problem in economy min φ K,φ 0 [1,2] 2 1 2 ( φ(x) x 2 φ(x)) d x K := {φ convex } [Rochet-Choné 98] φ # Leb [1,2] 2 Gradient flows in Wasserstein space [Otto] [Jordan-Kinderlehrer-Otto 98] min φ KX 1 2τ W2 2(ρ, φ # ρ) + E( φ # ρ) K X := {φ convex ; φ X} 3

Prior work: Discretizing the Space of Convex Functions Finite elements: piecewise-linear convex functions over a fixed mesh non-density result [Choné-Le Meur 99] h standard grid 4

Prior work: Discretizing the Space of Convex Functions Finite elements: piecewise-linear convex functions over a fixed mesh non-density result [Choné-Le Meur 99] h standard grid Convex interpolates: [Carlier-Lachand-Robert-Maury 01] [Ekeland Moreno-Bromberg 10] Def: Given a finite subset P R d, K(P ) := { restriction of convex functions to P } good convergence properties under convergence of P K(P ) is a finite-dimensional convex polytope defined by Card(P ) 2 linear inequalities P R d adaptive method exterior parameterization [Oberman 14] [Mirebeau 14] [Oudet-M. 14] 4

5 2. Space-discretization of Wasserstein gradient flows

Background: Optimal transport P 2 (R d ) = prob. measures with finite second moment P 2 ac(r d ) = P 2 (R d ) L 1 (R d ) ν R d p 2 π Wasserstein distance between µ, ν P 2 (R d ), Γ(µ, ν) := {π P(R d R d ); p 1# π = µ, p 2# π = ν} Definition: W 2 2(µ, ν) := min π Γ(µ,ν) x y 2 d π(x, y). µ p 1 R d Relation to convex functions: Def: K := convex functions on R d Theorem (Brenier): Given µ P 2 ac(r d ) the map φ K φ # µ P 2 (R d ) is surjective and moreover, W 2 2(µ, φ # µ) = R d x φ(x) 2 d µ(x) [Brenier 91] 6

Motivation 1: Crowd Motion Under Congestion JKO scheme for crowd motion with hard congestion: [Maury-Roudneff-Chupin-Santambrogio 10] ρ τ k+1 = min σ P 2 (X) 1 2τ W2 2(ρ τ k, σ) + E(σ) + U(σ) ( ) where X R d is convex and bounded, and U(ν) := 0 if ν P ac and d ν d x <= 1 E(ν) := V (x) d ν(x) R d + if not congestion potential energy Assuming σ = φ # ρ τ k with φ convex, the Wasserstein term becomes explicit: ( ) min φ 1 2τ R d x φ(x) 2 ρ τ k (x) d x + E( φ #ρ τ k ) + U( φ #ρ τ k ) On the other hand, the constraint becomes strongly nonlinear: U( φ # ρ τ k ) < + det D2 φ(x) ρ k (x) 7

Motivation 2: Nonlinear Diffusion ( ) ρ t = div [ρ (U (ρ) + V + W ρ)] ρ(0,.) = ρ 0 ρ(t,.) P ac (R d ) 8

Motivation 2: Nonlinear Diffusion ( ) ρ t = div [ρ (U (ρ) + V + W ρ)] ρ(0,.) = ρ 0 ρ(t,.) P ac (R d ) Formally, ( ) can be interpreted as the W 2 -gradient flow of U + E, with U(ν) := U( d ν R d x ) d x if ν P2 ac(r d ) d + if not internal energy, ex: U(r) = r log r = entropy 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] ρ τ k+1 = arg min σ P(R d ) 1 2τ W 2(ρ τ k, σ)2 + U(σ) + E(σ) 8

Displacement Convex Setting min σ P(R d ) 1 2τ W2 2(µ, σ) + E(σ) + U(σ) min φ K 1 2τ W2 2(µ, φ # µ) + U( φ # µ) + E( φ # µ) 9

Displacement Convex Setting min σ P(R d ) 1 2τ W2 2(µ, σ) + E(σ) + U(σ) min φ K 1 2τ W2 2(µ, φ # µ) + U( φ # µ) + E( φ # µ) When is this minimization problem convex in φ? 9

