From unbalanced Reminders: Static Formulation optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Kantorovich formulation (1942) Unbalanced optimal Let µ, ν ∈ P + ( M ), define D by transport An isometric embedding �� � M 2 c ( x , y ) d γ ( x , y ) : π 1 ∗ γ = µ and π 2 Euler-Arnold-Poincar´ e D ( µ, ν )= inf ∗ γ = ν equation γ ∈P ( M 2 ) The Camassa-Holm equation as an incompressible Euler Existence result: c lower semi-continuous and bounded from equation 1 Corresponding polar below. factorization Also valid in Polish spaces. 2 p | x − y | p , D 1 / p is the Wasserstein distance If c ( x , y ) = 1 3 denoted by W p .
From unbalanced Reminders: Static Formulation optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Kantorovich formulation (1942) Unbalanced optimal Let µ, ν ∈ P + ( M ), define D by transport An isometric embedding �� � M 2 c ( x , y ) d γ ( x , y ) : π 1 ∗ γ = µ and π 2 Euler-Arnold-Poincar´ e D ( µ, ν )= inf ∗ γ = ν equation γ ∈P ( M 2 ) The Camassa-Holm equation as an incompressible Euler Existence result: c lower semi-continuous and bounded from equation 1 Corresponding polar below. factorization Also valid in Polish spaces. 2 p | x − y | p , D 1 / p is the Wasserstein distance If c ( x , y ) = 1 3 denoted by W p . Linear optimization problem and associated numerical methods. Recently introduced, entropic regularization. (C. L´ eonard, M. Cuturi, J.C. Zambrini < - Schr¨ odinger)
From unbalanced Reminders: Dynamic formulation optimal transport to the Camassa-Holm equation (Benamou-Brenier) Fran¸ cois-Xavier For geodesic costs, for instance c ( x , y ) = 1 2 | x − y | 2 Vialard � 1 Unbalanced optimal inf E ( v ) = 1 � transport | v ( x ) | 2 ρ ( x ) d x d t , (14) An isometric 2 0 M embedding s.t. Euler-Arnold-Poincar´ e equation � ρ + ∇ · ( v ρ ) = 0 ˙ The Camassa-Holm (15) equation as an ρ (0) = µ 0 and ρ (1) = µ 1 . incompressible Euler equation Corresponding polar factorization
From unbalanced Reminders: Dynamic formulation optimal transport to the Camassa-Holm equation (Benamou-Brenier) Fran¸ cois-Xavier For geodesic costs, for instance c ( x , y ) = 1 2 | x − y | 2 Vialard � 1 Unbalanced optimal inf E ( v ) = 1 � transport | v ( x ) | 2 ρ ( x ) d x d t , (14) An isometric 2 0 M embedding s.t. Euler-Arnold-Poincar´ e equation � ρ + ∇ · ( v ρ ) = 0 ˙ The Camassa-Holm (15) equation as an ρ (0) = µ 0 and ρ (1) = µ 1 . incompressible Euler equation Convex reformulation: Change of variable: momentum m = ρ v , Corresponding polar factorization � 1 | m ( x ) | 2 inf E ( m ) = 1 � d x d t , (16) 2 ρ ( x ) 0 M s.t. � ρ + ∇ · m = 0 ˙ (17) ρ (0) = µ 0 and ρ (1) = µ 1 . where ( ρ, m ) ∈ M ([0 , 1] × M , R × R d ).
From unbalanced Reminders: Dynamic formulation optimal transport to the Camassa-Holm equation (Benamou-Brenier) Fran¸ cois-Xavier For geodesic costs, for instance c ( x , y ) = 1 2 | x − y | 2 Vialard � 1 Unbalanced optimal inf E ( v ) = 1 � transport | v ( x ) | 2 ρ ( x ) d x d t , (14) An isometric 2 0 M embedding s.t. Euler-Arnold-Poincar´ e equation � ρ + ∇ · ( v ρ ) = 0 ˙ The Camassa-Holm (15) equation as an ρ (0) = µ 0 and ρ (1) = µ 1 . incompressible Euler equation Convex reformulation: Change of variable: momentum m = ρ v , Corresponding polar factorization � 1 | m ( x ) | 2 inf E ( m ) = 1 � d x d t , (16) 2 ρ ( x ) 0 M s.t. � ρ + ∇ · m = 0 ˙ (17) ρ (0) = µ 0 and ρ (1) = µ 1 . where ( ρ, m ) ∈ M ([0 , 1] × M , R × R d ). Existence of minimizers: Fenchel-Rockafellar. Numerics: First-order splitting algorithm: Douglas-Rachford.
From unbalanced Starting point and initial motivation optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding • Extend the Wasserstein L 2 distance to positive Radon Euler-Arnold-Poincar´ e equation measures. The Camassa-Holm equation as an incompressible Euler • Develop associated numerical algorithms. equation Corresponding polar factorization Possible applications: Imaging, machine learning, gradient flows, ...
From unbalanced Unbalanced optimal transport optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization Figure – Optimal transport between bimodal densities
From unbalanced Unbalanced optimal transport optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization Figure – Another transformation
From unbalanced Bibliography before (june) 2015 optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Taking into account locally the change of mass: Unbalanced optimal transport Two directions: Static and dynamic. An isometric embedding • Static, Partial Optimal Transport [Figalli & Gigli, 2010] Euler-Arnold-Poincar´ e equation • Static, Hanin 1992, Benamou and Brenier 2001. The Camassa-Holm equation as an incompressible Euler • Dynamic, Numerics, Metamorphoses [Maas et al. , 2015] equation • Dynamic, Numerics, Growth model Corresponding polar factorization [Lombardi & Maitre, 2013] • Dynamic and static, [Piccoli & Rossi, 2013, Piccoli & Rossi, 2014] • . . .
From unbalanced Bibliography before (june) 2015 optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Taking into account locally the change of mass: Unbalanced optimal transport Two directions: Static and dynamic. An isometric embedding • Static, Partial Optimal Transport [Figalli & Gigli, 2010] Euler-Arnold-Poincar´ e equation • Static, Hanin 1992, Benamou and Brenier 2001. The Camassa-Holm equation as an incompressible Euler • Dynamic, Numerics, Metamorphoses [Maas et al. , 2015] equation • Dynamic, Numerics, Growth model Corresponding polar factorization [Lombardi & Maitre, 2013] • Dynamic and static, [Piccoli & Rossi, 2013, Piccoli & Rossi, 2014] • . . . No equivalent of L 2 Wasserstein distance on positive Radon measures.
From unbalanced Bibliography after june 2015 optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal More than 300 pages on the same model! transport An isometric Starting point: Dynamic formulation embedding Euler-Arnold-Poincar´ e equation • Dynamic, Numerics, Imaging [Chizat et al. , 2015] The Camassa-Holm equation as an incompressible Euler • Dynamic, Geometry and Static [Chizat et al. , 2015] equation • Dynamic, Gradient flow [Kondratyev et al. , 2015] Corresponding polar factorization • Dynamic, Gradient flow [Liero et al. , 2015b] • Static and more [Liero et al. , 2015a] • Optimal transport for contact forms [Rezakhanlou, 2015] • Static relaxation of OT, machine learning [Frogner et al. , 2015]
From unbalanced Two possible directions optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal Pros and cons: transport An isometric • Extend static formulation: Frogner et al. embedding Euler-Arnold-Poincar´ e equation min λ KL (Proj 1 ∗ γ, ρ 1 ) + λ KL (Proj 2 ∗ γ, ρ 2 ) The Camassa-Holm equation as an � incompressible Euler M 2 γ ( x , y ) d ( x , y ) 2 d x d y + (18) equation Corresponding polar factorization Good for numerics, but is it a distance ? • Extend dynamic formulation: on the tangent space of a density, choose a metric on the transverse direction. Built-in metric property but does there exist a static formulation ?
