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 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 p 1 ◮ Wasserstein distance between µ, ν ∈ P 2 ( R d ) , R d Γ( µ, ν ) := { π ∈ P ( R d × R d ); p 1# π = µ, p 2# π = ν } 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 p 1 ◮ Wasserstein distance between µ, ν ∈ P 2 ( R d ) , R d Γ( µ, ν ) := { π ∈ P ( R d × R d ); p 1# π = µ, p 2# π = ν } 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 ) and ν ∈ P 2 ( R d ) , [Brenier ’91] R d � x − ∇ φ ( x ) � 2 d µ ( x ) ∃ φ ∈ K such that ∇ φ # µ = ν , and W 2 � 2 ( 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 p 1 ◮ Wasserstein distance between µ, ν ∈ P 2 ( R d ) , R d Γ( µ, ν ) := { π ∈ P ( R d × R d ); p 1# π = µ, p 2# π = ν } 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 ) and ν ∈ P 2 ( R d ) , [Brenier ’91] R d � x − ∇ φ ( x ) � 2 d µ ( x ) ∃ φ ∈ K such that ∇ φ # µ = ν , and W 2 � 2 ( µ, ν ) = Given any µ ∈ P ac 2 ( R d ) , we get a ”parameterization” of P 2 ( R d ) , or more precisely, an onto map K �→ P 2 ( R d ) , φ �→ ∇ φ # µ. 3

Gas equilibrium and displacement convexity ◮ Equilibrium states of gazes: min ν ∈P 2 ( R d ) U ( ν ) + E ( ν ) �  ν = particle distribution R d U ( σ ( x )) d x if d ν = σ d H d  U ( ν ) := internal energy + ∞ if not  � � E ( ν ) := R d V ( x ) d ν ( x ) + R d W ( x − y ) d[ ν ⊗ ν ]( x, y ) interaction energy potential energy 4

Gas equilibrium and displacement convexity ◮ Equilibrium states of gazes: min ν ∈P 2 ( R d ) U ( ν ) + E ( ν ) �  ν = particle distribution R d U ( σ ( x )) d x if d ν = σ d H d  U ( ν ) := internal energy + ∞ if not  � � E ( ν ) := R d V ( x ) d ν ( x ) + R d W ( x − y ) d[ ν ⊗ ν ]( x, y ) interaction energy potential energy ◮ Displacement convexity Definition: F is displacement-convex if for any W 2 -geodesic ( ν t ) in P ac 2 ( R d ) , the function t �→ F ( ν t ) is convex. Theorem: V, W : R d → R are convex functions = ⇒ E is displacement-convex r d U ( r − d ) is convex non-increasing, U (0) = 0 = ⇒ U is displacement-convex − → strict convexity = ⇒ uniqueness of minimum [McCann ’94] 4

Convexity under generalized displacements ◮ Generalized displacement convexity Definition: F is convex under generalized displacement (u.g.d.) if for any µ in P ac 2 ( R d ) , the function φ ∈ H �→ F ( ∇ φ # µ ) is convex. 5

Convexity under generalized displacements ◮ Generalized displacement convexity Definition: F is convex under generalized displacement (u.g.d.) if for any µ in P ac 2 ( R d ) , the function φ ∈ H �→ F ( ∇ φ # µ ) is convex. µ t = ((1 − t )id + t ∇ φ ) # µ geodesic for W 2 5

Convexity under generalized displacements ◮ Generalized displacement convexity Definition: F is convex under generalized displacement (u.g.d.) if for any µ in P ac 2 ( R d ) , the function φ ∈ H �→ F ( ∇ φ # µ ) is convex. µ t = ((1 − t )id + t ∇ φ ) # µ µ t = ((1 − t ) ∇ φ 0 + t ∇ φ 1 ) # µ geodesic for W 2 ”generalized geodesic” 5

Convexity under generalized displacements ◮ Generalized displacement convexity Definition: F is convex under generalized displacement (u.g.d.) if for any µ in P ac 2 ( R d ) , the function φ ∈ H �→ F ( ∇ φ # µ ) is convex. µ t = ((1 − t )id + t ∇ φ ) # µ µ t = ((1 − t ) ∇ φ 0 + t ∇ φ 1 ) # µ geodesic for W 2 ”generalized geodesic” Theorem: V, W : R d → R are convex functions = ⇒ E is convex u.g.d r d U ( r − d ) is convex non-increasing, U (0) = 0 = ⇒ U is convex u.g.d [McCann ’94] 5

Convexity under generalized displacements ◮ Generalized displacement convexity Definition: F is convex under generalized displacement (u.g.d.) if for any µ in P ac 2 ( R d ) , the function φ ∈ H �→ F ( ∇ φ # µ ) is convex. µ t = ((1 − t )id + t ∇ φ ) # µ µ t = ((1 − t ) ∇ φ 0 + t ∇ φ 1 ) # µ geodesic for W 2 ”generalized geodesic” Theorem: V, W : R d → R are convex functions = ⇒ E is convex u.g.d r d U ( r − d ) is convex non-increasing, U (0) = 0 = ⇒ U is convex u.g.d [McCann ’94] ◮ Proof: Non-smooth change of variable formula: � � E ( ∇ φ # ρ ) = V ( ∇ φ ( x )) ρ ( x ) d x + W ( ∇ φ ( x ) − ∇ φ ( z )) ρ ( z ) ρ ( y ) d x d y 5

