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