Volume 48, Number 4, July-August 2014
|Page(s)||943 - 953|
|Published online||30 June 2014|
A weighted empirical interpolation method: a priori convergence analysis and applications
Modelling and Scientific Computing, CMCS, Mathematics Institute of
Computational Science and Engineering, MATHICSE, Ecole Polytechnique Fédérale de
Lausanne, EPFL, Station
firstname.lastname@example.org; email@example.com; firstname.lastname@example.org
2 Modellistica e Calcolo Scientifico, MOX, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy
3 SISSA MathLab, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy
Revised: 16 July 2013
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.
Mathematics Subject Classification: 65C20 / 65D05 / 97N50
Key words: Empirical interpolation method / a priori convergence analysis / greedy algorithm / Kolmogorov N-width / geometric Brownian motion / Karhunen–Loève expansion / reduced basis method
© EDP Sciences, SMAI, 2014
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