From unbalanced An extension of Benamou-Brenier formulation optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Add a source term in the constraint: (weak sense) Unbalanced optimal transport An isometric ρ = −∇ · ( ρ v ) + αρ , ˙ embedding Euler-Arnold-Poincar´ e equation where α can be understood as the growth rate. The Camassa-Holm equation as an incompressible Euler equation � 1 WF( m , α ) 2 = 1 � Corresponding polar | v ( x , t ) | 2 ρ ( x , t ) d x d t factorization 2 0 M � 1 + δ 2 � α ( x , t ) 2 ρ ( x , t ) d x d t . 2 0 M where δ is a length parameter.
From unbalanced An extension of Benamou-Brenier formulation optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Add a source term in the constraint: (weak sense) Unbalanced optimal transport An isometric ρ = −∇ · ( ρ v ) + αρ , ˙ embedding Euler-Arnold-Poincar´ e equation where α can be understood as the growth rate. The Camassa-Holm equation as an incompressible Euler equation � 1 WF( m , α ) 2 = 1 � Corresponding polar | v ( x , t ) | 2 ρ ( x , t ) d x d t factorization 2 0 M � 1 + δ 2 � α ( x , t ) 2 ρ ( x , t ) d x d t . 2 0 M where δ is a length parameter. Remark: very natural and not studied before.
From unbalanced Convex reformulation optimal transport to the Camassa-Holm equation Add a source term in the constraint: (weak sense) Fran¸ cois-Xavier Vialard ρ = −∇ · m + µ . ˙ Unbalanced optimal transport An isometric The Wasserstein-Fisher-Rao metric: embedding Euler-Arnold-Poincar´ e � 1 � 1 | m ( x , t ) | 2 d x d t + δ 2 µ ( x , t ) 2 equation WF( m , µ ) 2 = 1 � � ρ ( x , t ) d x d t . The Camassa-Holm 2 ρ ( x , t ) 2 equation as an 0 M 0 M incompressible Euler equation Corresponding polar factorization
From unbalanced Convex reformulation optimal transport to the Camassa-Holm equation Add a source term in the constraint: (weak sense) Fran¸ cois-Xavier Vialard ρ = −∇ · m + µ . ˙ Unbalanced optimal transport An isometric The Wasserstein-Fisher-Rao metric: embedding Euler-Arnold-Poincar´ e � 1 � 1 | m ( x , t ) | 2 d x d t + δ 2 µ ( x , t ) 2 equation WF( m , µ ) 2 = 1 � � ρ ( x , t ) d x d t . The Camassa-Holm 2 ρ ( x , t ) 2 equation as an 0 M 0 M incompressible Euler equation Corresponding polar factorization • Fisher-Rao metric: Hessian of the Boltzmann entropy/ Kullback-Leibler divergence and reparametrization invariant. Wasserstein metric on the space of variances in 1D. • Convex and 1-homogeneous: convex analysis (existence and more) • Numerics: First-order splitting algorithm: Douglas-Rachford. • Code available at https://github.com/lchizat/optimal-transport/
From unbalanced A general framework optimal transport to the Camassa-Holm equation Definition (Infinitesimal cost) Fran¸ cois-Xavier Vialard An infinitesimal cost is f : M × R × R d × R → R + ∪ { + ∞} such Unbalanced optimal that for all x ∈ M , f ( x , · , · , · ) is convex, positively 1-homogeneous, transport lower semicontinuous and satisfies An isometric embedding Euler-Arnold-Poincar´ e = 0 if ( m , µ ) = (0 , 0) and ρ ≥ 0 equation The Camassa-Holm f ( x , ρ, m , µ ) > 0 if | m | or | µ | > 0 equation as an incompressible Euler = + ∞ if ρ < 0 . equation Corresponding polar factorization Definition (Dynamic problem) For ( ρ, m , µ ) ∈ M ([0 , 1] × M ) 1+ d +1 , let � 1 � f ( x , d ρ d λ , d µ def. d λ , d m J ( ρ, m , µ ) = d λ ) d λ ( t , x ) (19) 0 M The dynamic problem is, for ρ 0 , ρ 1 ∈ M + ( M ), def. C ( ρ 0 , ρ 1 ) = inf 0 ( ρ 0 ,ρ 1 ) J ( ρ, ω, ζ ) . (20) ( ρ,ω,ζ ) ∈CE 1
From unbalanced Existence of minimizers optimal transport to the Camassa-Holm equation Proposition (Fenchel-Rockafellar) Fran¸ cois-Xavier Vialard Let B ( x ) be the polar set of f ( x , · , · , · ) for all x ∈ M and assume it Unbalanced optimal transport is a lower semicontinuous set-valued function. Then the minimum An isometric of (20) is attained and it holds embedding Euler-Arnold-Poincar´ e equation � � C D ( ρ 0 , ρ 1 ) = sup ϕ (1 , · ) d ρ 1 − ϕ (0 , · ) d ρ 0 (21) The Camassa-Holm equation as an ϕ ∈ K M M incompressible Euler equation def. with K = Corresponding polar factorization ϕ ∈ C 1 ([0 , 1] × M ) : ( ∂ t ϕ, ∇ ϕ, ϕ ) ∈ B ( x ) , ∀ ( t , x ) ∈ [0 , 1] × M � � .
From unbalanced Existence of minimizers optimal transport to the Camassa-Holm equation Proposition (Fenchel-Rockafellar) Fran¸ cois-Xavier Vialard Let B ( x ) be the polar set of f ( x , · , · , · ) for all x ∈ M and assume it Unbalanced optimal transport is a lower semicontinuous set-valued function. Then the minimum An isometric of (20) is attained and it holds embedding Euler-Arnold-Poincar´ e equation � � C D ( ρ 0 , ρ 1 ) = sup ϕ (1 , · ) d ρ 1 − ϕ (0 , · ) d ρ 0 (21) The Camassa-Holm equation as an ϕ ∈ K M M incompressible Euler equation def. with K = Corresponding polar factorization ϕ ∈ C 1 ([0 , 1] × M ) : ( ∂ t ϕ, ∇ ϕ, ϕ ) ∈ B ( x ) , ∀ ( t , x ) ∈ [0 , 1] × M � � . | y | 2 + δ 2 z 2 if x > 0, 2 x WF( x , y , z ) = 0 if ( x , | y | , z ) = (0 , 0 , 0) + ∞ otherwise and the corresponding Hamilton-Jacobi equation is |∇ ϕ | 2 + ϕ 2 � � ∂ t ϕ + 1 ≤ 0 . δ 2 2
From unbalanced Numerical simulations optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization Figure – WFR geodesic between bimodal densities
From unbalanced Numerical simulations optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation ρ 0 ρ 1 Corresponding polar factorization • t = 0 t = 0 . 5 t = 1 Figure – Geodesics between ρ 0 and ρ 1 for (1st row) Hellinger, (2nd row) W 2 , (3rd row) partial OT, (4th row) WF. An Interpolating Distance between Optimal Transport and Fisher-Rao , L. Chizat, B. Schmitzer, G. Peyr´ e, and F.-X. Vialard, FoCM, 2016.
