Block Iterative Algorithms for the Solution of Parabolic Optimal Control Problems

Abstract

In this paper, we describe block matrix algorithms for the iterative solution of large scale linear-quadratic optimal control problems arising from the control of parabolic partial differential equations over a finite control horizon. After spatial discretization, by finite element or finite difference methods, the original problem reduces to an optimal control problem for n coupled ordinary differential equations, where n can be quite large. As a result, its solution by conventional control algorithms can be prohibitively expensive in terms of computational cost and memory requirements. We describe two iterative algorithms. The first algorithm employs a CG method to solve a symmetric positive definite reduced linear system for the unknown control variable. A preconditioner is described, which we prove has a rate of convergence independent of the space and time discretization parameters, however, double iteration is required. The second algorithm is designed to avoid double iteration by introducing an auxiliary variable. It yields a symmetric indefinite system, and for this system a positive definite block preconditioner is described. We prove that the resulting rate of convergence is independent of the space and time discretization parameters, when MINRES acceleration is used. Numerical results are presented for test problems.