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A certified reduced basis method for parametrized elliptic optimal control problems

Published online by Cambridge University Press:  07 March 2014

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
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Abstract

In this paper, we employ the reduced basis method as a surrogate model for the solution of linear-quadratic optimal control problems governed by parametrized elliptic partial differential equations. We present a posteriori error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the assumption of affine parameter dependence, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. Numerical results are presented to confirm the validity of our approach.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2014

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