From unbalanced Numerical simulations optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation ρ 0 ρ 1 Corresponding polar factorization • t = 0 t = 0 . 5 t = 1 Figure – Geodesics between ρ 0 and ρ 1 for (1st row) Hellinger, (2nd row) W 2 , (3rd row) partial OT, (4th row) WF. An Interpolating Distance between Optimal Transport and Fisher-Rao , L. Chizat, B. Schmitzer, G. Peyr´ e, and F.-X. Vialard, FoCM, 2016.
From unbalanced Numerical simulations optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation ρ 0 ρ 1 Corresponding polar factorization • t = 0 t = 0 . 5 t = 1 Figure – Geodesics between ρ 0 and ρ 1 for (1st row) Hellinger, (2nd row) W 2 , (3rd row) partial OT, (4th row) WF. An Interpolating Distance between Optimal Transport and Fisher-Rao , L. Chizat, B. Schmitzer, G. Peyr´ e, and F.-X. Vialard, FoCM, 2016.
From unbalanced Numerical simulations optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation ρ 0 ρ 1 Corresponding polar factorization • t = 0 t = 0 . 5 t = 1 Figure – Geodesics between ρ 0 and ρ 1 for (1st row) Hellinger, (2nd row) W 2 , (3rd row) partial OT, (4th row) WF. An Interpolating Distance between Optimal Transport and Fisher-Rao , L. Chizat, B. Schmitzer, G. Peyr´ e, and F.-X. Vialard, FoCM, 2016.
From unbalanced Numerical simulations optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation ρ 0 ρ 0 ρ 1 ρ 1 Corresponding polar factorization • • t = 0 t = 0 t = 0 . 5 t = 0 . 5 t = 1 t = 1 Figure – Geodesics between ρ 0 and ρ 1 for (1st row) Hellinger, (2nd row) W 2 , (3rd row) partial OT, (4th row) WF. An Interpolating Distance between Optimal Transport and Fisher-Rao , L. Chizat, B. Schmitzer, G. Peyr´ e, and F.-X. Vialard, FoCM, 2016.
From unbalanced From dynamic to static optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Group action Unbalanced optimal transport Mass can be moved and changed: consider m ( t ) δ x ( t ) . An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced From dynamic to static optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Group action Unbalanced optimal transport Mass can be moved and changed: consider m ( t ) δ x ( t ) . An isometric embedding Euler-Arnold-Poincar´ e equation Infinitesimal action The Camassa-Holm equation as an � incompressible Euler x ( t ) = v ( t , x ( t )) ˙ equation ρ = −∇ · ( v ρ ) + µ ⇔ ˙ Corresponding polar m ( t ) = µ ( t , x ( t )) ˙ factorization
From unbalanced From dynamic to static optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Group action Unbalanced optimal transport Mass can be moved and changed: consider m ( t ) δ x ( t ) . An isometric embedding Euler-Arnold-Poincar´ e equation Infinitesimal action The Camassa-Holm equation as an � incompressible Euler x ( t ) = v ( t , x ( t )) ˙ equation ρ = −∇ · ( v ρ ) + µ ⇔ ˙ Corresponding polar m ( t ) = µ ( t , x ( t )) ˙ factorization A cone metric m 2 m )) = 1 x 2 + ˙ WF 2 ( x , m ) ((˙ x , ˙ m ) , (˙ x , ˙ 2( m ˙ m ) , Change of variable: r 2 = m ...
From unbalanced Riemannian cone optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Definition Vialard Let ( M , g ) be a Riemannian manifold. The cone over ( M , g ) is Unbalanced optimal transport the Riemannian manifold ( M × R ∗ + , r 2 g + d r 2 ) . An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced Riemannian cone optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Definition Vialard Let ( M , g ) be a Riemannian manifold. The cone over ( M , g ) is Unbalanced optimal transport the Riemannian manifold ( M × R ∗ + , r 2 g + d r 2 ) . An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar α factorization r For M = S 1 ( r ), radius r ≤ 1. One has sin( α ) = r .
From unbalanced Geometry of a cone optimal transport to the Camassa-Holm • Change of variable: WF 2 = 1 equation 2 r 2 g + 2 d r 2 . Fran¸ cois-Xavier • Non complete metric space: add the vertex M × { 0 } . Vialard • The distance: Unbalanced optimal transport d (( x 1 , m 1 ) , ( x 2 , m 2 )) 2 = An isometric embedding m 2 + m 1 − 2 √ m 1 m 2 cos � 1 � Euler-Arnold-Poincar´ e 2 d M ( x 1 , x 2 ) ∧ π . (22) equation The Camassa-Holm equation as an • Curvature tensor: R ( ˜ X , e ) = 0 and incompressible Euler equation R ( ˜ X , ˜ Y )˜ Z = ( R g ( X , Y ) Z − g ( Y , Z ) X + g ( X , Z ) Y , 0). • M = R then ( x , m ) �→ √ me ix / 2 ∈ C local isometry. Corresponding polar factorization Corollary If ( M , g ) has sectional curvature greater than 1 , then ( M × R ∗ 4 m d m 2 ) has non-negative sectional curvature. 1 + , m g + For X , Y two orthornormal vector fields on M, K ( ˜ X , ˜ Y ) = ( K g ( X , Y ) − 1) (23) where K and K g denote respectively the sectional curvatures of M × R ∗ + and M.
Visualize geodesics for r 2 g + d r 2 From unbalanced optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization Figure – Geodesics on the cone
From unbalanced Distance between Diracs optimal transport to the Camassa-Holm y equation P 2 Fran¸ cois-Xavier Vialard P 3 Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e x P 1 equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization 1 4 WF ( m 1 δ x 1 , m 2 δ x 2 ) 2 = m 2 + m 1 − 2 √ m 1 m 2 cos � 1 � 2 d M ( x 1 , x 2 ) ∧ π/ 2 . Proof: prove that an explicit geodesic is a critical point of the convex functional. Properties: positively 1-homogeneous and convex in ( m 1 , m 2 ).
From unbalanced Generalization of Otto’s Riemannian submersion optimal transport to the Camassa-Holm equation Idea of a left group action: Fran¸ cois-Xavier Vialard Diff( M ) ⋉ C ∞ ( M , R ∗ � � π : + ) × Dens( M ) �→ Dens( M ) Unbalanced optimal π (( ϕ, λ ) , ρ ) := ϕ ∗ ( λ 2 ρ ) transport An isometric embedding Group law: Euler-Arnold-Poincar´ e equation The Camassa-Holm ( ϕ 1 , λ 1 ) · ( ϕ 2 , λ 2 ) = ( ϕ 1 ◦ ϕ 2 , ( λ 1 ◦ ϕ 2 ) λ 2 ) (24) equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced Generalization of Otto’s Riemannian submersion optimal transport to the Camassa-Holm equation Idea of a left group action: Fran¸ cois-Xavier Vialard Diff( M ) ⋉ C ∞ ( M , R ∗ � � π : + ) × Dens( M ) �→ Dens( M ) Unbalanced optimal π (( ϕ, λ ) , ρ ) := ϕ ∗ ( λ 2 ρ ) transport An isometric embedding Group law: Euler-Arnold-Poincar´ e equation The Camassa-Holm ( ϕ 1 , λ 1 ) · ( ϕ 2 , λ 2 ) = ( ϕ 1 ◦ ϕ 2 , ( λ 1 ◦ ϕ 2 ) λ 2 ) (24) equation as an incompressible Euler equation Corresponding polar Theorem (P1) factorization Let ρ 0 ∈ Dens( M ) and π 0 : Diff( M ) ⋉ C ∞ ( M , R ∗ + ) �→ Dens( M ) defined by π 0 ( ϕ, λ ) := ϕ ∗ ( λ 2 ρ 0 ) . It is a Riemannian submersion π 0 (Diff( M ) ⋉ C ∞ ( M , R ∗ + ) , L 2 ( M , M × R ∗ + )) − → (Dens( M ) , WF) (where M × R ∗ + is endowed with the cone metric). O’Neill’s formula: sectional curvature of (Dens( M ) , WF).
