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Numerical analysis of lognormal diffusions on the sphere

Journal article
Authors L. Herrmann
Annika Lang
C. Schwab
Published in Stochastics and Partial Differential Equations: Analysis and Computations
Volume 6
Issue 1
Pages 1-44
ISSN 2194-0401
Publication year 2018
Published at Department of Mathematical Sciences
Pages 1-44
Language en
Keywords Isotropic Gaussian random fields, Lognormal random fields, Karhunen-Loeve expansion, Spherical harmonic functions, Stochastic partial differentia, partial-differential-equations, monte-carlo methods, simulation, Mathematics
Subject categories Probability Theory and Statistics, Mathematical Analysis


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.

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