One-shot solution of a time-dependent time-periodic PDE-constrained optimization problem
Martin Stoll · IMA Journal of Numerical Analysis · 2013
In this paper we describe the efficient solution of a partial differential equation (PDE)-constrained optimization problem subject to the time-periodic heat equation. We propose a space-time formulation for which we develop a monolithic solver. We present preconditioners well suited to approximating the Schur complement of the saddle point system associated with the first-order conditions. This means that, in addition to a Richardson iteration-based preconditioner, we also introduce a preconditioner based on the tensor product structure of the PDE discretization, which allows the use of an FFT-based preconditioner. We also consider additional bound constraints that can be treated using a semismooth Newton method. Moreover, we introduce robust preconditioners with respect to the regularization parameter. Numerical results will illustrate the competitiveness and flexibility of our approach.