The Domain Decomposition Methods Of Hamilton-Jacobi-Bellman Equations

The Hamilton-Jacobi-Bellman equations (here-in-after called HJB equations) havebeen widely used in engineering and economy, and the theory as well as the numericalsolutions for them have drawn much attention. This thesis will mainly discuss the domaindecomposition of the discrete problem of a kind of HJB equation.The thesis first introduces the iterative method proposed by Lions-Mercier. On thebasis of the iterative algorithm, the thesis generalizes the two-subdomain decompositionmethod into multi-subdomain decomposition method.This paper proposes another domain decomposition method of discrete HJB equa-tions. On the base of iterative algorithm by Cheng Xiaoliang, Xu Yuanji and MengBingquan, this thesis will combine the iterative algorithm with domain decompositionmethod. It first divides definition domain into several subdomains and then uses thisiterative algorithm to solve discrete HJB equations in each subdomain.This paper also presents the third domain decomposition method. Basing on themonotone iterative algorithm of Jacobi type given by Zhou Shuzi and Chen Guanghua.This thesis will use this iterative algorithm to solve subproblem. The advantage of theiterative algorithm lies that it does not need to solve any linearity equation system andinequality in the subproblem solving.The fourth domain decomposition method in this paper is based on alternating direc-tion algorithms. On the basis of the alternating direction algorithms that Sun proposedto solve HJB equations, this thesis first utilizes domain decomposition method to lowerthe degree of the discrete HJB equations and then use alternating direction Algorithmsto solve subproblems.Under proper conditions, we proves the monotone convergence of above algorithms.In the end, this thesis gives the numerical experiment results of the above algorithmsto indicate that above algorithms are e?ective.