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Preprint-No.: <   402   >   Published in: July 2014   PDF-File: IGPM402.pdf
Title:Certified Reduced Basis Methods for Parametrized Distributed Optimal Control Problems
Authors:Mark Kärcher, Martin Grepl, Karen Veroy
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 (BNB) theory and present numerical results for two model problems: a Graetz flow problem and a heat transfer problem.
Keywords:optimal control, reduced basis method, a posteriori error estimation, model order reduction, parameter-dependent systems, partial differential equations, elliptic problems