A weighted empirical interpolation method: a priori convergence analysis and applications
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 4, pp. 943-953.

We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.

DOI : 10.1051/m2an/2013128
Classification : 65C20, 65D05, 97N50
Mots clés : empirical interpolation method, a priori convergence analysis, greedy algorithm, Kolmogorov N-width, geometric brownian motion, Karhunen-Loève expansion, reduced basis method
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     title = {A weighted empirical interpolation method: \protect\emph{a priori }convergence analysis and applications},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {943--953},
     publisher = {EDP-Sciences},
     volume = {48},
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     year = {2014},
     doi = {10.1051/m2an/2013128},
     zbl = {1304.65097},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2013128/}
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Chen, Peng; Quarteroni, Alfio; Rozza, Gianluigi. A weighted empirical interpolation method: a priori convergence analysis and applications. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 4, pp. 943-953. doi : 10.1051/m2an/2013128. http://www.numdam.org/articles/10.1051/m2an/2013128/

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