Surface finite element approximation of spherical Whittle--Mat\'ern Gaussian random fields

TL;DR

This paper develops a surface finite element method to approximate spherical Whittle--Matérn Gaussian random fields, providing error analysis and computational complexity bounds, with numerical validation of the theoretical results.

Contribution

It introduces a novel finite element approach for fractional elliptic SPDEs on the sphere, combining recursive and quadrature schemes, with rigorous error and complexity analysis.

Findings

Numerical results confirm theoretical error bounds

Method effectively approximates fractional SPDE solutions on the sphere

Computational complexity is bounded in terms of desired accuracy

Abstract

Spherical Whittle--Mat\'ern Gaussian random fields are considered as solutions to fractional elliptic stochastic partial differential equations on the sphere. Approximation is done with surface finite elements. While the non-fractional part of the operator is solved by a recursive scheme, a quadrature of the Dunford--Taylor integral representation is employed for the fractional part. Strong error analysis is performed, and the computational complexity is bounded in terms of the accuracy. Numerical experiments for different choices of parameters confirm the theoretical findings.