Qi Ye
Abstract: In this paper, we improve and complete the theoretical results of the kernel-based approximation (collocation) method for solving the high-dimensional stochastic partial differential equations (SPDEs) given in our previous papers. According to the extended theorems, we can use more general positive definite kernels to construct the kernel-based estimators to approximate the numerical solutions of the SPDEs. Because a parabolic SPDE driven by Levy noises can be discretized into serval elliptic SPDEs by the implicit Euler scheme at time. We mainly focus on how to solve a system of elliptic SPDEs driven by various kinds of right-hand-side random noises. The kernel-based approximate solution of the elliptic SPDEs is a linear combination of the positive definite kernel with the differential and boundary operators of the SPDEs centered at the chosen collocation points, and its random coefficients are obtained by solving a system of random linear equations, whose random parts are simulated by the elliptic SPDEs. Moreover, we introduce the error bounds -- confident intervals -- of the kernel-based approximate solutions of the elliptic (parabolic) SPDEs in terms of fill distances (or possible time distances) in the probability sense. We also give a well coding algorithm to compute the kernel-based solutions of the second-order parabolic SPDEs driven by time and space Poisson noises. The two-dimensional numerical experiments show that the approximate probability distributions of the kernel-based solutions are well-behave for the Sobolev-spline kernels and the compact support kernels.
ArXiv: 1303.5381