Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case, the least-squares method is quasi-optimal in expectation in [A. Cohen, M A. Davenport and D. Leviatan. Found. Comput. Math. 13 (2013) 819–834] and in probability in [G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, Found. Comput. Math. 14 (2014) 419–456], under suitable conditions that relate the number of samples with respect to the dimension of the polynomial space. Here “quasi-optimal” means that the accuracy of the least-squares approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the quasi-optimality of the polynomial least-squares method in arbitrary dimension. Our analysis applies to any arbitrary multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space, independently of the anisotropic shape and of the number of variables. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss “inclusion type” elliptic PDE models, and derive an exponential convergence estimate for the least-squares method. Numerical results confirm our estimate, yet pointing out a gap between the condition necessary to achieve optimality in the theory, and the condition that in practice yields the optimal convergence rate.
DOI : 10.1051/m2an/2014050
Mots-clés : Approximation theory, polynomial approximation, least squares, parametric and stochastic PDEs, high-dimensional approximation
@article{M2AN_2015__49_3_815_0,
author = {Chkifa, Abdellah and Cohen, Albert and Migliorati, Giovanni and Nobile, Fabio and Tempone, Raul},
title = {Discrete least squares polynomial approximation with random evaluations \ensuremath{-} application to parametric and stochastic elliptic {PDEs}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {815--837},
publisher = {EDP-Sciences},
volume = {49},
number = {3},
year = {2015},
doi = {10.1051/m2an/2014050},
zbl = {1318.41004},
mrnumber = {3342229},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2014050/}
}
TY - JOUR AU - Chkifa, Abdellah AU - Cohen, Albert AU - Migliorati, Giovanni AU - Nobile, Fabio AU - Tempone, Raul TI - Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 815 EP - 837 VL - 49 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2014050/ DO - 10.1051/m2an/2014050 LA - en ID - M2AN_2015__49_3_815_0 ER -
%0 Journal Article %A Chkifa, Abdellah %A Cohen, Albert %A Migliorati, Giovanni %A Nobile, Fabio %A Tempone, Raul %T Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 815-837 %V 49 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2014050/ %R 10.1051/m2an/2014050 %G en %F M2AN_2015__49_3_815_0
Chkifa, Abdellah; Cohen, Albert; Migliorati, Giovanni; Nobile, Fabio; Tempone, Raul. Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 815-837. doi : 10.1051/m2an/2014050. http://www.numdam.org/articles/10.1051/m2an/2014050/
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