From unbalanced Horizontal lift optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric Proposition (Horizontal lift) embedding Euler-Arnold-Poincar´ e equation Let ρ ∈ Dens s ( M ) be a smooth density and X ρ ∈ H s ( M , R ) be a The Camassa-Holm tangent vector at the density ρ . The horizontal lift at ( Id , 1) of X ρ equation as an incompressible Euler is given by ( 1 2 ∇ Z , Z ) where Z is the solution to the elliptic partial equation Corresponding polar differential equation: factorization − div( ρ ∇ Z ) + 2 Z ρ = X ρ . (25) By elliptic regularity, the unique solution Z belongs to H s +2 ( M ) .
From unbalanced Geometric consequence optimal transport to the Camassa-Holm equation The sectional curvature of Dens( M ) at point ρ is: ( Z being the Fran¸ cois-Xavier horizontal lift) Vialard Unbalanced optimal transport � K ( ρ )( X 1 , X 2 ) = k ( x , 1)( Z 1 ( x ) , Z 2 ( x )) w ( Z 1 ( x ) , Z 2 ( x )) ρ ( x ) d ν ( x ) An isometric embedding M Euler-Arnold-Poincar´ e + 3 � 2 equation � [ Z 1 , Z 2 ] V � � (26) 4 The Camassa-Holm equation as an where incompressible Euler equation Corresponding polar w ( Z 1 ( x ) , Z 2 ( x )) = g ( x )( Z 1 ( x ) , Z 1 ( x )) g ( x )( Z 2 ( x ) , Z 2 ( x )) factorization − g ( x )( Z 1 ( x ) , Z 2 ( x )) 2 and [ Z 1 , Z 2 ] V denotes the vertical projection of [ Z 1 , Z 2 ] at identity and � · � denotes the norm at identity.
From unbalanced Geometric consequence optimal transport to the Camassa-Holm equation The sectional curvature of Dens( M ) at point ρ is: ( Z being the Fran¸ cois-Xavier horizontal lift) Vialard Unbalanced optimal transport � K ( ρ )( X 1 , X 2 ) = k ( x , 1)( Z 1 ( x ) , Z 2 ( x )) w ( Z 1 ( x ) , Z 2 ( x )) ρ ( x ) d ν ( x ) An isometric embedding M Euler-Arnold-Poincar´ e + 3 � 2 equation � � [ Z 1 , Z 2 ] V � (26) 4 The Camassa-Holm equation as an where incompressible Euler equation Corresponding polar w ( Z 1 ( x ) , Z 2 ( x )) = g ( x )( Z 1 ( x ) , Z 1 ( x )) g ( x )( Z 2 ( x ) , Z 2 ( x )) factorization − g ( x )( Z 1 ( x ) , Z 2 ( x )) 2 and [ Z 1 , Z 2 ] V denotes the vertical projection of [ Z 1 , Z 2 ] at identity and � · � denotes the norm at identity. Corollary Let ( M , g ) be a compact Riemannian manifold of sectional curvature bounded below by 1 , then the sectional curvature of (Dens( M ) , WF) is non-negative.
From unbalanced Consequences optimal transport to the Camassa-Holm equation Monge formulation Fran¸ cois-Xavier Vialard Unbalanced optimal � � ( ϕ, λ ) − ( Id , 1) � L 2 ( ρ 0 ) : ϕ ∗ ( λ 2 ρ 0 ) = ρ 1 � WF ( ρ 0 , ρ 1 ) = inf transport ( ϕ,λ ) An isometric (27) embedding Euler-Arnold-Poincar´ e equation Under existence and smoothness of the minimizer, there exists a The Camassa-Holm function p ∈ C ∞ ( M , R ) such that equation as an incompressible Euler equation � 1 � ( ϕ ( x ) , λ ( x )) = exp C ( M ) 2 ∇ p ( x ) , p ( x ) , (28) Corresponding polar x factorization Equivalent to Monge-Amp` ere equation def. With z = log(1 + p ) one has (1 + |∇ z | 2 ) e 2 z ρ 0 = det( D ϕ ) ρ 1 ◦ ϕ (29) and � ∇ z ( x ) � � 1 � ϕ ( x ) = exp M arctan 2 |∇ z | . ( x , 1) |∇ z ( x ) |
From unbalanced Equivalence static/dynamic optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal Definition transport An isometric The path-based cost c s is embedding Euler-Arnold-Poincar´ e � 1 equation def. f ( x ( t ) , m ( t ) , m ( t ) x ′ ( t ) , m ′ ( t )) d t c s ( x 0 , m 0 , x 1 , m 1 ) = inf The Camassa-Holm equation as an ( x ( t ) , m ( t )) 0 incompressible Euler (30) equation for ( x ( t ) , m ( t )) ∈ C 1 ([0 , 1] , Ω × [0 , + ∞ [) such that Corresponding polar factorization ( x ( i ) , m ( i )) = ( x i , m i ) for i ∈ { 0 , 1 } . Consequence: c d ≤ c s .
