Efficient iterative solvers for a complex valued two-by-two block linear system with application to parabolic optimal control problems

Abstract

In this paper, we are concerned with the efficient iterative solution of time-harmonic parabolic optimal control problems. A robust parameterized preconditioner is proposed for the arising complex valued two-by-two block linear system related to the first-order optimality conditions. Practical parameter choice strategies are considered for the new preconditioner to improve the performance of the original preconditioner within Krylov subspace acceleration. Moreover, a nonstationary second-order iteration method is devised from the parameterized preconditioner within Chebyshev acceleration. Based on a detailed spectral analysis of the preconditioned matrix, convergence rates are analyzed for both the established Krylov subspace and Chebyshev acceleration methods. Due to the tight and problem independent eigenvalue distributions of the preconditioned matrix, the implementation of the Chebyshev acceleration method is parameter free and the obtained iteration error bounds of both methods result in almost parameter independent convergence rates. Numerical experiments are presented to confirm the robustness and effectiveness of the parameterized preconditioner for both Krylov subspace and Chebyshev accelerations and improvement compared to earlier results.