Numerical Methods for Multi-Marginal Optimal Transportation.

Summary In this thesis, our goal is to give a general numerical framework to approximate the solutions of optimal transport (OT) problems. The general idea is to introduce an entropy regularization of the initial problem. The regularized problem corresponds to minimize a relative entropy with respect to a given reference measure. Indeed, this is equivalent to finding the projection of a coupling with respect to the Kullback-Leibler divergence. This allows us to use the Bregman/Dykstra algorithm and to solve several variational problems related to TO. We are particularly interested in the solution of multi-marginal optimal transport (MMOT) problems that arise in the context of fluid dynamics (incompressible Euler equations à la Brenier) and quantum physics (the density functional theory). In these cases, we show that entropy regularization plays a more important role than simple numerical stabilization. Moreover, we give results concerning the existence of optimal transports (e.g. fractal transports) for the TOMM problem.