Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables - with Application to Lognormal Elliptic Diffusion Problems

Oliver G. Ernst, Björn Sprungk, Lorenzo Tamellini

Published 2016-11-22Version 1

We give a convergence proof of sparse collocation to approximate Hilbert space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions or quantities of interest associated with elliptic PDEs with lognormal diffusion coefficients. We outline a general $L^2$-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.