Applied Mathematics, University of Waterloo
Numerical Solutions of Hamilton-Jacobi-Bellman Equations with Applications
This research proposal focuses on Hamilton-Jacobi-Bellman (HJB) equa- tions, which are nonlinear controlled partial differential equations (PDEs). We are interested in constructing finite difference schemes that converge to the viscosity solutions of the HJB equations, and developing solvers, and furthermore, fast solvers, for the discretized equations.
We discuss two specific applications of the HJB equations. One is to solve a Monge-Amp`ere equation by converting it to an equivalent HJB equation. Wide stencil scheme is applied to discretize the HJB equation. We prove that the numerical scheme is consistent, stable and monotone, and thus con- verges to the viscosity solution. We apply this numerical scheme to image registration problem.
Another application of the HJB equations is the oligopolistic mean field game model in economics. The optimal lifetime profits of the companies in a mean field game can be determined by a system of PDEs that contains an HJB equation. Multigrid method is employed as the fast solver for the discretized equations.
In the end of the research proposal, we summarize our progress and pro- pose some future research topics.