Multigrid Methods for Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Bellman-Isaacs Equations

multigrid methods;full approximation scheme;relaxation scheme;policy iteration;Hamilton-Jacobi-Bellman Equations;Hamilton-Jacobi-Bellman-Isaacs Equations;jump in control;Computer Science

Han, Dong

University of Waterloo

关键词: multigrid methods;    full approximation scheme;    relaxation scheme;    policy iteration;    Hamilton-Jacobi-Bellman Equations;    Hamilton-Jacobi-Bellman-Isaacs Equations;    jump in control;    Computer Science;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/6021/1/Han_Dong.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

We propose multigrid methods for solving Hamilton-Jacobi-Bellman (HJB) and Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations. The methods are based on the full approximation scheme. We propose a damped-relaxation method as smoother for multigrid. In contrast with policy iteration, the relaxation scheme is convergent for both HJB and HJBI equations. We show by local Fourier analysis that the damped-relaxation smoother effectively reduces high frequency error. For problems where the control has jumps, restriction and interpolation methods are devised to capture the jump on the coarse grid as well as during coarse grid correction. We will demonstrate the effectiveness of the proposed multigrid methods for solving HJB and HJBI equations arising from option pricing as well as problems where policy iteration does not converge or converges slowly.