Computation of optimal transport on discrete metric measure spaces

In this paper we investigate the numerical approximation of an analogue of the Wasserstein distance for optimal transport on graphs that is defined via a discrete modification of the Benamou–Brenier formula. This approach involves the logarithmic mean of measure densities on adjacent nodes of the graph. For this model a variational time discretization of the probability densities on graph nodes and the momenta on graph edges is proposed. A robust descent algorithm for the action functional is derived, which in particular uses a proximal splitting with an edgewise nonlinear projection on the convex subgraph of the logarithmic mean. Thereby, suitable chosen slack variables avoid a global coupling of probability densities on all graph nodes in the projection step. For the time discrete action functional Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence to the time continuous action is established. Numerical results for a selection of test cases show qualitative and quantitative properties of the optimal transport on graphs. Finally, we use our algorithm to implement a JKO scheme for the gradient flow of the entropy in discrete transportation distance, which is known to coincide with underlying Markov semigroup, and test our results against a classical backward Euler discretization of this discrete heat flow.