From unbalanced Equivalence static/dynamic optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal Definition transport An isometric The path-based cost c s is embedding Euler-Arnold-Poincar´ e � 1 equation def. f ( x ( t ) , m ( t ) , m ( t ) x ′ ( t ) , m ′ ( t )) d t c s ( x 0 , m 0 , x 1 , m 1 ) = inf The Camassa-Holm equation as an ( x ( t ) , m ( t )) 0 incompressible Euler (30) equation for ( x ( t ) , m ( t )) ∈ C 1 ([0 , 1] , Ω × [0 , + ∞ [) such that Corresponding polar factorization ( x ( i ) , m ( i )) = ( x i , m i ) for i ∈ { 0 , 1 } . Consequence: c d ≤ c s . Theorem If C K weak ∗ continuous and c d l.s.c. then c d = c ∗∗ and C K = C D . s
From unbalanced Kantorovich formulation optimal transport to the Camassa-Holm equation Recall Fran¸ cois-Xavier Vialard 1 4 c 2 d ( x 1 , m 1 , x 2 , m 2 ) = m 2 + m 1 Unbalanced optimal transport − 2 √ m 1 m 2 cos � 1 � An isometric 2 d M ( x 1 , x 2 ) ∧ π/ 2 . embedding Euler-Arnold-Poincar´ e equation then The Camassa-Holm equation as an � � ( x , d γ 1 d γ ) , ( y , d γ 2 � WF ( ρ 1 , ρ 2 ) 2 = M 2 c 2 incompressible Euler inf d γ ) d γ ( x , y ) , equation d ( γ 1 ,γ 2 ) ∈ Γ( ρ 1 ,ρ 2 ) Corresponding polar factorization
From unbalanced Kantorovich formulation optimal transport to the Camassa-Holm equation Recall Fran¸ cois-Xavier Vialard 1 4 c 2 d ( x 1 , m 1 , x 2 , m 2 ) = m 2 + m 1 Unbalanced optimal transport − 2 √ m 1 m 2 cos � 1 � An isometric 2 d M ( x 1 , x 2 ) ∧ π/ 2 . embedding Euler-Arnold-Poincar´ e equation then The Camassa-Holm equation as an � � ( x , d γ 1 d γ ) , ( y , d γ 2 � WF ( ρ 1 , ρ 2 ) 2 = M 2 c 2 incompressible Euler inf d γ ) d γ ( x , y ) , equation d ( γ 1 ,γ 2 ) ∈ Γ( ρ 1 ,ρ 2 ) Corresponding polar factorization Theorem (Dual formulation) � � WF 2 ( ρ 0 , ρ 1 ) = sup φ ( x ) d ρ 0 + ψ ( y ) d ρ 1 ( φ,ψ ) ∈ C ( M ) 2 M M subject to ∀ ( x , y ) ∈ M 2 , � φ ( x ) ≤ 1 , ψ ( y ) ≤ 1 , (1 − φ ( x ))(1 − ψ ( y )) ≥ cos 2 ( | x − y | / 2 ∧ π/ 2)
From unbalanced A relaxed static OT formulation optimal transport to the Camassa-Holm Define equation � d γ � d γ � Fran¸ cois-Xavier KL ( γ, ν ) = d ν log d ν + | ν | − | γ | Vialard d ν Unbalanced optimal Theorem (Dual formulation, P1) transport An isometric � � embedding WF 2 ( ρ 0 , ρ 1 ) = sup φ ( x ) d ρ 0 + ψ ( y ) d ρ 1 Euler-Arnold-Poincar´ e ( φ,ψ ) ∈ C ( M ) 2 M M equation subject to ∀ ( x , y ) ∈ M 2 , φ ( x ) ≤ 1 , ψ ( y ) ≤ 1 and The Camassa-Holm equation as an (1 − φ ( x ))(1 − ψ ( y )) ≥ cos 2 ( | x − y | / 2 ∧ π/ 2) incompressible Euler equation Corresponding polar factorization The corresponding primal formulation WF 2 ( ρ 1 , ρ 2 ) = inf γ KL (Proj 1 ∗ γ, ρ 1 ) + KL (Proj 2 ∗ γ, ρ 2 ) � M 2 γ ( x , y ) log(cos 2 ( d ( x , y ) / 2 ∧ π/ 2)) d x d y − Theorem (P2) On a Riemannian manifold (compact without boundary), the static and dynamic formulations are equal.
From unbalanced New algorithm optimal transport to the Camassa-Holm equation Scaling Algorithms for Unbalanced Transport Problems , L. Chizat, Fran¸ cois-Xavier Vialard G. Peyr´ e, B. Schmitzer, F.-X. Vialard. Unbalanced optimal • Use of entropic regularization. transport An isometric embedding Euler-Arnold-Poincar´ e equation WF 2 ( ρ 1 , ρ 2 ) = inf γ KL (Proj 1 ∗ γ, ρ 1 ) + KL (Proj 2 ∗ γ, ρ 2 ) The Camassa-Holm equation as an incompressible Euler � M 2 γ ( x , y ) log(cos 2 ( d ( x , y ) / 2 ∧ π/ 2)) d x d y + ε KL ( γ, µ 0 ) . equation − Corresponding polar factorization • Alternate projection algorithm (contraction for a Hilbert type metric). • Applications to color transfer, Fr´ echet-Karcher mean (barycenters). • Similarity measure in inverse problems. (Optimal transport for diffeomorphic registration, MICCAI 2017). • Simulations for gradient flows.
From unbalanced Application to color transfer optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e Figure – Transporting the color histograms: initial and final image equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization Optimal transport Range constraint Kullback-Leibler Total variation
From unbalanced Contents optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal Unbalanced optimal transport transport 1 An isometric embedding Euler-Arnold-Poincar´ e An isometric embedding equation 2 The Camassa-Holm equation as an incompressible Euler equation Euler-Arnold-Poincar´ e equation 3 Corresponding polar factorization The Camassa-Holm equation as an incompressible Euler 4 equation Corresponding polar factorization 5
From unbalanced The Riemannian submersion for WFR optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Diff( M ) ⋉ C ∞ ( M , R ∗ + ) Vialard L 2 ( M , C ( M )) Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Isotropy Corresponding polar factorization subgroup of µ π ( ϕ, λ ) = ϕ ∗ ( λ 2 µ ) µ (Dens( M ) , WFR) Figure – The same picture in our case: what is the corresponding equation to Euler?
From unbalanced The isotropy subgroup for unbalanced optimal optimal transport to the Camassa-Holm equation transport Fran¸ cois-Xavier Vialard Recall that Unbalanced optimal π − 1 0 ( { ρ 0 } ) = { ( ϕ, λ ) ∈ Diff( M ) ⋉ C ∞ ( M , R ∗ + ) : ϕ ∗ ( λ 2 ρ 0 ) = ρ 0 } transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced The isotropy subgroup for unbalanced optimal optimal transport to the Camassa-Holm equation transport Fran¸ cois-Xavier Vialard Recall that Unbalanced optimal π − 1 0 ( { ρ 0 } ) = { ( ϕ, λ ) ∈ Diff( M ) ⋉ C ∞ ( M , R ∗ + ) : ϕ ∗ ( λ 2 ρ 0 ) = ρ 0 } transport An isometric embedding Euler-Arnold-Poincar´ e π − 1 � Jac( ϕ )) ∈ Diff( M ) ⋉ C ∞ ( M , R ∗ equation 0 ( { ρ 0 } ) = { ( ϕ, + ) : ϕ ∈ Diff( M ) } . The Camassa-Holm equation as an The vertical space is incompressible Euler equation Vert ( ϕ,λ ) = { ( v , α ) ◦ ( ϕ, λ ) ; div( ρ v ) = 2 αρ } , (31) Corresponding polar factorization where ( v , α ) ∈ Vect( M ) × C ∞ ( M , R ). The horizontal space is �� 1 � � ◦ ( ϕ, λ ) ; p ∈ C ∞ ( M , R ) Hor ( ϕ,λ ) = 2 ∇ p , p . (32)
From unbalanced The isotropy subgroup for unbalanced optimal optimal transport to the Camassa-Holm equation transport Fran¸ cois-Xavier Vialard Recall that Unbalanced optimal π − 1 0 ( { ρ 0 } ) = { ( ϕ, λ ) ∈ Diff( M ) ⋉ C ∞ ( M , R ∗ + ) : ϕ ∗ ( λ 2 ρ 0 ) = ρ 0 } transport An isometric embedding Euler-Arnold-Poincar´ e π − 1 � Jac( ϕ )) ∈ Diff( M ) ⋉ C ∞ ( M , R ∗ equation 0 ( { ρ 0 } ) = { ( ϕ, + ) : ϕ ∈ Diff( M ) } . The Camassa-Holm equation as an The vertical space is incompressible Euler equation Vert ( ϕ,λ ) = { ( v , α ) ◦ ( ϕ, λ ) ; div( ρ v ) = 2 αρ } , (31) Corresponding polar factorization where ( v , α ) ∈ Vect( M ) × C ∞ ( M , R ). The horizontal space is �� 1 � � ◦ ( ϕ, λ ) ; p ∈ C ∞ ( M , R ) Hor ( ϕ,λ ) = 2 ∇ p , p . (32) The induced metric is | v | 2 d µ + 1 � � | div v | 2 d µ . G ( v , div v ) = (33) 4 M M The H div right-invariant metric on the group of diffeomorphisms.
