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A weighted empirical interpolation method: a prioriconvergence 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 efficientreduced-basis discretization of partial differential equations. Compt. Rend. Math.Anal. Num. 339 (2004) 667–672] to a weighted empiricalinterpolation method in order to approximate nonlinear parametric functions with weightedparameters, e.g. random variables obeying various probabilitydistributions. A priori convergence analysis is provided for the proposedmethod 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 interpolationprocedure: the magic points. Commun. Pure Appl. Anal. 8(2009) 383–404]. We apply our method to geometric Brownian motion, exponentialKarhunen–Loève expansion and reduced basis approximation of non-affine stochastic ellipticequations. We demonstrate its improved accuracy and efficiency over the empiricalinterpolation 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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