Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations

TL;DR

This paper studies the regularity, simulation, and stochastic PDEs of isotropic Gaussian random fields on the sphere, providing theoretical insights and efficient algorithms relevant for environmental science models.

Contribution

It establishes convergence rates for Karhunen-Loève expansions, analyzes sample regularity, and develops fast simulation methods using spherical harmonics and Fourier transforms.

Findings

Decay of angular power spectrum relates to smoothness and regularity.

Fast algorithms for sample generation on the sphere are proposed.

Spectral discretizations of the stochastic heat equation achieve strong convergence.

Abstract

Isotropic Gaussian random fields on the sphere are characterized by Karhunen-Lo\`{e}ve expansions with respect to the spherical harmonic functions and the angular power spectrum. The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample H\"{o}lder continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen-Lo\`{e}ve expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated. The relevance of the results on sample regularity for isotropic Gaussian random fields and the corresponding lognormal random fields on the sphere for several models from environmental sciences is indicated. Finally, the stochastic heat equation on the sphere driven by additive, isotropic Wiener noise is considered, and strong convergence rates for spectral discretizations based on the spherical harmonic functions are proven.