[Bornes dʼerreur a posterori par bases réduites pour des problèmes linéaires-quadratiques elliptiques de contrôle optimal]
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.
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.
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@article{CRMATH_2011__349_15-16_873_0,
author = {Martin A. Grepl and Mark K\"archer},
title = {Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems},
journal = {Comptes Rendus. Math\'ematique},
pages = {873--877},
publisher = {Elsevier},
volume = {349},
number = {15-16},
year = {2011},
doi = {10.1016/j.crma.2011.07.010},
language = {en},
}
TY - JOUR AU - Martin A. Grepl AU - Mark Kärcher TI - Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems JO - Comptes Rendus. Mathématique PY - 2011 SP - 873 EP - 877 VL - 349 IS - 15-16 PB - Elsevier DO - 10.1016/j.crma.2011.07.010 LA - en ID - CRMATH_2011__349_15-16_873_0 ER -
%0 Journal Article %A Martin A. Grepl %A Mark Kärcher %T Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems %J Comptes Rendus. Mathématique %D 2011 %P 873-877 %V 349 %N 15-16 %I Elsevier %R 10.1016/j.crma.2011.07.010 %G en %F CRMATH_2011__349_15-16_873_0
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/
[1] Adaptive finite element methods for optimal control of partial differential equations: basis concepts, SIAM J. Control Optim., Volume 39 (2000) no. 1, pp. 113-132
[2] Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems, SIAM J. Sci. Comp., Volume 32 (2010) no. 2, pp. 997-1019
[3] M.A. Grepl, M. Kärcher, A certified reduced basis approach for linear-quadratic optimal control problems, in preparation.
[4] Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Meth. Eng., Volume 15 (2007) no. 3, pp. 229-275
[5] Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, American Mathematical Society, 2010
[6] POD a-posteriori error estimates for linear-quadratic optimal control problems, Comp. Opt. Appl., Volume 44 (2009) no. 1, pp. 83-115
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