Entropy Dissipation Semi-Discretization Schemes for Fokker–Planck Equations

Abstract

We propose a new semi-discretization scheme to approximate nonlinear Fokker-Planck equations, by exploiting the gradient flow structures with respect to the 2-Wasserstein metric in the space of probability densities. We discretize the underlying state by a finite graph and define a discrete 2-Wasserstein metric in the discrete probability space. Based on such metric, we introduce a gradient flow of the discrete free energy as semi discretization scheme. We prove that the scheme maintains dissipativity of the free energy and converges to a discrete Gibbs measure at exponential dissipation rate. We exhibit these properties on several numerical examples.1991 Mathematics Subject Classification. 65L07, 65L12.

A discrete Schrödinger equation via optimal transport on graphs

“…Let (v(t), ρ(t)) be a critical point of(14), and S(t) a function on G that induces v(t), then S(t)) and ρ(t) satisfy (3) if and only if…”

Algorithm for Hamilton–Jacobi Equations in Density Space Via a Generalized Hopf Formula

“…We further discretize the above variational problem following the same discretization as in optimal transport on graphs [9,10,11,16,23]. We then apply a multi-level block stochastic gradient descent method to optimize the discretized problem.…”

Dynamical optimal transport on discrete surfaces

“…The (finite-dimensional) space of distributions in this case inherits a Riemannian metric with some structure preserved from the infinite-dimensional definition; for instance, the gradient flow of entropy corresponds to a notion of heat flow along the graph. A similar structure is proposed by Chow et al [2016], but they recover a different heat flow. Erbar et al [2017] propose a numerical algorithm for approximating the discrete Wasserstein distance introduced by Maas, but the distributions they produce have a tendency to diffuse along the graph.…”

“…To take full advantage of the triangulation, we want to use triangles and not only edges to define our objective functional. The latter choice leads to formulas similar to [Chow et al 2016;Maas 2011], which, as we say above, exhibit strongly diffuse geodesics. We prefer to represent vector fields on triangles.…”