Block ω-Circulant Preconditioners for Parabolic Optimal Control Problems

In this work, we propose a class of novel preconditioned Krylov subspace methods for solving an optimal control problem of parabolic equations. Namely, we develop a family of block ω-circulant based preconditioners for the all-at-once linear system arising from the concerned optimal control problem, where both first order and second order time discretization methods are considered. The proposed preconditioners can be efficiently diagonalized by fast Fourier transforms in a parallel-in-time fashion, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix are clustered around ±1. This clustering leads to rapid convergence when the minimal residual method is used, particularly when the regularization parameter is sufficiently small. When the generalized minimal residual method is deployed, the efficacy of the proposed preconditioners is justified in the way that the singular values of the preconditioned matrices are proven clustered around unity. Numerical results are provided to demonstrate our proposed solvers, especially the effectiveness of the preconditioned generalized minimal residual approach.