From unbalanced Contents optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal Unbalanced optimal transport transport 1 An isometric embedding Euler-Arnold-Poincar´ e An isometric embedding equation 2 The Camassa-Holm equation as an incompressible Euler equation Euler-Arnold-Poincar´ e equation 3 Corresponding polar factorization The Camassa-Holm equation as an incompressible Euler 4 equation Corresponding polar factorization 5
From unbalanced Right-invariant metric on a Lie group optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Definition (Right-invariant metric) Unbalanced optimal transport Let g 1 , g 2 ∈ G be two group elements, the distance between g 1 An isometric and g 2 can be defined by: embedding Euler-Arnold-Poincar´ e �� 1 equation � d 2 ( g 1 , g 2 ) = inf � v ( t ) � 2 g dt | g (0) = g 0 and g (1) = g 1 The Camassa-Holm equation as an g ( t ) 0 incompressible Euler equation where ∂ t g ( t ) g ( t ) − 1 = v ( t ) ∈ g , with g the Lie algebra. Corresponding polar factorization
From unbalanced Right-invariant metric on a Lie group optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Definition (Right-invariant metric) Unbalanced optimal transport Let g 1 , g 2 ∈ G be two group elements, the distance between g 1 An isometric and g 2 can be defined by: embedding Euler-Arnold-Poincar´ e �� 1 equation � d 2 ( g 1 , g 2 ) = inf � v ( t ) � 2 g dt | g (0) = g 0 and g (1) = g 1 The Camassa-Holm equation as an g ( t ) 0 incompressible Euler equation where ∂ t g ( t ) g ( t ) − 1 = v ( t ) ∈ g , with g the Lie algebra. Corresponding polar factorization Right-invariance means: d 2 ( g 1 g , g 2 g ) = d ( g 1 , g 2 ) . It comes from: ∂ t ( g ( t ) g 0 )( g ( t ) g 0 ) − 1 = ∂ t g ( t ) g 0 g − 1 0 g ( t ) − 1 = ∂ t g ( t ) g ( t ) − 1 .
From unbalanced Euler-Arnold-Poincar´ e equation optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Compute the Euler-Lagrange equation of the distance functional: Vialard ∂ g − d ∂ L ∂ L Unbalanced optimal g = 0 transport dt ∂ ˙ An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced Euler-Arnold-Poincar´ e equation optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Compute the Euler-Lagrange equation of the distance functional: Vialard ∂ g − d ∂ L ∂ L Unbalanced optimal g = 0 transport dt ∂ ˙ An isometric embedding Euler-Arnold-Poincar´ e � 1 � 1 0 � u � 2 dt , equation In the case of 0 L ( g , ˙ g ) dt = The Camassa-Holm Euler-Poincar´ e-Arnold equation equation as an incompressible Euler equation � g = u ◦ g ˙ Corresponding polar (34) factorization u + ad ∗ ˙ u u = 0 where ad ∗ u is the (metric) adjoint of ad u v = [ v , u ]. Proof. Compute variations of v ( t ) in terms of u ( t ) = δ g ( t ) g ( t ) − 1 . Find that admissible variations on g can be written as: δ v ( t ) = ˙ u − ad v u for any u vanishing at 0 and 1.
From unbalanced Fluid dynamics examples of Euler-Arnold optimal transport to the Camassa-Holm equation equations Fran¸ cois-Xavier Vialard • Incompressible Euler equation. Unbalanced optimal transport • Korteweg-de-Vries equation. An isometric embedding • Camassa-Holm equation 1981/1993. An integrable shallow Euler-Arnold-Poincar´ e water equation with peaked solitons equation Consider Diff( S 1 ) endowed with the H 1 right-invariant metric The Camassa-Holm equation as an incompressible Euler � v � 2 L 2 + 1 4 � ∂ x v � 2 L 2 . One has equation Corresponding polar factorization � ∂ t u − 1 4 ∂ txx u u + 3 ∂ x u u − 1 2 ∂ xx u ∂ x u − 1 4 ∂ xxx u u = 0 (35) ∂ t ϕ ( t , x ) = u ( t , ϕ ( t , x )) . • Model for waves in shallow water. • Completely integrable system (bi-Hamiltonian). • Exhibits particular solutions named as peakons. (geodesics as collective Hamiltonian). • Blow-up of solutions which gives a model for wave breaking.
From unbalanced Ebin-Marsden and Michor-Mumford optimal transport to the Camassa-Holm equation Rewrite the metric in Lagrangian coordinates ϕ and a tangent Fran¸ cois-Xavier vector X ϕ and realize that it is smooth... Vialard • The right-invariant H div metric: Unbalanced optimal transport � a 2 | X ϕ ◦ ϕ − 1 | 2 + b 2 div( X ϕ ◦ ϕ − 1 ) 2 d µ . G ϕ ( X ϕ , X ϕ ) = (36) An isometric embedding M Euler-Arnold-Poincar´ e Smooth weak metric on an infinite dimensional Riemannian equation manifold when M = S 1 . The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced Ebin-Marsden and Michor-Mumford optimal transport to the Camassa-Holm equation Rewrite the metric in Lagrangian coordinates ϕ and a tangent Fran¸ cois-Xavier vector X ϕ and realize that it is smooth... Vialard • The right-invariant H div metric: Unbalanced optimal transport � a 2 | X ϕ ◦ ϕ − 1 | 2 + b 2 div( X ϕ ◦ ϕ − 1 ) 2 d µ . G ϕ ( X ϕ , X ϕ ) = (36) An isometric embedding M Euler-Arnold-Poincar´ e Smooth weak metric on an infinite dimensional Riemannian equation manifold when M = S 1 . Consequences: The Camassa-Holm equation as an • Geodesic equations is a simple ODE (No need for a incompressible Euler equation Riemannian connection) Corresponding polar • Gauss lemma on H s for s > d / 2 + 2 factorization • Geodesics are minimizing within H s topology. Theorem (Consequence of Ebin and Marsden) Local well-posedness of the geodesics for the H 1 ( S 1 ) right-invariant metric on Diff s ( S 1 ) for s > 1 / 2 + 2 . Theorem (Michor-Mumford) Local well-posedness of the geodesics for the H div right-invariant metric on Diff s ( R d ) for s high enough.
From unbalanced Metric properties optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport Theorem (Michor and Mumford, 2005) An isometric embedding The distance on Diff( M ) endowed with the right-invariant metric Euler-Arnold-Poincar´ e L 2 is degenerate; i.e. d ( ϕ 0 , ϕ 1 ) = 0 for every ϕ 0 , ϕ 1 ∈ Diff( M ) . equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced Metric properties optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport Theorem (Michor and Mumford, 2005) An isometric embedding The distance on Diff( M ) endowed with the right-invariant metric Euler-Arnold-Poincar´ e L 2 is degenerate; i.e. d ( ϕ 0 , ϕ 1 ) = 0 for every ϕ 0 , ϕ 1 ∈ Diff( M ) . equation The Camassa-Holm equation as an incompressible Euler Theorem (Michor and Mumford, 2005) equation Corresponding polar factorization The distance on Diff( M ) endowed with the right-invariant metric H Div is non degenerate. Proof. Direct using the isometric injection.
