Numerical Approximation of Gaussian random fields on Closed Surfaces

TL;DR

This paper presents a novel numerical method for approximating Gaussian random fields on closed surfaces using a fractional SPDE, combining sinc quadrature with surface finite elements, and provides rigorous error analysis and numerical validation.

Contribution

It introduces a new eigenpair-free numerical approach for Gaussian fields on surfaces, leveraging the Balakrishnan integral and sinc quadrature.

Findings

Method achieves high accuracy in numerical experiments.

Error bounds are rigorously established.

Approach efficiently handles smoothness and correlation length parameters.

Abstract

We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances by several numerical experiments.