A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations

摘要

A domain decomposition method (DDM) is presented to solve the distributed optimal control problem. The optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable. The proposed algorithm, called SA–GP algorithm, consists of two iterative stages. In the inner loops, the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of the computational cost is used to solve PDEs. In this paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of L2-error of control variable is derived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results.