Approximation of nonlinear stochastic partial differential equations by a kernel-based collocation method. (1st January 2014)

• Record Type: Journal Article
• Authors: Ye, Qi
• Is Part Of: International journal of applied nonlinear science. Volume 1:Number 2(2014)
• Page Start: 156
• Page End: 172
• Publication Date: 2014-01-01
• Subjects: nonlinear stochastic partial differential equation -- SPDE -- kernel-based collocation method -- meshfree approximation method -- reproducing kernel -- Gaussian field -- Matérn function -- Sobolev-spline kernel; Science -- Mathematical models -- Periodicals; Engineering -- Mathematical models -- Periodicals; Nonlinear theories -- Periodicals; 501.5118
• Languages: English
• ISSNs: 1752-2862

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.