Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm

TL;DR

This paper introduces a numerical method using Sinkhorn's algorithm for entropic regularization in optimal transport, applied to incompressible flows, with promising results in low dimensions.

Contribution

It develops a novel numerical approach based on Sinkhorn's algorithm for generalized incompressible flows, connecting Brenier's formulation with entropic regularization.

Findings

Successful implementation in 1D and 2D

Demonstrates feasibility of the method

Provides comparison with Bredinger entropic interpolation

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. Numerical results in dimension one and two illustrate the feasibility of the method.