Comptes Rendus
Partial Differential Equations/Optimal Control
Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems
Comptes Rendus. Mathématique, Volume 349 (2011) no. 15-16, pp. 873-877.

We employ the reduced basis method as a surrogate model for the solution of optimal control problems governed by parametrized partial differential equations (PDEs) and develop rigorous a posteriori error bounds for the error in the optimal control and the associated error in the cost functional. The proposed bounds can be efficiently evaluated in an offline–online computational procedure. We present numerical results that confirm the validity of our approach.

Nous employons la méthode des bases réduites comme modèle de substitution pour la solution de problèmes de contrôle optimal, qui sont régis par des équations aux dérivées partielles paramétrisée, et nous développons ainsi des bornes dʼerreur rigoureuses a posteriori pour lʼerreur dans le contrôle optimal et lʼerreur associée dans la fonctionnelle du coût. Les bornes proposées peuvent être efficacement évaluées avec une procédure de calcul en-ligne/hors-ligne. Nous présentons des résultats numériques qui confirment le bien-fondé de notre méthode.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2011.07.010

Martin A. Grepl 1; Mark Kärcher 2

1 Institut für Geometrie und Praktische Mathematik (IGPM), RWTH Aachen University, 52056 Aachen, Germany
2 Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, 52056 Aachen, Germany
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     title = {Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems},
     journal = {Comptes Rendus. Math\'ematique},
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Martin A. Grepl; Mark Kärcher. Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. Comptes Rendus. Mathématique, Volume 349 (2011) no. 15-16, pp. 873-877. doi : 10.1016/j.crma.2011.07.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.07.010/

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[3] M.A. Grepl, M. Kärcher, A certified reduced basis approach for linear-quadratic optimal control problems, in preparation.

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