Convexity under generalized displacements ◮ Generalized displacement convexity Definition: F is convex under generalized displacement (u.g.d.) if for any µ in P ac 2 ( R d ) , the function φ ∈ H �→ F ( ∇ φ # µ ) is convex. µ t = ((1 − t )id + t ∇ φ ) # µ µ t = ((1 − t ) ∇ φ 0 + t ∇ φ 1 ) # µ geodesic for W 2 ”generalized geodesic” Theorem: V, W : R d → R are convex functions = ⇒ E is convex u.g.d r d U ( r − d ) is convex non-increasing, U (0) = 0 = ⇒ U is convex u.g.d [McCann ’94] ◮ Proof: Non-smooth change of variable formula: � � E ( ∇ φ # ρ ) = V ( ∇ φ ( x )) ρ ( x ) d x + W ( ∇ φ ( x ) − ∇ φ ( z )) ρ ( z ) ρ ( y ) d x d y � � ρ ( x ) � U ( ∇ φ # ρ ) = U MA[ φ ]( x ) d x MA[ φ ]( x ) := det(D 2 φ ( x )) MA[ φ ]( x ) Minkowski determinant inequality: A ∈ SDP( R d ) → det( A ) 1 /d is concave 5

Heat equation as a Wasserstein gradient flow ∂ρ ρ ( t, . ) ∈ P ac ( R d ) ρ (0 , . ) = ρ 0 Heat equation ∂t = ∆ ρ �∇ ρ ( x ) � 2 d x . ◮ Solution ρ ( t, . ) = gradient flow in L 2 ( R d ) of D ( ρ ) := 1 � 2 6

Heat equation as a Wasserstein gradient flow ∂ρ ρ ( t, . ) ∈ P ac ( R d ) ρ (0 , . ) = ρ 0 Heat equation ∂t = ∆ ρ �∇ ρ ( x ) � 2 d x . ◮ Solution ρ ( t, . ) = gradient flow in L 2 ( R d ) of D ( ρ ) := 1 � 2 Time-discretization using an implicit Euler scheme: for τ > 0 , 1 ρ τ 2 τ � ρ τ k − σ � 2 L 2 ( R d ) + D ( σ ) . k +1 = arg min σ ∈P ac ( R d ) 6

Heat equation as a Wasserstein gradient flow ∂ρ ρ ( t, . ) ∈ P ac ( R d ) ρ (0 , . ) = ρ 0 Heat equation ∂t = ∆ ρ �∇ ρ ( x ) � 2 d x . ◮ Solution ρ ( t, . ) = gradient flow in L 2 ( R d ) of D ( ρ ) := 1 � 2 Time-discretization using an implicit Euler scheme: for τ > 0 , 1 ρ τ 2 τ � ρ τ k − σ � 2 L 2 ( R d ) + D ( σ ) . k +1 = arg min σ ∈P ac ( R d ) ◮ Jordan, Kinderleherer, Otto: the heat equation is a gradient flow, for W 2 ( R d ) , of � the functional U ( ρ ) := ρ ( x ) log ρ ( x ) d x = − entropy of ρ . 6

Heat equation as a Wasserstein gradient flow ∂ρ ρ ( t, . ) ∈ P ac ( R d ) ρ (0 , . ) = ρ 0 Heat equation ∂t = ∆ ρ �∇ ρ ( x ) � 2 d x . ◮ Solution ρ ( t, . ) = gradient flow in L 2 ( R d ) of D ( ρ ) := 1 � 2 Time-discretization using an implicit Euler scheme: for τ > 0 , 1 ρ τ 2 τ � ρ τ k − σ � 2 L 2 ( R d ) + D ( σ ) . k +1 = arg min σ ∈P ac ( R d ) ◮ Jordan, Kinderleherer, Otto: the heat equation is a gradient flow, for W 2 ( R d ) , of � the functional U ( ρ ) := ρ ( x ) log ρ ( x ) d x = − entropy of ρ . Corresponding time-discrete scheme: k , σ ) 2 + U ( σ ) . 1 ρ τ 2 τ W 2 ( ρ τ k +1 = arg min σ ∈P ( R d ) 6

Heat equation as a Wasserstein gradient flow ∂ρ ρ ( t, . ) ∈ P ac ( R d ) ρ (0 , . ) = ρ 0 Heat equation ∂t = ∆ ρ �∇ ρ ( x ) � 2 d x . ◮ Solution ρ ( t, . ) = gradient flow in L 2 ( R d ) of D ( ρ ) := 1 � 2 Time-discretization using an implicit Euler scheme: for τ > 0 , 1 ρ τ 2 τ � ρ τ k − σ � 2 L 2 ( R d ) + D ( σ ) . k +1 = arg min σ ∈P ac ( R d ) ◮ Jordan, Kinderleherer, Otto: the heat equation is a gradient flow, for W 2 ( R d ) , of � the functional U ( ρ ) := ρ ( x ) log ρ ( x ) d x = − entropy of ρ . Corresponding time-discrete scheme: k , σ ) 2 + U ( σ ) . 1 ρ τ 2 τ W 2 ( ρ τ k +1 = arg min σ ∈P ( R d ) ◮ Convergence analysis for the linear Fokker-Planck equation. [Jordan, Kinderlehrer, Otto ’99] 6

Diffusive PDEs as Wasserstein gradient flows ◮ Generalization to some evolution PDEs, where ρ ( t, . ) ∈ P ac ( R d ) ∂ρ ∂t = div [ ρ ∇ ( U ′ ( ρ ) + V + W ∗ ρ )] ρ (0 , . ) = ρ 0 ( ∗ ) 7