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

Peng Chen; Alfio Quarteroni; Gianluigi Rozza

- Volume: 48, Issue: 4, page 943-953
- ISSN: 0764-583X

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topChen, Peng, Quarteroni, Alfio, and Rozza, Gianluigi. "A weighted empirical interpolation method: a priori convergence analysis and applications." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.4 (2014): 943-953. <http://eudml.org/doc/273279>.

@article{Chen2014,

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.},

author = {Chen, Peng, Quarteroni, Alfio, Rozza, Gianluigi},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {empirical interpolation method; a priori convergence analysis; greedy algorithm; Kolmogorov N-width; geometric brownian motion; Karhunen–Loève expansion; reduced basis method; empirical interpolation methods; geometric Brownian motion; error bounds; Kolmogorov -width; Karhunen-Loève expansion; weighted interpolation; random variables; algorithm},

language = {eng},

number = {4},

pages = {943-953},

publisher = {EDP-Sciences},

title = {A weighted empirical interpolation method: a priori convergence analysis and applications},

url = {http://eudml.org/doc/273279},

volume = {48},

year = {2014},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 4

SP - 943

EP - 953

AB - 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.

LA - eng

KW - empirical interpolation method; a priori convergence analysis; greedy algorithm; Kolmogorov N-width; geometric brownian motion; Karhunen–Loève expansion; reduced basis method; empirical interpolation methods; geometric Brownian motion; error bounds; Kolmogorov -width; Karhunen-Loève expansion; weighted interpolation; random variables; algorithm

UR - http://eudml.org/doc/273279

ER -

## References

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