Entropic Approximation Of Wasserstein Gradient Flows

This article details a novel numerical scheme to approximate gradient flows for optimal transport (i.e., Wasserstein) metrics. These flows have proved useful to tackle theoretically and numerically nonlinear diffusion equations that model, for instance, porous media or crowd evolutions. These gradient flows define a suitable notion of weak solutions for these evolutions and they can be approximated in a stable way using discrete flows. These discrete flows are implicit Euler time stepping according to the Wasserstein metric. A bottleneck of these approaches is the high computational load induced by the resolution of each step. Indeed, this corresponds to the resolution of a convex optimization problem involving a Wasserstein distance to the previous iterate. Following several recent works on the approximation of Wasserstein distances, we consider a discrete flow induced by an entropic regularization of the transportation coupling. This entropic regularization allows one to trade the initial Wasserstein fidelity term for a Kullback-Leibler divergence, which is easier to deal with numerically. We show how Kullback-Leibler first order proximal schemes and, in particular, Dykstra's algorithm, can be used to compute each step of the regularized flow. The resulting algorithm is both fast, parallelizable, and versatile, because it only requires multiplications by the Gibbs kernel e(-c/gamma), where c is the ground cost and gamma > 0 the regularization strength. On Euclidean domains discretized on a uniform grid, this corresponds to a linear filtering (for instance, a Gaussian filtering when c is the squared Euclidean distance) which can be computed in nearly linear time. On more general domains, such as (possibly nonconvex) shapes or on manifolds discretized by a triangular mesh, following a recently proposed numerical scheme for optimal transport, this Gibbs kernel multiplication is approximated by a short-time heat diffusion. We show numerical illustrations of this method to approximate crowd motions with congestion on complicated domains as well as extensions to take into account interactions between multiple densities.