Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method

TL;DR

This paper develops a theoretical framework for a kernel-based collocation method to numerically solve high-dimensional stochastic PDEs, transforming them into elliptic equations and approximating solutions with reproducing kernels.

Contribution

It introduces a meshfree collocation approach for high-dimensional SPDEs, providing a rigorous justification and demonstrating feasibility through numerical experiments.

Findings

Feasibility of the kernel-based collocation method for high-dimensional SPDEs

Transformation of stochastic parabolic to elliptic equations via implicit time stepping

Numerical experiments confirm the effectiveness of the proposed approach

Abstract

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the SPDE centered at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method.