Approximation of nonlinear stochastic partial differential equations by a kernel-based collocation method
type: Item
instance of: scholarly article
title: Approximation of nonlinear stochastic partial differential equations by a kernel-based collocation method
zbMATH Open document ID: 1330.65019
DOI: 10.1504/IJANS.2014.061018
author: Qi Ye
published in: International Journal of Applied Nonlinear Science
publication date: 2016-01-04T00:00:00Z
review text: Summary: 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.
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H35
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 65N12
zbMATH DE Number: 6526075
zbMATH Keywords: nonlinear stochastic partial differential equations
zbMATH Keywords: kernel-based collocation method
zbMATH Keywords: meshfree approximation method
zbMATH Keywords: reproducing kernel
zbMATH Keywords: Gaussian field
zbMATH Keywords: Matérn function
zbMATH Keywords: Sobolev-spline kernel
zbMATH Keywords: Klein-Gordon equation
zbMATH Keywords: convergence
zbMATH Keywords: numerical experiment
MaRDI profile type: Publication
full work available at URL: ijans.2014.061018
OpenAlex ID: W2023364597
description: scientific article; zbMATH DE number 6526075