Kernel-based Methods for Stochastic Partial Differential Equations
For researchers in scientific computing, this provides a new framework for solving stochastic PDEs with kernel methods, though it is an incremental extension of existing techniques.
This work extends kernel-based methods from deterministic to stochastic data to solve high-dimensional stochastic partial differential equations. Numerical examples on stochastic Poisson and heat equations demonstrate well-posed approximate probability distributions using compactly supported and Sobolev-spline kernels.
This article gives a new insight of kernel-based (approximation) methods to solve the high-dimensional stochastic partial differential equations. We will combine the techniques of meshfree approximation and kriging interpolation to extend the kernel-based methods for the deterministic data to the stochastic data. The main idea is to endow the Sobolev spaces with the probability measures induced by the positive definite kernels such that the Gaussian random variables can be well-defined on the Sobolev spaces. The constructions of these Gaussian random variables provide the kernel-based approximate solutions of the stochastic models. In the numerical examples of the stochastic Poisson and heat equations, we show that the approximate probability distributions are well-posed for various kinds of kernels such as the compactly supported kernels (Wendland functions) and the Sobolev-spline kernels (Matérn functions).