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Numerical analysis of lognormal diffusions on the sphere

  • Lukas HerrmannEmail author
  • Annika Lang
  • Christoph Schwab
Article

Abstract

Numerical solutions of stationary diffusion equations on the unit 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 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 Multilevel Monte Carlo methods 

Mathematics Subject Classification

60G60 60G15 60G17 33C55 41A25 60H15 60H35 65C30 65N30 

Notes

Acknowledgements

The work was supported in part by the European Research Council under ERC AdG 247277, the Swiss National Science Foundation under SNF 200021_159940/1, the Knut and Alice Wallenberg foundation, and the Swedish Research Council under Reg. No. 621-2014-3995. We thank three anonymous referees for comments that improve the presentation.

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Authors and Affiliations

  1. 1.Seminar für Angewandte Mathematik, ETH ZürichZürichSwitzerland
  2. 2.Department of Mathematical SciencesChalmers University of Technology and University of GothenburgGöteborgSweden

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