Numerical analysis of lognormal diffusions on the sphere
Journal article, 2018

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

Karhunen-Loeve expansion

Spherical harmonic functions

Isotropic Gaussian random fields

Lognormal random fields

Stochastic partial differentia

Author

Lukas Herrmann

Swiss Federal Institute of Technology in Zürich (ETH)

Annika Lang

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

University of Gothenburg

Christoph Schwab

Swiss Federal Institute of Technology in Zürich (ETH)

Stochastics and Partial Differential Equations: Analysis and Computations

2194-0401 (ISSN) 2194-041X (eISSN)

Vol. 6 1 1-44

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

DOI

10.1007/s40072-017-0101-x

More information

Latest update

3/5/2019 1