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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.
Keywords: empirical interpolation method, a priori convergence analysis, greedy algorithm, Kolmogorov N-width, geometric brownian motion, Karhunen-Loève expansion, reduced basis method
@article{M2AN_2014__48_4_943_0,
author = {Chen, Peng and Quarteroni, Alfio and Rozza, Gianluigi},
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},
number = {4},
year = {2014},
doi = {10.1051/m2an/2013128},
zbl = {1304.65097},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013128/}
}
TY - JOUR AU - Chen, Peng AU - Quarteroni, Alfio AU - Rozza, Gianluigi TI - A weighted empirical interpolation method: a priori convergence analysis and applications JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 943 EP - 953 VL - 48 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013128/ DO - 10.1051/m2an/2013128 LA - en ID - M2AN_2014__48_4_943_0 ER -
%0 Journal Article %A Chen, Peng %A Quarteroni, Alfio %A Rozza, Gianluigi %T A weighted empirical interpolation method: a priori convergence analysis and applications %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 943-953 %V 48 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013128/ %R 10.1051/m2an/2013128 %G en %F M2AN_2014__48_4_943_0
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
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