From unbalanced An isometric embedding optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard We have Unbalanced optimal transport inj : (Diff( M ) , H div ) ֒ → L 2 ( M , C ( M )) An isometric embedding � ϕ �→ ( ϕ, Jac( ϕ )) . Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced An isometric embedding optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard We have Unbalanced optimal transport inj : (Diff( M ) , H div ) ֒ → L 2 ( M , C ( M )) An isometric embedding � ϕ �→ ( ϕ, Jac( ϕ )) . Euler-Arnold-Poincar´ e equation The geodesic equations can be written in Lagrangian coordinates The Camassa-Holm equation as an incompressible Euler � ˙ equation D λ ϕ = −∇ g P ◦ ϕ Dt ˙ ϕ + 2 λ ˙ (37) Corresponding polar ¨ factorization λ r − λ rg ( ˙ ϕ, ˙ ϕ ) = − 2 λ rP ◦ ϕ . In Eulerian coordinates, � v + ∇ g v v + 2 v α = −∇ g P ˙ (38) α + �∇ α, v � + α 2 − g ( v , v ) = − 2 P , ˙ λ ◦ ϕ − 1 and v = ∂ t ϕ ◦ ϕ − 1 . ˙ λ where α =
From unbalanced Consequences of the isometric embedding optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding (Diff( M ) , H div ) ֒ → L 2 ( M , C ( M )) (39) Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced Consequences of the isometric embedding optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding (Diff( M ) , H div ) ֒ → L 2 ( M , C ( M )) (39) Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an Using Gauss-Codazzi formula, it generalizes a curvature 1 incompressible Euler equation formula by Khesin et al. obtained on Diff( S 1 ). Corresponding polar factorization Smooth geodesics are length minimizing for a short enough 2 time under mild conditions (generalization of Brenier’s proof). The Camassa-Holm equation as incompressible Euler. 3 A new polar factorization theorem. 4
From unbalanced In short: optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Gain w.r.t. Ebin and Marsden Unbalanced optimal transport • Ebin and Marsden proved that: Smooth solutions are An isometric minimizing in a H d / 2+2+ ε neighborhood. embedding Euler-Arnold-Poincar´ e • We have: Smooth solutions are minimizing in a W 1 , ∞ equation neighborhood. The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced In short: optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Gain w.r.t. Ebin and Marsden Unbalanced optimal transport • Ebin and Marsden proved that: Smooth solutions are An isometric minimizing in a H d / 2+2+ ε neighborhood. embedding Euler-Arnold-Poincar´ e • We have: Smooth solutions are minimizing in a W 1 , ∞ equation neighborhood. The Camassa-Holm equation as an incompressible Euler equation Theorem (P2) Corresponding polar factorization When M = S 1 , smooth solutions to the Camassa-Holm equation � ∂ t u − 1 4 ∂ txx u + 3 ∂ x u u − 1 2 ∂ xx u ∂ x u − 1 4 ∂ xxx u u = 0 (40) ∂ t ϕ ( t , x ) = u ( t , ϕ ( t , x )) . are length minimizing for short times.
From unbalanced Generalisation of Brenier’s proof optimal transport to the Camassa-Holm equation Theorem (P2) Fran¸ cois-Xavier Vialard Let ( ϕ ( t ) , r ( t )) be a smooth solution to the geodesic equations on the time interval [ t 0 , t 1 ] . If ( t 1 − t 0 ) 2 � w , ∇ 2 Ψ P ( t ) ( x , r ) w � < π 2 � w � 2 holds for all Unbalanced optimal transport t ∈ [ t 0 , t 1 ] and ( x , r ) ∈ C ( M ) and w ∈ T ( x , r ) C ( M ) , then for every smooth An isometric curve ( ϕ 0 ( t ) , r 0 ( t )) ∈ Aut vol ( C ( M )) satisfying ( ϕ 0 ( t i ) , r 0 ( t i )) = ( ϕ ( t i ) , r ( t i )) embedding for i = 0 , 1 and the condition ( ∗ ) , one has Euler-Arnold-Poincar´ e equation � t 1 � t 1 r ) � 2 d t ≤ r 0 ) � 2 d t , � ( ˙ ϕ, ˙ � ( ˙ ϕ 0 , ˙ (41) The Camassa-Holm equation as an t 0 t 0 incompressible Euler equation with equality if and only if the two paths coincide on [ t 0 , t 1 ] . Corresponding polar def. Define δ 0 = min { r ( x , t ) : injectivity radius at ( ϕ ( t , x ) , r ( t , x )) } , then the factorization condition ( ∗ ) is: If the sectional curvature of C ( M ) can assume both signs or if 1 diam( M ) ≥ π , there exists δ satisfying 0 < δ < δ 0 such that the curve ( ϕ 0 ( t ) , r 0 ( t )) has to belong to a δ -neighborhood of ( ϕ ( t ) , r ( t )) , namely d C ( M ) (( ϕ 0 ( t , x ) , r 0 ( t , x )) , ( ϕ ( t , x ) , r ( t , x )))) ≤ δ for all ( x , t ) ∈ M × [ t 0 , t 1 ] where d C ( M ) is the distance on the cone. If C ( M ) has non positive sectional curvature, then, for every δ as above, 2 there exists a short enough time interval on which the geodesic will be length minimizing. If M = S d (1) , the result is valid for every path ( ˙ ϕ 0 , ˙ r 0 ) . 3
From unbalanced Contents optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal Unbalanced optimal transport transport 1 An isometric embedding Euler-Arnold-Poincar´ e An isometric embedding equation 2 The Camassa-Holm equation as an incompressible Euler equation Euler-Arnold-Poincar´ e equation 3 Corresponding polar factorization The Camassa-Holm equation as an incompressible Euler 4 equation Corresponding polar factorization 5
From unbalanced Toward the incompressible Euler equation optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Why? Unbalanced OT is linked to standard OT on the cone. Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced Toward the incompressible Euler equation optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Why? Unbalanced OT is linked to standard OT on the cone. Unbalanced optimal transport An isometric Question embedding Euler-Arnold-Poincar´ e Understand Diff( M ) ⋉ C ∞ ( M , R ∗ + ) as a subgroup of Diff( C ( M )) ? equation The Camassa-Holm equation as an incompressible Euler Answer equation Corresponding polar The cone C ( M ) is a trivial principal fibre bundle over M . factorization The automorphism group Aut( C ( M )) ⊂ Diff( C ( M )) can be identified with Diff( M ) ⋉ C ∞ ( M , R ∗ + ). One has ( ϕ, λ ) : ( x , r ) �→ ( ϕ ( x ) , λ ( x ) r ). Recall that ψ ∈ Aut( C ( M )) if ψ ∈ Diff( C ( M )) and ∀ λ ∈ R ∗ + one def. has ψ ( λ · ( x , r )) = λ · ψ ( x , r ) where λ · ( x , r ) = ( x , λ r ).
