Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs)

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TitleConvergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs)
Publication TypeIMATI_Reports
MonthApril
Year of Publication2017
AuthorsErnst, O.G., Sprungk B., and Tamellini L.
Series TitleIMATI Report Series
nr.17-10
Pagination30
CityPavia
InstitutionCNR-IMATI
Keywordsbest-N -term approximation, lognormal diffusion coefficient, parameteric PDEs, Random PDEs, Sparse grids, stochastic collocation
Abstract

We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients. We outline a general L2-convergence theory based on previous work by Bachmayr et al. (2016) and Chen (2016) and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We verify specifically for Gauss-Hermite nodes that this assumption holds and also show algebraic convergence w.r.t. the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate

URLhttp://bibliograzia.imati.cnr.it/reports/bibrep17-10
Citation Keybibrep17-10
Access DateApril 13, 2017
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