Iterative Algorithms For Solving Discrete HJB Obstacle Problems

Discrete HJB obstacle problems arise from finite difference or finite elementdiscretization of differential equations. These problems have many applications inoptimal control, hydrology, mechanics, finance and so on. Because of thenon-convexity and non-smoothness of the discrete problems, some existing iterativealgorithms for systems of nonlinear equations can not be used to solve them directly.It is of important practical significance and theoretical value to develop fast iterativealgorithms for solving these problems. By summarizing the recent work, we proposenonlinear Jacobi iterative method and nonlinear Gauss-Seidel iterative method tosolve discrete HJB obstacle problems and study the convergence of the proposedalgorithms. Additionally, nonlinear additive Schwarz method is proposed to solvediscrete HJB obstacle problems and the monotone convergence is verified.This paper is divided into four chapters. In the first chapter, we shall give thebackground and introduce some current research results. In the second chapter, weintroduce some important concepts and interesting conclusions. In the third chapter,we propose nonlinear Jacobi iterative method and nonlinear Gauss-Seidel iterativemethod to solve discrete HJB obstacle problems. Under proper conditions, we verifythat the monotone convergence of the proposed algorithms. Preliminary numericalresults show it. Finally, we present nonlinear additive Schwarz method to solvediscrete HJB obstacle problems, and under proper conditions, we verify that thesequence of iterates converges monotonically to the solution.