From unbalanced CH as an incompressible Euler equation optimal transport to the Camassa-Holm The geodesic equation on Diff( M ) ⋉ C ∞ ( M , R ∗ equation + ) Fran¸ cois-Xavier Vialard ˙ � D λ ϕ = −∇ g P ◦ ϕ Dt ˙ ϕ + 2 λ ˙ (42) Unbalanced optimal ¨ transport λ r − λ rg ( ˙ ϕ, ˙ ϕ ) = − 2 λ rP ◦ ϕ . An isometric embedding can be extended to Aut( C ( M )) as Euler-Arnold-Poincar´ e equation The Camassa-Holm D ϕ, ˙ equation as an Dt ( ˙ λ r ) = −∇ Ψ P ◦ ( ϕ, λ r ) , (43) incompressible Euler equation Corresponding polar def. = r 2 P ( x ). factorization where Ψ P ( x , r )
From unbalanced CH as an incompressible Euler equation optimal transport to the Camassa-Holm The geodesic equation on Diff( M ) ⋉ C ∞ ( M , R ∗ equation + ) Fran¸ cois-Xavier Vialard ˙ � D λ ϕ = −∇ g P ◦ ϕ Dt ˙ ϕ + 2 λ ˙ (42) Unbalanced optimal ¨ transport λ r − λ rg ( ˙ ϕ, ˙ ϕ ) = − 2 λ rP ◦ ϕ . An isometric embedding can be extended to Aut( C ( M )) as Euler-Arnold-Poincar´ e equation The Camassa-Holm D ϕ, ˙ equation as an Dt ( ˙ λ r ) = −∇ Ψ P ◦ ( ϕ, λ r ) , (43) incompressible Euler equation Corresponding polar def. = r 2 P ( x ). factorization where Ψ P ( x , r ) Question Does there exist a density ˜ µ on the cone such that inj(Diff( M )) ⊂ SDiff ˜ µ ( C ( M )) ? (answer: yes)
From unbalanced CH as an incompressible Euler equation optimal transport to the Camassa-Holm The geodesic equation on Diff( M ) ⋉ C ∞ ( M , R ∗ equation + ) Fran¸ cois-Xavier Vialard ˙ � D λ ϕ = −∇ g P ◦ ϕ Dt ˙ ϕ + 2 λ ˙ (42) Unbalanced optimal ¨ transport λ r − λ rg ( ˙ ϕ, ˙ ϕ ) = − 2 λ rP ◦ ϕ . An isometric embedding can be extended to Aut( C ( M )) as Euler-Arnold-Poincar´ e equation The Camassa-Holm D ϕ, ˙ equation as an Dt ( ˙ λ r ) = −∇ Ψ P ◦ ( ϕ, λ r ) , (43) incompressible Euler equation Corresponding polar def. = r 2 P ( x ). factorization where Ψ P ( x , r ) Question Does there exist a density ˜ µ on the cone such that inj(Diff( M )) ⊂ SDiff ˜ µ ( C ( M )) ? (answer: yes) Proof. = r − 3 d r d µ where µ denotes the volume form on def. The measure ˜ µ M .
From unbalanced A new geometric picture optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Aut( C ( M ) ) Aut( C ( M ) ) L 2 ( M , C ( M )) Unbalanced optimal transport An isometric Diff( C ( M )) embedding L 2 ( C ( M )) Diff ˜ ν ( C ( M ) ) Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an Aut vol ( C ( M ) ) incompressible Euler Aut vol ( C ( M ) ) equation Corresponding polar factorization π ( ϕ, λ ) = ϕ ∗ ( λ 2 vol) π ( ψ ) = ψ ∗ (˜ ˜ ν ) (Dens( M ) , WFR) (Dens( C ( M )) , W 2 ) ν = r − 3 d vol d r vol ˜ Figure – On the left, the picture represents the Riemannian submersion between Aut( C ( M )) and the space of positive densities on M and the fiber above the volume form is Aut vol ( C ( M )). On the right, the picture represents the automorphism group Aut( C ( M )) isometrically embedded in Diff( C ( M )) and the intersection of Diff ˜ ν ( C ( M )) and Aut( C ( M )) is equal to Aut vol ( C ( M )).
From unbalanced Results optimal transport to the Camassa-Holm equation Theorem (P2) Fran¸ cois-Xavier Vialard Let ϕ be the flow of a smooth solution to the Camassa-Holm Unbalanced optimal def. � equation then Ψ( θ, r ) = ( ϕ ( θ ) , Jac( ϕ ( θ )) r ) is the flow of a transport An isometric solution to the incompressible Euler equation for the density embedding 1 r 4 r d r d θ . Euler-Arnold-Poincar´ e equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
From unbalanced Results optimal transport to the Camassa-Holm equation Theorem (P2) Fran¸ cois-Xavier Vialard Let ϕ be the flow of a smooth solution to the Camassa-Holm Unbalanced optimal def. � equation then Ψ( θ, r ) = ( ϕ ( θ ) , Jac( ϕ ( θ )) r ) is the flow of a transport An isometric solution to the incompressible Euler equation for the density embedding 1 r 4 r d r d θ . Euler-Arnold-Poincar´ e equation The Camassa-Holm � ∂ x ϕ ( θ ) e i ϕ ( θ ) ] then the Case where M = S 1 , M ( ϕ ) = [( θ, r ) �→ r equation as an incompressible Euler CH equation is equation Corresponding polar factorization � ∂ t u − 1 4 ∂ txx u u + 3 ∂ x u u − 1 2 ∂ xx u ∂ x u − 1 4 ∂ xxx u u = 0 (44) ∂ t ϕ ( t , x ) = u ( t , ϕ ( t , x )) . The Euler equation on the cone, C ( M ) = R 2 \ { 0 } for the density ρ = 1 r 4 Leb is � v + ∇ v v = −∇ p , ˙ (45) ∇ · ( ρ v ) = 0 . def. u ( θ ) , r � � where v ( θ, r ) = 2 ∂ x u ( θ ) .
From unbalanced Conclusion on this link with CH: optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal transport An isometric embedding Euler-Arnold-Poincar´ e equation Reformulation of CH The Camassa-Holm equation as an CH is a geodesic equation for an L 2 metric on the subgroup incompressible Euler equation Aut vol ( C ( M )): automorphisms of C ( M ) which preserve Corresponding polar 1 factorization r 3 d r d vol M .
From unbalanced Contents optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Unbalanced optimal Unbalanced optimal transport transport 1 An isometric embedding Euler-Arnold-Poincar´ e An isometric embedding equation 2 The Camassa-Holm equation as an incompressible Euler equation Euler-Arnold-Poincar´ e equation 3 Corresponding polar factorization The Camassa-Holm equation as an incompressible Euler 4 equation Corresponding polar factorization 5
From unbalanced Toward polar factorization optimal transport to the Camassa-Holm equation Definition Fran¸ cois-Xavier Vialard The generalized automorphism semigroup of C ( M ) is the set of Unbalanced optimal mesurable maps ( ϕ, λ ) from M to C ( M ) transport An isometric ( ϕ, λ ) ∈ M es ( M , M ) ⋉ M es ( M , R ∗ � � Aut( C ( M )) = + ) , (46) embedding Euler-Arnold-Poincar´ e equation endowed with the semigroup law The Camassa-Holm equation as an incompressible Euler ( ϕ 1 , λ 1 ) · ( ϕ 2 , λ 2 ) = ( ϕ 1 ◦ ϕ 2 , ( λ 1 ◦ ϕ 2 ) λ 2 ) . equation Corresponding polar factorization The stabilizer of the volume measure in the automorphisms of C ( M ) is � � Aut vol ( C ( M )) = ( s , λ ) ∈ Aut( C ( M )) : π (( s , λ ) , vol) = vol . (47) By abuse of notation, any ( s , λ ) ∈ Aut vol ( C ( M )) will be denoted � � � s , Jac( s ) i.e. f ∈ C ( M , R ) � 2 d vol( x ) = � � f ( s ( x )) Jac( s ) f ( x ) d vol( x ) . (48) M M
From unbalanced Toward polar factorization optimal transport to the Camassa-Holm equation Fran¸ cois-Xavier Vialard Definition (Admissible measures) Unbalanced optimal transport We say that a positive Radon measure ρ on M is admissible (with An isometric embedding respect to vol) if for any x ∈ M , there exists y ∈ Supp( ρ ) such Euler-Arnold-Poincar´ e that d ( x , y ) < π/ 2. equation The Camassa-Holm equation as an incompressible Euler equation Corresponding polar factorization
Recommend
More recommend
