Block $\omega$-circulant preconditioners for parabolic optimal control problems

TL;DR

This paper introduces a new class of block ω-circulant preconditioners for Krylov subspace methods to efficiently solve linear systems from parabolic optimal control problems, improving convergence rates.

Contribution

The paper develops novel block ω-circulant preconditioners that can be diagonalized efficiently and are theoretically proven to cluster eigenvalues, enhancing solver performance for parabolic control problems.

Findings

Eigenvalues of preconditioned matrices cluster around ±1

Singular values cluster around 1 for GMR methods

Numerical results confirm improved convergence

Abstract

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 $\omega$-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 $\pm 1$, which leads to rapid convergence when the minimal residual method is used. When the generalized minimal residual method is deployed, the efficacy of the proposed preconditioners are justified in the way that the singular values of the preconditioned matrices are proven clustered around unity. Numerical results are provided to demonstrate the effectiveness of our proposed solvers.