Numerical analysis of lognormal diffusions on the sphere
Preprint, 2016

Numerical solutions of stationary diffusion equations on the sphere with isotropic lognormal diffusion coefficients are considered. Hölder 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 on the sphere 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.

Finite Element Methods

spherical harmonic functions

Spectral Galerkin Methods

random partial differential equations

lognormal random fields

Karhunen–Loève expansion

Isotropic Gaussian random fields

stochastic partial differential equations

regularity of random fields


Lukas Herrmann

Annika Lang

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematical Statistics

Christoph Schwab

Subject Categories

Computational Mathematics

Probability Theory and Statistics


Basic sciences

More information