A weighted empirical interpolation method: a priori convergence analysis and applications
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) no. 4, pp. 943-953.

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.

DOI : https://doi.org/10.1051/m2an/2013128
Classification : 65C20,  65D05,  97N50
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: a priori convergence analysis and applications},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {943--953},
publisher = {EDP-Sciences},
volume = {48},
number = {4},
year = {2014},
doi = {10.1051/m2an/2013128},
zbl = {1304.65097},
language = {en},
url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) no. 4, pp. 943-953. doi : 10.1051/m2an/2013128. http://www.numdam.org/item/M2AN_2014__48_4_943_0/

[1] 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. C. R. Math. Anal. Numér. 339 (2004) 667-672. | MR 2103208 | Zbl 1061.65118

[2] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10 (2001) 1-102. | MR 2009692 | Zbl 1105.65349

[3] P. Binev, A. Cohen, W. Dahmen, R. Devore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457-1472. | MR 2821591 | Zbl 1229.65193

[4] S. Chaturantabut and D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32 (2010) 2737-2764. | MR 2684735 | Zbl 1217.65169

[5] P. Chen and A. Quarteroni, 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 3127301 | Zbl 1286.65156

[6] P. Chen, A. Quarteroni and G. Rozza, Comparison between reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput. 59 (2014) 187-216. | MR 3167732 | Zbl 1301.65007

[7] P. Chen, A. Quarteroni and G. Rozza, A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 51 (2013) 3163-3185. | MR 3129759 | Zbl 1288.65007

[8] R.A. Devore and G.G. Lorentz, Constructive Approximation. Springer (1993). | MR 1261635 | Zbl 0797.41016

[9] M.B. Giles and E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica 11 (2002) 145-236. | MR 2009374 | Zbl 1105.65350

[10] M.A. Grepl, Y. Maday, N.C. Nguyen and A.T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41 (2007) 575-605. | Numdam | MR 2355712 | Zbl 1142.65078

[11] T. Lassila, A. Manzoni and G. Rozza, 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 2996340 | Zbl 1276.65069

[12] T. Lassila and G. Rozza, Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Eng. 199 (2010) 1583-1592. | MR 2630164 | Zbl 1231.76245

[13] 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. | MR 2449115 | Zbl 1184.65020

[14] A. Manzoni, A. Quarteroni and G. Rozza, Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomedical Eng. 28 (2012) 604-625. | MR 2946552

[15] F. Nobile, R. Tempone and C.G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2309-2345. | MR 2421037 | Zbl 1176.65137

[16] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications. Springer (2010). | MR 1619188 | Zbl 0567.60055

[17] A. Pinkus, N-widths in Approximation Theory. Springer (1985). | MR 774404 | Zbl 0551.41001

[18] A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Industry 1 (2011) 1-49. | MR 2824231 | Zbl 1273.65148

[19] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics. Springer (2007). | MR 2265914 | Zbl 1136.65001

[20] G. Rozza, Reduced basis methods for Stokes equations in domains with non-affine parameter dependence. Comput. Vis. Sci. 12 (2009) 23-35. | MR 2489209

[21] G. Rozza, D.B.P. Huynh and A.T. Patera, 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 2430350

[22] K. Urban and B. Wieland, Affine decompositions of parametric stochastic processes for application within reduced basis methods. In Proc. MATHMOD, 7th Vienna International Conference on Mathematical Modelling (2012).