An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations

Maxime Barrault; Yvon Maday; Ngoc Cuong Nguyen; Anthony T. Patera

doi:10.1016/j.crma.2004.08.006

Numerical Analysis

An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations

Presented by: Olivier Pironneau

Maxime Barrault ¹; Yvon Maday ²; Ngoc Cuong Nguyen ³; Anthony T. Patera ⁴

¹ CERMICS – ENPC, cité Descartes, Champs sur Marne, 77455 Marne la Vallée cedex 2, France
² Laboratoire J.-L. Lions, université Pierre et Marie Curie, B.C. 187, 75242 Paris cedex 05, France
³ National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576
⁴ Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-264, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA

Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 667-672.

Abstract
Résumé

We present an efficient reduced-basis discretization procedure for partial differential equations with nonaffine parameter dependence. The method replaces nonaffine coefficient functions with a collateral reduced-basis expansion which then permits an (effectively affine) offline–online computational decomposition. The essential components of the approach are (i) a good collateral reduced-basis approximation space, (ii) a stable and inexpensive interpolation procedure, and (iii) an effective a posteriori estimator to quantify the newly introduced errors. Theoretical and numerical results respectively anticipate and confirm the good behavior of the technique.

Nous présentons dans cette Note une méthode rapide de base réduite pour la résolution d'équations aux dérivées partielles ayant une dépendance non affine en ses paramètres. L'approche propose de remplacer le calcul des fonctionelles non affines par un développement en base réduite annexe qui conduit à une évaluation en ligne effectivement affine. Les points essentiels de cette approche sont (i) un bon système de base réduite annexe, (ii) une méthode stable et peu coûteuse d'interpolation dans cette base, et (iii) un estimateur a posteriori pertinent pour quantifier les nouvelles erreurs introduites. Des résultats théoriques et numériques viennent anticiper puis confirmer le bon comportement de cette technique.

Received: 2004-06-11
Accepted: 2004-08-28
Published online: 2004-10-14

DOI: 10.1016/j.crma.2004.08.006

Author's affiliations:

Maxime Barrault ¹; Yvon Maday ²; Ngoc Cuong Nguyen ³; Anthony T. Patera ⁴

¹ CERMICS – ENPC, cité Descartes, Champs sur Marne, 77455 Marne la Vallée cedex 2, France
² Laboratoire J.-L. Lions, université Pierre et Marie Curie, B.C. 187, 75242 Paris cedex 05, France
³ National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576
⁴ Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-264, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA

@article{CRMATH_2004__339_9_667_0,
     author = {Maxime Barrault and Yvon Maday and Ngoc Cuong Nguyen and Anthony T. Patera},
     title = {An {\textquoteleft}empirical interpolation{\textquoteright} method: application to efficient reduced-basis discretization of partial differential equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {667--672},
     publisher = {Elsevier},
     volume = {339},
     number = {9},
     year = {2004},
     doi = {10.1016/j.crma.2004.08.006},
     language = {en},
}

TY  - JOUR
AU  - Maxime Barrault
AU  - Yvon Maday
AU  - Ngoc Cuong Nguyen
AU  - Anthony T. Patera
TI  - An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 667
EP  - 672
VL  - 339
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2004.08.006
LA  - en
ID  - CRMATH_2004__339_9_667_0
ER  -

%0 Journal Article
%A Maxime Barrault
%A Yvon Maday
%A Ngoc Cuong Nguyen
%A Anthony T. Patera
%T An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations
%J Comptes Rendus. Mathématique
%D 2004
%P 667-672
%V 339
%N 9
%I Elsevier
%R 10.1016/j.crma.2004.08.006
%G en
%F CRMATH_2004__339_9_667_0

Maxime Barrault; Yvon Maday; Ngoc Cuong Nguyen; Anthony T. Patera. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 667-672. doi : 10.1016/j.crma.2004.08.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.006/

References
Cited by

[1] B.O. Almroth; P. Stern; F.A. Brogan Automatic choice of global shape functions in structural analysis, AIAA J., Volume 16 (1978), pp. 525-528

[2] J.P. Fink; W.C. Rheinboldt On the error behavior of the reduced basis technique for nonlinear finite element approximations, Z. Angew. Math. Mech., Volume 63 (1983), pp. 21-28

[3] L. Machiels; Y. Maday; I.B. Oliveira; A.T. Patera; D.V. Rovas Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, C. R. Acad. Sci. Paris, Ser. I, Volume 331 (2000) no. 2, pp. 153-158

[4] Y. Maday; A.T. Patera; G. Turinici Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002) no. 3, pp. 289-294

[5] A.K. Noor; J.M. Peters Reduced basis technique for nonlinear analysis of structures, AIAA J., Volume 18 (1980) no. 4, pp. 455-462

[6] C. Prud'homme; D. Rovas; K. Veroy; Y. Maday; A.T. Patera; G. Turinici Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods, J. Fluids Engrg., Volume 124 (2002) no. 1, pp. 70-80

[7] A. Quarteroni; R. Sacco; F. Saleri Numer. Math., Texts Appl. Math., vol. 37, Springer, New York, 1991

[8] K. Veroy; C. Prud'homme; D.V. Rovas; A.T. Patera A Posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, June, 2003 (AIAA Paper 2003-3847)

Cited by Sources:

Comments - Policy