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A Posteriori Error Estimation for ReducedOrder Solutions of Parametrized Parabolic Optimal Control Problems∗

Published online by Cambridge University Press:  09 September 2014

Mark Kärcher and
Martin A. Grepl
Mark Kärcher
Affiliation:
Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany.. kaercher@aices.rwth-aachen.de
Martin A. Grepl
Affiliation:
Numerical Mathematics, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany. ; grepl@igpm.rwth-aachen.de

Abstract

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

Keywords

Optimal controlreduced basis methoda posteriori error estimationmodel order reductionparameter-dependent systemspartial differential equationsparabolic problems

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 6 , November 2014, pp. 1615 - 1638
Copyright
© EDP Sciences, SMAI 2014

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