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Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs). (English) Zbl 06864017

Summary: 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 \(L^2\)-convergence theory based on previous work by M. Bachmayr et al. [ESAIM, Math. Model. Numer. Anal. 51, No. 1, 341–363 (2017; Zbl 1366.41005)] and P. Chen [“Sparse quadrature for high-dimensional integration with Gaussian measure”, ESAIM, Math. Model. Numer. Anal. (to appear), doi:10.1051/m2an/2018012] 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 specifically verify for Gauss-Hermite nodes that this assumption holds and also show algebraic convergence with respect to the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate.

MSC:

65D05 Numerical interpolation
65D15 Algorithms for approximation of functions
65C30 Numerical solutions to stochastic differential and integral equations
60H25 Random operators and equations (aspects of stochastic analysis)

Citations:

Zbl 1366.41005

Software:

HRMSYM; PATSYM
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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