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Numerical Analysis of Lognormal Diffusions on the Sphere

by L. Herrmann and A. Lang and Ch. Schwab

(Report number 2016-02)

Abstract
Numerical solutions of stationary diffusion equations on \(\mathbb{S}^2\) 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 \(\mathbb{S}^2\) 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.

Keywords: Isotropic Gaussian random fields, lognormal random fields, Karhunen-Loève expansion, spherical harmonic functions, stochastic partial differential equations, random partial differential equations, regularity of random fields, Finite Element Methods, Spectral Galerkin Methods

@Techreport{HLS16_639,
  author = {L. Herrmann and A. Lang and Ch. Schwab},
  title = {Numerical Analysis of Lognormal Diffusions on the Sphere},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-02},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-02.pdf },
  year = {2016}
}

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