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Isotropic Gaussian random fields on the sphere

by A. Lang

(Report number 2013-26)

Abstract
Sample regularity and fast simulation of isotropic Gaussian random fields on the sphere are for example of interest for the numerical analysis of stochastic partial differential equations and for the simulation of ice crystals or Saharan dust particles as lognormal random fields. We recall the results from SAM Report 2013-15, which include the approximation of isotropic Gaussian random fields with convergence rates as well as the regularity of the samples in relation to the smoothness of the covariance expressed in terms of the decay of the angular power spectrum. As example we construct isotropic Q-Wiener processes out of isotropic Gaussian random fields and discretize the stochastic heat equation with spectral methods.

Keywords: Gaussian random fields, isotropic random fields, Karhunen-Loève expansion, spherical harmonic functions, Kolmogorov-Chentsov theorem, sample Hölder continuity, sample differentiability, stochastic heat equation, spectral Galerkin methods, strong convergence rates

@Techreport{L13_523,
  author = {A. Lang},
  title = {Isotropic Gaussian random fields on the sphere},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-26},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-26.pdf },
  year = {2013}
}

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