Minimizing Flows for the Monge--Kantorovich Problem

Abstract

In this work, we formulate a new minimizing flow for the optimal mass transport (Monge-Kantorovich) problem. We study certain properties of the flow, including weak solutions as well as short- and long-term existence. Optimal transport has found a number of applications, includ- ing econometrics, fluid dynamics, cosmology, image processing, automatic control, transportation, statistical physics, shape optimization, expert systems, and meteorology. 1. Introduction. In this paper, we derive a novel gradient descent flow for the computation of the optimal transport map (when it exists) in the Monge-Kantorovich framework. Besides being quite useful for the efficient computation of the transport map, we believe that the flow presented here is quite interesting from a theoretical point of view as well. In the present work, we undertake a study of some of its key