Multilevel representations of isotropic Gaussian random fields on the sphere

IMA Journal of Numerical Analysis, vol. 43, pp. 1970–2000

Abstract

Abstract Series expansions of isotropic Gaussian random fields on $\mathbb {S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations with multilevel localized structure provide an alternative to the standard Karhunen–Loève expansions of isotropic random fields in terms of spherical harmonics. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field. In addition, numerical illustrations are given and an application to random elliptic PDEs on the sphere is analysed.

Authors 2

1. Markus Bachmayr corresponding Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University , Templergraben 55, 52056 Aachen, Germany

2. Ana Djurdjevac

   Freie Universität Berlin

   Affiliation as printed

   Institut für Mathematik, Freie Universität Berlin , Arnimallee 6, 14195 Berlin, Germany