Numerical analysis of lognormal diffusions on the sphere
Artikel i vetenskaplig tidskrift, 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.

Lognormal random fields

Spherical harmonic functions

Isotropic Gaussian random fields

Stochastic partial differentia

Karhunen-Loeve expansion

Författare

Lukas Herrmann

Eidgenössische Technische Hochschule Zürich (ETH)

Annika Lang

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Christoph Schwab

Eidgenössische Technische Hochschule Zürich (ETH)

Stochastics and Partial Differential Equations: Analysis and Computations

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

Vol. 6 1-44

Ämneskategorier

Beräkningsmatematik

Sannolikhetsteori och statistik

Matematisk analys

DOI

10.1007/s40072-017-0101-x