Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls

Abstract

In this paper, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable. We also propose two different error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline–online computational procedure, thus making our approach relevant in the many-query or real-time context. We compare our bounds with a previously proposed bound based on the Banach–Necas–Babuska theory and present numerical results for two model problems: a Graetz flow problem and a heat transfer problem. Finally, we also apply and test the performance of our newly proposed bound on a hyperthermia treatment planning problem.