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Discrete least squares polynomial approximation with random evaluations - Application to parametric and stochastic elliptic PDEs

By Moulay Abdellah Chkifa, Albert Cohen, Giovanni Migliorati, Fabio Nobile and Raul Tempone

Abstract

International audienceMotivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares methodfor polynomial approximation of multivariate functionsbased 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 \cite{CDL} and in probability in \cite{MNST2011}, under suitable conditions that relate the number of sampleswith 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 approximationin the given polynomial space.In this paper, we discuss the quasi-optimality of thepolynomial 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 betweenthe 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 themto 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

Topics: [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA]
Publisher: EDP sciences SMAI
Year: 2015
DOI identifier: 10.1051/m2an
OAI identifier: oai:HAL:hal-01352276v1
Provided by: Hal-Diderot

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