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

• 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
• 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

• Entropic OT, Sinkhorn algorithm and its convergence
• Matching problems, barycenters,
• Wasserstein distances as a loss, Wasserstein GANs