Variational problems and optimal transport
Enseignant responsable :
Volume horaire : 24Description 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 forrmulations 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
Chapter 3: Dynamic optimal transport, Wasserstein spaces, gradient flows
-Wasserstein spaces -Benamou-Brenier formula and geodesics, displacement convexity -gradient flows, a starter: the Fokker-Planck equation, general theory for lambda-convex functionals
Chapter 4: Computational OT and applications
-Entropic OT, Sinkhorn algorithm and its convergence -Matching problems, barycenters, -Wasserstein distances as a loss, Wasserstein GANs
Compétence à acquérir :
Mastering of variational and optimal transport methods used in economy.