Comptes Rendus
no. 8
Numerical Analysis
A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants
[Une méthode d'optimisation linéaire de contraintes successives pour les bornes inférieures des constantes paramétriques de coercivité et de stabilité inf–sup]

Présenté par : Olivier Pironneau

D.B.P. Huynh1 ; G. Rozza2 ; S. Sen2 ; A.T. Patera2
1 Singapore-MIT Alliance, National University of Singapore, Singapore 117576, Singapore
2 Mechanical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 473-478.

Nous présentons une méthode pour le calcul de bornes inférieures pour les constantes de stabilité (de coercivité ou d'inf–sup) nécessaires pour les estimateurs d'erreur a posteriori, associées à l'approximation par base réduite d'équations aux dérivées partielles ayant une dépendance affine en les paramétres. La méthode—basée sur une stratégie hors-ligne/en-ligne intéressante pour le calcul temps réel et les cas d'évaluations multiples—réduit le calcul en-ligne à un problème d'optimisation linéaire peu coûteux. La fonction objectif est un développement paramétrique du quotient de Rayleigh. Les contraintes traduisent la stabilité pour un ensemble optimal de paramétres. Nous présentons des résultats numériques pour un problème d'élasticité (coercif) ainsi que pour un problème d'acoustique de type Helmholtz (non-coercif).

We present an approach to the construction of lower bounds for the coercivity and inf–sup stability constants required in a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. The method, based on an Offline–Online strategy relevant in the reduced basis many-query and real-time context, reduces the Online calculation to a small Linear Program: the objective is a parametric expansion of the underlying Rayleigh quotient; the constraints reflect stability information at optimally selected parameter points. Numerical results are presented for coercive elasticity and non-coercive acoustics Helmholtz problems.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2007.09.019
D.B.P. Huynh 1 ; G. Rozza 2 ; S. Sen 2 ; A.T. Patera 2

1 Singapore-MIT Alliance, National University of Singapore, Singapore 117576, Singapore
2 Mechanical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA 
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D.B.P. Huynh; G. Rozza; S. Sen; A.T. Patera. A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants. Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 473-478. doi : 10.1016/j.crma.2007.09.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.09.019/

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[3] D.B.P. Huynh, G. Rozza, S. Sen, A.T. Patera, Analysis of a successive constraint method for efficient approximation of lower bounds of parametric coercivity and inf–sup stability constants, M3AS, in preparation

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