Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs

Handle URI:
http://hdl.handle.net/10754/550822
Title:
Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs
Authors:
Chkifa, Abdellah; Cohen, Albert; Migliorati, Giovanni; Nobile, Fabio; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
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.
KAUST Department:
Applied Mathematics and Computational Science Program; Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
Citation:
Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs 2015, 49 (3):815 ESAIM: Mathematical Modelling and Numerical Analysis
Publisher:
EDP Sciences
Journal:
ESAIM: Mathematical Modelling and Numerical Analysis
Issue Date:
8-Apr-2015
DOI:
10.1051/m2an/2014050
Type:
Article
ISSN:
0764-583X; 1290-3841
Additional Links:
http://www.esaim-m2an.org/10.1051/m2an/2014050
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program

Full metadata record

DC FieldValue Language
dc.contributor.authorChkifa, Abdellahen
dc.contributor.authorCohen, Alberten
dc.contributor.authorMigliorati, Giovannien
dc.contributor.authorNobile, Fabioen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-04-28T12:20:48Zen
dc.date.available2015-04-28T12:20:48Zen
dc.date.issued2015-04-08en
dc.identifier.citationDiscrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs 2015, 49 (3):815 ESAIM: Mathematical Modelling and Numerical Analysisen
dc.identifier.issn0764-583Xen
dc.identifier.issn1290-3841en
dc.identifier.doi10.1051/m2an/2014050en
dc.identifier.urihttp://hdl.handle.net/10754/550822en
dc.description.abstractMotivated 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.en
dc.publisherEDP Sciencesen
dc.relation.urlhttp://www.esaim-m2an.org/10.1051/m2an/2014050en
dc.rightsArchived with thanks to ESAIM: Mathematical Modelling and Numerical Analysis. © EDP Sciences, SMAI 2015en
dc.subjectApproximation theoryen
dc.subjectpolynomial approximationen
dc.subjectleast squaresen
dc.subjectparametric and stochastic PDEsen
dc.subjecthigh-dimensional approximationen
dc.titleDiscrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEsen
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)en
dc.identifier.journalESAIM: Mathematical Modelling and Numerical Analysisen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionInstitut Universitaire de France, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, Franceen
dc.contributor.institutionMATHICSE-CSQI, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerlanden
kaust.authorTempone, Raulen
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