Multilevel representations of isotropic Gaussian random fields on the sphere

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.

On the convergence of a finite volume method for the Navier–Stokes–Fourier system

“…Zhang and Karniadakis [44]. Rigourous convergence analysis of these uncertainty quantification methods typically relies on the existence of a unique regular solution and pathwise arguments to analyse the convergence in random space, see, e.g., Abgrall and Mishra [1], Babuška et al [2], Bachmayr et al [3,4], Badwaik et al [5], Fjordholm et al [22], