PhD Comprehensive Seminar | Yangang Chen, Numerical solutions of Hamilton-Jacobi-Bellman equations with applications

Numerical solutions of Hamilton-Jacobi-Bellman equations with applications

Abstract

Hamilton-Jacobi-Bellman (HJB) equation is a nonlinear controlled partial differential equation (PDE). We are interested in constructing a finite difference scheme that converges to the viscosity solution of the HJB equation, and developing fast solvers for the discretized equations.

We show two specific applications of the HJB equations. One is to solve a Monge-Ampere equation, which can be applied in image registration, etc. We convert the Monge-Ampere equation to an equivalent HJB equation. Wide stencil scheme is used to discretize the HJB equation. We prove the convergence of the numerical scheme in the viscosity sense.

The other application is the oligopolistic competition in economy modelled by a mean field differential game. The payoffs of the players can be determined by a system of PDEs that contain an HJB equation. Multigrid method is employed as the fast solver of the discretized equations.