A Kernel-Based Collocation Method for Elliptic Partial Differential Equations With Random Coefficients

Abstract

This paper is an extension of previous work where we laid the foundation for the kernel-based collocation solution of stochastic partial differential equations (SPDEs), but dealt only with the simpler problem of right-hand-side Gaussian noises. In the present paper we show that kernel-based collocation methods can be used to approximate the solutions of high-dimensional elliptic partial differential equations with potentially non-Gaussian random coefficients on the left-hand-side. The kernel-based method is a meshfree approximation method, which does not require an underlying computational mesh. The kernel-based solution is a linear combination of a reproducing kernel derived from the related random differential and boundary operators of SPDEs centered at collocation points to be chosen by the user. The random expansion coefficients are obtained by solving a system of random linear equations. For a given kernel function, we show that the convergence of our estimator depends only on the fill distance of the collocation points for the bounded domain of the SPDEs when the random coefficients in the differential operator are random variables. According to our numerical experiments, the kernel-based method produces well-behaved approximate probability distributions of the solutions of SPDEs.