Displacement Convex Setting min σ P(R d ) 1 2τ W2 2(µ, σ) + E(σ) + U(σ) min φ K 1 2τ W2 2(µ, φ # µ) + U( φ # µ) + E( φ # µ) When is this minimization problem convex in φ? Formally, the pushforward σ = φ # µ satisfies the Monge-Ampère equation σ( φ(x))ma[φ](x) = µ(x) MA[φ](x) := det(d 2 φ(x)) With a change-of-variable formula, U( φ # µ) = R d U(σ(x)) d x = R d U ( ) µ(x) MA[φ](x) MA[φ](x) d x Using concavity of det 1/d over SDP, this gives: Theorem: Given µ P 2 ac(r 2 ), for the functional φ K U( φ #µ ) to be convex, it suffices that r > 0 r d U(r d ) is convex non-increasing and U(0) = 0. [McCann 94] 9

Discretizing the Space of Convex Functions We replace K with K X := {φ convex ; φ X} Definition: Given a finite subset P X, let K X (P ) := {ψ P ; ψ K X } R P 10

Discretizing the Space of Convex Functions We replace K with K X := {φ convex ; φ X} Definition: Given a finite subset P X, let K X (P ) := {ψ P ; ψ K X } R P = φ : P R P X 10

Discretizing the Space of Convex Functions We replace K with K X := {φ convex ; φ X} Definition: Given a finite subset P X, let K X (P ) := {ψ P ; ψ K X } R P K X envelope of a function φ on P : φ : P R ˆφ : R d R ˆφ = φ : P R ˆφ := max{ψ; ψ K X and ψ P φ} K X P X Given µ = p P µ pδ p on P and φ K X (P ) we need to define φ # µ. 10

Discretizing the Space of Convex Functions We replace K with K X := {φ convex ; φ X} Definition: Given a finite subset P X, let K X (P ) := {ψ P ; ψ K X } R P K X envelope of a function φ on P : φ : P R ˆφ : R d R ˆφ = φ : P R ˆφ := max{ψ; ψ K X and ψ P φ} K X V p P X Given µ = p P µ pδ p on P and φ K X (P ) we need to define φ # µ. Definition: For φ K X (P ), let V p := ˆφ(p) and V p p X 10

Discretizing the Space of Convex Functions We replace K with K X := {φ convex ; φ X} Definition: Given a finite subset P X, let K X (P ) := {ψ P ; ψ K X } R P K X envelope of a function φ on P : φ : P R ˆφ : R d R ˆφ = φ : P R ˆφ := max{ψ; ψ K X and ψ P φ} K X V p P X Given µ = p P µ pδ p on P and φ K X (P ) we need to define φ # µ. Definition: For φ K X (P ), let V p := ˆφ(p) and V p p X Gφ # µ := p P µ p Leb(V p ) Leb V p P ac (X) 10

Internal Energy and Discrete Monge-Ampère Operator Discrete Monge-Ampère operator: MA[φ](p) := Leb( ˆφ(p)). Related to Alexandrov s Monge-Ampère measure of ˆφ, i.e. p P MA[φ](p)δ p 11

Internal Energy and Discrete Monge-Ampère Operator Discrete Monge-Ampère operator: MA[φ](p) := Leb( ˆφ(p)). Related to Alexandrov s Monge-Ampère measure of ˆφ, i.e. p P MA[φ](p)δ p Internal energy of the push-forward Gφ # µ = p P (µ p/ma[φ](p)) Leb ˆφ(p) : U(Gφ # µ) = U(Gφ # µ(x)) d x, = p P ˆφ(p) U (µ p/ma[φ](p))) d x = p P U (µ p/ma[φ](p)) MA[φ](p) (Compare to the continuous formula: U( φ # µ) = R d U ( ) µ(x) MA[φ](x) MA[φ](x) d x) 11

