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

TL;DR

This paper proves convergence of sparse collocation methods for functions of infinitely many Gaussian variables, with applications to elliptic PDEs, demonstrating algebraic convergence rates and dimension-independent performance.

Contribution

It provides a rigorous convergence proof and algebraic rate analysis for sparse collocation applied to functions of countably many Gaussian variables, especially in PDE contexts.

Findings

Established $L^2$-convergence for sparse collocation of infinite-dimensional Gaussian functions.

Proved algebraic convergence rates under smoothness and growth assumptions.

Numerical experiments confirm dimension-independent convergence rates.

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 $L^2$-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. Numerical experiments illustrate the dimension-independent convergence rate.