Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm

Abstract

Starting from Brenier’s relaxed formulation of the incompressible Euler equation in terms of geodesics in the group of measure-preserving diffeomorphisms, we propose a numerical method based on Sinkhorn’s algorithm for the entropic regularization of optimal transport. We also make a detailed comparison of this entropic regularization with the so-called Bredinger entropic interpolation problem (see Arnaudon et al. in An entropic interpolation problem for incompressible viscid fluids, 2017, arXiv preprint arXiv:1704.02126 ). Numerical results in dimension one and two illustrate the feasibility of the method.