Internal Energy and Discrete Monge-Ampère Operator Discrete Monge-Ampère operator: MA[φ](p) := Leb( ˆφ(p)). 11 Related to Alexandrov s Monge-Ampère measure of ˆφ, i.e. p P MA[φ](p)δ p Internal energy of the push-forward Gφ # µ = p P (µ p/ma[φ](p)) Leb ˆφ(p) : U(Gφ # µ) = U(Gφ # µ(x)) d x, = p P ˆφ(p) U (µ p/ma[φ](p))) d x = p P U (µ p/ma[φ](p)) MA[φ](p) (Compare to the continuous formula: U( φ # µ) = R d U ( ) µ(x) MA[φ](x) MA[φ](x) d x) Concavity of discrete MA operator: φ K(P ) MA[φ](p) 1/d is concave Brunn-Minkowski Leb( ˆφ t ) 1/d Leb((1 t) ˆφ 0 (p) t ˆφ 1 (p)) 1/d. (1 t)leb( ˆφ 0 (p)) 1/d + tleb( ˆφ 1 (p)) 1/d. Proposition: Under McCann s assumption, φ K X (P ) U(Gφ # µ P ) is convex.

Convex Space-Discretization of One Gradient Step Theorem: Under McCann s hypotheses the discrete problem min φ KX (P ) 1 2τ W2 2(µ, Hφ # µ) + E(Hφ # µ) + U(Gφ # µ) is convex, and the minimum is unique if r d U(r d ) is strictly convex. [Benamou, Carlier, M., Oudet 14] Unfortunately, two different definitions of push-forward seem necessary to get a convex discretisation (i.e. φ E(Gφ # µ) is non-convex). If lim r U(r)/r = +, the internal energy U is a barrier for convexity, i.e. U(Gφ # µ P ) < + = p P, MA[φ](p) = Leb( ˆφ(p)) > 0 = φ is in the interior of K X (P ). NB: P non-linear constraints vs P 2 linear constraints 12

Convergence of the Space-Discretization Setting: X bounded and convex, µ P ac (X) with density c 1 ρ c min σ P(X) 1 2τ W2 2(µ, σ) + E(σ) + U(σ) ( ) 13

Convergence of the Space-Discretization Setting: X bounded and convex, µ P ac (X) with density c 1 ρ c min σ P(X) 1 2τ W2 2(µ, σ) + E(σ) + U(σ) ( ) (C1) E continuous, U l.s.c on (P(X), W 2 ) 13

Convergence of the Space-Discretization Setting: X bounded and convex, µ P ac (X) with density c 1 ρ c min σ P(X) 1 2τ W2 2(µ, σ) + E(σ) + U(σ) ( ) (C1) E continuous, U l.s.c on (P(X), W 2 ) (C2) U(ρ) = U(ρ(x)) d x, where U is convex with superlinear growth. McCann s condition for displacement-convexity Theorem: Let P n X finite, µ n P(P n ) with lim W 2 (µ n, µ) = 0, and: min φ KX (P n ) 1 2τ W 2(µ n, Hφ # µ n ) + E(Hφ # µ n ) + U(Gφ # µ n ) ( ) n If φ n minimizes ( ) n, then (Gφ n #µ n ) is a minimizing sequence for ( ). [Benamou, Carlier, M., Oudet 14] 13

14 3. Numerical results

Monge-Ampère operator and Computational Geometry U(Gφ # µ P ) = p P U (µ p/ma[φ](p)) MA[φ](p) with MA[φ](p) := Leb( ˆφ(p)). Goal: fast computation of MA[φ] and its 1st/2nd derivatives w.r.t φ We rely on the notion of Laguerre diagram in computational geometry Definition: Given a function φ : P R, Lag φ P (p) := {y Rd ; q P, φ(q) φ(p) + q p y } Lag φ P (p) X For φ K X (P ), ˆφ(p) = Lag φ P (p) X. p 15

