Friday, 05 September, 2014
SPEAKER: Prof. Max Jensen, Sussex
HOST: Xiaobing Feng
TITLE: Hamilton-Jacobi-Bellman equations and their numerical approximations
ABSTRACT: Hamilton-Jacobi-Bellman (HJB) equations is a class of second order fully nonlinear PDEs which describe how the cost of an optimal control problem changes as problem parameters vary. Applications can be found in engineering, finance and science as well as in geometric PDE theory. This talk will first introduce HJB equations in the context of stochastic optimal control using dynamic programming (Bellman Principle) and then briefly discuss PDE theory for HJB equations. The focus of the remaining talk will be on explaining how Galerkin methods can be adapted to solve these equations efficiently. In particular, it will be discussed how the convergence argument by Barles and Souganidis for finite difference schemes can be extended to Galerkin finite element methods to ensure convergence to viscosity solutions. A key question in this regard is the formulation of the consistency condition. Due to the Galerkin approach, coercivity properties of the HJB operator may also be satisfied by the numerical scheme. In this case one achieves besides uniform also strong H1 convergence of numerical solutions on unstructured meshes.