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A weighted empirical interpolation method: a priori convergence analysis and applications

Published online by Cambridge University Press:  30 June 2014

Peng Chen ,
Alfio Quarteroni  and
Gianluigi Rozza
Peng Chen
Affiliation:
Modelling and Scientific Computing, CMCS, Mathematics Institute of Computational Science and Engineering, MATHICSE, Ecole Polytechnique Fédérale de Lausanne, EPFL, Station 8, 1015 Lausanne, Switzerland. peng.chen@epfl.ch; cpempire@gmail.com; alfio.quarteroni@epfl.ch
Alfio Quarteroni
Affiliation:
Modelling and Scientific Computing, CMCS, Mathematics Institute of Computational Science and Engineering, MATHICSE, Ecole Polytechnique Fédérale de Lausanne, EPFL, Station 8, 1015 Lausanne, Switzerland. peng.chen@epfl.ch; cpempire@gmail.com; alfio.quarteroni@epfl.ch Modellistica e Calcolo Scientifico, MOX, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy
Gianluigi Rozza
Affiliation:
SISSA MathLab, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy; gianluigi.rozza@sissa.it
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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.

Keywords

Empirical interpolation methoda priori convergence analysisgreedy algorithmKolmogorov N-widthgeometric Brownian motionKarhunen–Loève expansionreduced basis method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 943 - 953
Copyright
© EDP Sciences, SMAI, 2014

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References

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