Example: semi-lagrangian scheme for crowd motion Gradient flow model of crowd motion with congestion, with a JKO scheme: [Maury-Roudneff-Chupin-Santambrogio 10] µ k+1 = min ν P(X) 1 2τ W2 2(µ k, ν) + E(ν) + U(ν) E(ν) := V (x) d ν(x) X { 0 if d ν/ d x 1, U(ν) := + if not Convex optimization problem if V is λ-convex (V + λ. 2 convex) and τ λ/2. We solve this problem with a relaxed hard congestion term: 6 r r α log(1 r 1/d ) U α (ρ) := ρ(x) α log(1 ρ(x) 1/d ) d x 4 2 α=1 α=5 α=10 16 r = 0 r = 1

Example: semi-lagrangian scheme for crowd motion Initial density on X = [ 2, 2] 2 Potential 0 1 (2, 2) ( 2, 2) P = 200 200 regular grid. V (x) = x (2, 0) 2 + 5 exp( 5 x 2 /2) Algorithm: Input: µ 0 P(P ), τ > 0, α > 0, β 1. For k {0,..., T } φ arg min φ K G X (P k ) 1 2τ W2 2(µ k, Hφ # µ k ) + E(Hφ # µ k ) + αu β (Gφ # µ k ) ν G φ# µ k ; µ k+1 projection of ν [ 2,2) [ 2,2] on P. 17

4. Application to Geometry? Ongoing work with J.M. Mirebeau 18

Support Functions of Convex Bodies Objective: Logarithmic barrier for the space of support functions of convex bodies. 19

Support Functions of Convex Bodies Objective: Logarithmic barrier for the space of support functions of convex bodies. interior point method for shape optimization problem: Minkowski, Meissner, etc. 19

Support Functions of Convex Bodies Objective: Logarithmic barrier for the space of support functions of convex bodies. Definition: Given a convex body K, h K : u S d 1 max p K u p. 19

Support Functions of Convex Bodies Objective: Logarithmic barrier for the space of support functions of convex bodies. Definition: Given a convex body K, h K : u S d 1 max p K u p. K (P ) := {h K P ; K bounded convex body } K(h) n 1 n 2 h 1 h 2 n 5 n 3 h 5 h 3 h 4 n 4 P = {n 1,..., n N } S d 1 R d K(h) := N i=1 {x; x n i h i } Prop: Φ(h) := N i=1 log(hd 1 (ith face of K(h))) is a convex barrier for K (P ). 19

Application: convex bodies with constant-width Definition: The width of K in direction u is w K (u) = h K (u) + h K ( u). K has constant width α if w K (u) = α. 20

Application: convex bodies with constant-width α Definition: The width of K in direction u is w K (u) = h K (u) + h K ( u). K has constant width α if w K (u) = α. Example: Reuleaux triangle in R 2 Theorem: Reuleaux triangles minimize the area over convex sets of the plane with constant width α. [Blaschke-Lebesgue] 20

Application: convex bodies with constant-width Definition: The width of K in direction u is α w K (u) = h K (u) + h K ( u). K has constant width α if w K (u) = α. Example: Reuleaux triangle in R 2 Theorem: Reuleaux triangles minimize the area over convex sets of the plane with constant width α. [Blaschke-Lebesgue] Bonnensen-Fenchel conjecture in R 3 : Meissner s body minimize the volume among convex sets with fixed constant width. 20

21 Appendix

c-concave Functions Definition: φ : X R is c-concave if ψ : Y R s.t. φ = min y c(x, y) + ψ(y) Cost function: c : X Y R 22

c-concave Functions Definition: φ : X R is c-concave if ψ : Y R s.t. φ = min y c(x, y) + ψ(y) K c (P ) := {φ P ; φ : X R is c-concave } Cost function: c : X Y R 22

c-concave Functions Definition: φ : X R is c-concave if ψ : Y R s.t. φ = min y c(x, y) + ψ(y) K c (P ) := {φ P ; φ : X R is c-concave } Cost function: c : X Y R Characterization of the costs functions such that the space K c is convex Application: generalized principal-agent problem [Figalli, Kim, McCann 10] Under the same hypotheses, there exists a convex logarithmic barrier for K c (P ): c-laguerre diagram: Lag ψ c (p) = {x X; q P, c(x, p) + ψ(p) c(x, q) + ψ(q)} Lag ψ c (p) X 22