Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)
Provides a rigorous convergence theory for sparse collocation in high-dimensional uncertainty quantification problems involving Gaussian random fields.
The paper proves algebraic convergence of sparse collocation for approximating Hilbert-space-valued functions of countably many Gaussian random variables, such as solutions to elliptic PDEs with lognormal coefficients, and demonstrates dimension-independent convergence rates in numerical experiments.
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.