On Approximation for Fractional Stochastic Partial Differential Equations on the Sphere

TL;DR

This paper derives an exact solution for a fractional stochastic PDE on the sphere using Karhunen-Loève expansion, analyzes approximation errors, and demonstrates results with numerical simulations related to cosmic microwave background.

Contribution

It provides a novel exact solution representation and convergence analysis for fractional SPDEs on the sphere, with applications to CMB simulations.

Findings

Convergence rates for truncation errors are established.

Numerical simulations validate theoretical error bounds.

Application to cosmic microwave background modeling.

Abstract

This paper gives the exact solution in terms of the Karhunen-Lo\`{e}ve expansion to a fractional stochastic partial differential equation on the unit sphere $\mathbb{S}^{2}\subset \mathbb{R}^{3}$ with fractional Brownian motion as driving noise and with random initial condition given by a fractional stochastic Cauchy problem. A numerical approximation to the solution is given by truncating the Karhunen-Lo\`{e}ve expansion. We show the convergence rates of the truncation errors in degree and the mean square approximation errors in time. Numerical examples using an isotropic Gaussian random field as initial condition and simulations of evolution of cosmic microwave background (CMB) are given to illustrate the theoretical results.