A weighted empirical interpolation method: a priori convergence analysis and applications. (30th June 2014)

• Record Type: Journal Article
• Authors: Chen, Peng; Quarteroni, Alfio; Rozza, Gianluigi
• Is Part Of: Mathematical modelling and numerical analysis. Volume 48:Part 4(2014)
• Page Start: 943
• Page End: 953
• Publication Date: 2014-06-30
• Subjects: Empirical interpolation method, -- a priori convergence analysis, -- greedy algorithm, -- Kolmogorov N-width, -- geometric Brownian motion, -- Karhunen–Loève expansion, -- reduced basis method; Numerical analysis -- Periodicals; Mathematical models -- Periodicals; 510
• DOI: 10.1051/m2an/2013128
• Languages: English
• ISSNs: 0764-583X

Abstract

Abstract : 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.