A new iterative method for discrete HJB equations
The goal of this paper is to propose a successive relaxation iterative algorithm for discrete Hamilton-Jacobi-Bellman equation: \((1) \max_{1\leq j\leq K} \{A^JU-F^J\}=0\) where \(A^j \in \mathbb R^{n \times n}, F^j \in \mathbb R^n, j=1,2,\dots K\). Equation (1) is a system of nonsmooth nonlinear equations. A successive iterative scheme, similar to the Jacobi iteration for a linear system is presented. Main result: Numerical tests show that the authors' iterative scheme is faster than algorithm A (a simultaneous iterative scheme). It may be a new choice for solving the subproblems of domain decomposition methods for (1). It is also a possible nonlinear smoother if a new multigrid method for (1) is constructed. The new iterative scheme and the convergence theorem (the monotone convergence for the algorithm) are proposed. Finally some theoretical discussions are also presented.
discrete Hamilton-Jacobi-Bellman equation
scientific article; zbMATH DE number 5376043