Damped Semismooth Newton Methods For Solving HJB Obstacle Problems

Discrete HJB(Hamilton-Jacobi-Bellman) obstacle problems have many important applications in finance, optimal control theory and so on. These problems arise from the discritization of a class of HJBI(Hamilton-Jacobi-Bellman-Isaacs)equations. Because of the high nonlinearity of the problems, the iterative methods proposed for smooth equations cannot be used to solve them directly. In this thesis, a damped semi-smooth Newton method is presented to solve discrete HJB obstacle problems. Under appropriate conditions, we verify that the algorithm is monotonously convergent.The thesis is divided into three chapters. In the first chapter, we shall give the background, introduce some current research results and introduce some important concepts and interesting conclusions; In the second chapter, we propose a damped semi-smooth Newton method to solve a class of discrete HJB obstacle problems of the form,Under proper conditions, we verify that the monotone convergence and finite termination properties of the proposed method. The numerical results show its efficiency. In the last chapter, we propose a damped semi-smooth Newton method to solve the HJB obstacle problem,and we study the convergence of the method. Preliminary numerical results show its efficiency.