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We give a convergence proof for the approximation by sparse collocation ofHilbert-space-valued functions depending on countably many Gaussian randomvariables. Such functions appear as solutions of elliptic PDEs with lognormaldiffusion coefficients. We outline a general $L^2$-convergence theory based onprevious work by Bachmayr et al. (2016) and Chen (2016) and establish analgebraic convergence rate for sufficiently smooth functions assuming a mildgrowth bound for the univariate hierarchical surpluses of the interpolationscheme applied to Hermite polynomials. We verify specifically for Gauss-Hermitenodes that this assumption holds and also show algebraic convergence w.r.t. theresulting number of sparse grid points for this case. Numerical experimentsillustrate the dimension-independent convergence rate.

Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)