Parameter-robust preconditioning for the optimal control of the wave equation

Abstract

In this paper, we propose and analyze a new matching-type Schur complement preconditioner for solving the discretized first-order necessary optimality conditions that characterize the optimal control of wave equations. Coupled with this is a recently developed second-order implicit finite difference scheme used for the full space-time discretization of the optimality system of PDEs. Eigenvalue bounds for the preconditioned system are derived, which provide insights into the convergence rates of the preconditioned Krylov subspace method applied. Numerical examples are presented to validate our theoretical analysis and demonstrate the effectiveness of the proposed preconditioner, in particular its robustness with respect to very small regularization parameters, and all mesh sizes in the spatial variables.