A multilevel sparse kernel-based stochastic collocation finite element method for elliptic problems with random coefficients

Abstract

A new stochastic collocation finite element method is proposed for the numerical solution of elliptic boundary value problems (BVP) with random coefficients, assuming that the randomness is well-approximated by a finite number of random variables with given probability distributions. The proposed method consists of a finite element approximation in physical space, along with a stochastic collocation quadrature approach utilizing the recent Multilevel Sparse Kernel-Based Interpolation (MuSIK) technique (Georgoulis et al., 2013). MuSIK is based on a multilevel sparse grid-type algorithm with the basis functions consisting of directionally anisotropic Gaussian radial basis functions (kernels) placed at directionally-uniform grid-points. We prove that MuSIK is interpolatory at these nodes, and, therefore, can be naturally used to define a quadrature scheme. Numerical examples are also presented, assessing the performance of the new algorithm in the context of high-dimensional stochastic collocation finite element methods.