A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems^∗

¹ Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany.
kaercher@aices.rwth-aachen.de
² Numerical Mathematics, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany.
grepl@igpm.rwth-aachen.de

Received: 18 April 2013
Revised: 14 January 2014

Abstract

We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to confirm the validity of our approach.