Variational problems and optimal transport

ECTS : 6

Volume horaire : 24

Description du contenu de l'enseignement :

Chapter 1: Convexity in the calculus of variations 

• separation theorems, Legendre transforms, subdifferentiability
• convex duality by a general perturbation argument, special cases (Fenchel-Rockafellar, linear programming, zero sum games, Lagrangian duality)
• calculus of variations: the role of convexity, relaxation, Euler-Lagrange equations

Chapter 2: The optimal transport problem of Monge and Kantorovich

• The formulations of Monge and Kantorovich, examples and special cases (dimension one, the assignment problem, Birkhoff theorem), Kantorovich as a relaxation of Monge
• Kantorovich duality
• Twisted costs, existence of Monge solutions, Brenier’s theorem, Monge-Ampère equation, OT proof of the isoperimetric inequality
• Introduction to entropic optimal transport

 Chapter 3: Dynamic optimal transport, Wasserstein spaces, gradient flows 

• The distance cost case and its connection with minimal flows
• Wasserstein spaces
• Benamou-Brenier formula and geodesics, displacement convexity
• gradient flows, a starter: the Fokker-Planck equation, general theory for lambda-convex functionals

Compétence à acquérir :

Maitrise des outils d'analyse convexe pour le calcul des variations et des éléments fondamentaux de la théorie du transport optimal 

Mode de contrôle des connaissances :

Examen écrit ou oral