Numerical analysis of lognormal diffusions on the sphere

TL;DR

This paper analyzes the regularity and convergence of numerical solutions for stationary diffusion equations on the sphere with lognormal coefficients, providing theoretical rates and numerical validation.

Contribution

It offers new regularity results for solutions and derives convergence rate estimates for multilevel Monte Carlo methods applied to these problems.

Findings

Regularity of solutions in Sobolev spaces is established.

Convergence rates depend on the decay of the angular power spectrum.

Numerical experiments confirm theoretical predictions.

Abstract

Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. H\"older regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the regularity of the driving lognormal coefficients. This yields regularity in $L^p$ sense of the solution to the diffusion problem in Sobolev spaces. Convergence rate estimates of multilevel Monte Carlo Finite and Spectral Element discretizations of these problems are then deduced. Specifically, a convergence analysis is provided with convergence rate estimates in terms of the number of Monte Carlo samples of the solution to the considered diffusion equation and in terms of the total number of degrees of freedom of the spatial discretization, and with bounds for the total work required by the algorithm in the case of Finite Element discretizations. The obtained convergence rates are solely in terms of the decay of the angular power spectrum of the (logarithm) of the diffusion coefficient. Numerical examples confirm the presented theory.