Optimal Transport and Gradient Flows: A Discrete-to-Continuum Approximation Problem

Abstract:
In the seminal work of Jordan, Kinderlehrer and Otto (’98) the authors showed that the heat flow on R^d can be seen as gradient flow of the relative entropy functional in the space of probability measures with respect to the Wasserstein distance W2. The correspondent discrete counterpart is represented by the work of Maas (2011) and Mielke (2011), where a new notion of discrete dynamical optimal transport has been introduced and a similar result has been obtained. In this talk we first recall the classical gradient flow structure of the Fokker-Planck equation in R^d and the connection with the optimal transport problem. Secondly, we describe a classical finite volume method-type discretization of the equation, to which we are able to associate a natural graph structure, together with the correspondent discrete transport distance. We then analyze the correspondent discrete gradient flow structure and from that we discuss some convergence results, both concerning the discrete transport costs and the evolutionary G-convergence of the discrete gradient flow structures to the one associated to the continuous Fokker-Planck equation.
This project is based on joint works with Dominik Forkert, Eva Kopfer, Peter Gladbach and Jan Maas.