Wuchen Li

Computations of Optimal Transport Distance with Fisher Information Regularization

We propose a fast algorithm to approximate the optimal transport distance. The main idea is to add a Fisher information regularization into the dynamical setting of the problem, originated by Benamou and Brenier. The regularized problem is shown to be smooth and strictly convex, thus many classical fast algorithms are available. In this paper, we adopt Newton’s method, which converges to the minimizer with a quadratic rate. Several numerical examples are provided.

Entropy Dissipation Semi-Discretization Schemes for Fokker–Planck Equations

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.

Population Games and Discrete Optimal Transport

We propose an evolutionary dynamics for population games with discrete strategy sets, inspired by optimal transport theory and mean field games. The proposed dynamics is the Smith dynamics with strategy graph structure, in which payoffs are modified by logarithmic terms. The dynamics can be described as a Fokker–Planck equation on a discrete strategy set. For potential games, the dynamics is a gradient flow system under a Riemannian metric from optimal transport theory. The stability of the dynamics is studied through optimal transport metric tensor, free energy and Fisher information.

Method of evolving junctions: A new approach to optimal control with constraints ☆

Abstract We propose a new strategy, called method of evolving junctions (MEJ), to compute the solutions for a class of optimal control problems with constraints on both state and control variables. Our main idea is that by leveraging the geometric structures of the optimal solutions, we recast the infinite dimensional optimal control problem into an optimization problem depending on a finite number of points, called junctions. Then, using a modified gradient flow method, whose dimension can change dynamically, we find local solutions for the optimal control problem. We also employ intermittent diffusion, a global optimization method based on stochastic differential equations, to obtain the global optimal solution. We demonstrate, via a numerical example, that MEJ can effectively solve path planning problems in dynamical environments.

Method of evolving junctions: A new approach to optimal path-planning in 2D environments with moving obstacles

We propose a novel algorithm to find the global optimal path in 2D environments with moving obstacles, where the optimality is understood relative to a general convex continuous running cost. By leveraging the geometric structures of optimal solutions and using gradient flows, we convert the path-planning problem into a system of finite dimensional ordinary differential equations, whose dimensions change dynamically. Then a stochastic differential equation based optimization method, called intermittent diffusion, is employed to obtain the global optimal solution. We demonstrate, via numerical examples, that the new algorithm can solve the problem efficiently.