Title: Approximation of nonlinear stochastic partial differential equations by a kernel-based collocation method

Authors: Qi Ye

Addresses: The Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA

Abstract: In this paper, we present a new idea to approximate high-dimensional nonlinear stochastic partial differential equations (SPDEs) by a kernel-based collocation method, which is a meshfree approximation method. A reproducing kernel is used to construct an approximate basis. The kernel-based collocation solution is a linear combination of the reproducing kernel with the differential and boundary operators of SPDEs at the given collocation points placed in the related high-dimensional domains. Its random expansion coefficients are computed by a random optimisation problem with constraint conditions induced by the nonlinear SPDEs. For a fixed kernel function, the convergence of kernel-based collocation solutions only depends on the fill distance of the chosen collocation points for the bounded domain of SPDEs. The numerical experiments of Sobolev-spline kernels for Klein-Gordon SPDEs show that the kernel-based collocation method produces the well-behaved approximate probability distributions of the SPDE solutions.

Keywords: nonlinear stochastic PDEs; partial differential equations; SPDEs; kernel-based collocation; mesh free approximation; reproducing kernel; Gaussian field; Matern function; Sobolev-spline kernels.

DOI: 10.1504/IJANS.2014.061018

International Journal of Applied Nonlinear Science, 2014 Vol.1 No.2, pp.156 - 172

Received: 06 Nov 2012
Accepted: 20 Mar 2013

Published online: 12 Jul 2014 *

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