Comptes Rendus
Numerical Analysis
A posteriori error bounds for the empirical interpolation method
Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 575-579.

We present rigorous a posteriori error bounds for the Empirical Interpolation Method (EIM). The essential ingredients are (i) analytical upper bounds for the parametric derivatives of the function to be approximated, (ii) the EIM “Lebesgue constant,” and (iii) information concerning the EIM approximation error at a finite set of points in parameter space. The bound is computed “off-line” and is valid over the entire parameter domain; it is thus readily employed in (say) the “on-line” reduced basis context. We present numerical results that confirm the validity of our approach.

On introduit des bornes d'erreur a posteriori rigoureuses pour la méthode d'interpolation empirique, EIM en abrégé (pour Empirical Interpolation Method). Les ingrédients essentiels sont (i) des bornes analytiques des dérivées par rapport au paramètre de la fonction à interpoler, (ii) une « constante de Lebesgue » de EIM, et (iii) de l'information sur l'erreur d'approximation commise par EIM en un nombre fini de points dans l'espace des paramètres. La borne, une fois pré-calculée « hors-ligne », est valable sur tout l'espace des paramètres ; elle peut donc être utilisée directement telle quelle dans les applications (étape « en ligne » des calculs dans le contexte de la méthode des bases réduites). On montre des résultats numériques qui confirment la validité de notre approche.

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DOI: 10.1016/j.crma.2010.03.004
Jens L. Eftang 1; Martin A. Grepl 2; Anthony T. Patera 3

1 Norwegian University of Science and Technology, Department of Mathematical Sciences, NO-7491 Trondheim, Norway
2 RWTH Aachen University, Numerical Mathematics, Templergraben 55, 52056 Aachen, Germany
3 Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-264, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
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Jens L. Eftang; Martin A. Grepl; Anthony T. Patera. A posteriori error bounds for the empirical interpolation method. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 575-579. doi : 10.1016/j.crma.2010.03.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.004/

[1] M. Barrault; Y. Maday; N.C. Nguyen; A.T. Patera An ‘empirical interpolation’ method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 667-672

[2] J.L. Eftang, M.A. Grepl, A.T. Patera, A posteriori error estimation for the empirical interpolation method, 2010, in preparation

[3] J.L. Eftang, A.T. Patera, E.M. Rønquist, An “hp” certified reduced basis method for parametrized elliptic partial differential equations, SIAM J. Sci. Comput. (2009), submitted for publication

[4] M. Grepl; Y. Maday; N.C. Nguyen; A.T. Patera Efficient reduced basis treatment of nonaffine and nonlinear partial differential equations, M2AN, Volume 41 (2007), pp. 575-605

[5] J. Hesthaven, Y. Maday, B. Stamm, Reduced basis method for the parametrized electrical field integral equation, in preparation

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

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