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.
Mots-clés : empirical interpolation method, a priori convergence analysis, greedy algorithm, Kolmogorov N-width, geometric brownian motion, Karhunen-Loève expansion, reduced basis method
@article{M2AN_2014__48_4_943_0,
author = {Chen, Peng and Quarteroni, Alfio and Rozza, Gianluigi},
title = {A weighted empirical interpolation method: \protect\emph{a priori }convergence analysis and applications},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {943--953},
publisher = {EDP-Sciences},
volume = {48},
number = {4},
year = {2014},
doi = {10.1051/m2an/2013128},
zbl = {1304.65097},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013128/}
}
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 PY - 2014 SP - 943 EP - 953 VL - 48 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013128/ DO - 10.1051/m2an/2013128 LA - en ID - M2AN_2014__48_4_943_0 ER -
%0 Journal Article %A Chen, Peng %A Quarteroni, Alfio %A Rozza, Gianluigi %T A weighted empirical interpolation method: a priori convergence analysis and applications %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 943-953 %V 48 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013128/ %R 10.1051/m2an/2013128 %G en %F M2AN_2014__48_4_943_0
Chen, Peng; Quarteroni, Alfio; Rozza, Gianluigi. A weighted empirical interpolation method: a priori convergence analysis and applications. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 4, pp. 943-953. doi : 10.1051/m2an/2013128. http://archive.numdam.org/articles/10.1051/m2an/2013128/
[1] , , and , An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Anal. Numér. 339 (2004) 667-672. | MR | Zbl
[2] and , An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10 (2001) 1-102. | MR | Zbl
[3] , , , , and , Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457-1472. | MR | Zbl
[4] and , Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32 (2010) 2737-2764. | MR | Zbl
[5] and , Accurate and efficient evaluation of failure probability for partial differential equations with random input data. Comput. Methods Appl. Mech. Eng. 267 (2013) 233-260. | MR | Zbl
[6] , and , Comparison between reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput. 59 (2014) 187-216. | MR | Zbl
[7] , and , A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 51 (2013) 3163-3185. | MR | Zbl
[8] and , Constructive Approximation. Springer (1993). | MR | Zbl
[9] and , Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica 11 (2002) 145-236. | MR | Zbl
[10] , , and , Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41 (2007) 575-605. | Numdam | MR | Zbl
[11] , and , On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition. ESAIM: M2AN 46 (2012) 1555-1576. | Numdam | MR | Zbl
[12] and , Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Eng. 199 (2010) 1583-1592. | MR | Zbl
[13] , , and , A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404. | MR | Zbl
[14] , and , Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomedical Eng. 28 (2012) 604-625. | MR
[15] , and , A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2309-2345. | MR | Zbl
[16] , Stochastic Differential Equations: An Introduction with Applications. Springer (2010). | MR | Zbl
[17] , N-widths in Approximation Theory. Springer (1985). | MR | Zbl
[18] , and , Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Industry 1 (2011) 1-49. | MR | Zbl
[19] , and , Numerical Mathematics. Springer (2007). | MR | Zbl
[20] , Reduced basis methods for Stokes equations in domains with non-affine parameter dependence. Comput. Vis. Sci. 12 (2009) 23-35. | MR
[21] , and , Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives Comput. Meth. Eng. 15 (2008) 229-275. | MR
[22] and , Affine decompositions of parametric stochastic processes for application within reduced basis methods. In Proc. MATHMOD, 7th Vienna International Conference on Mathematical Modelling (2012).
Cité par Sources :
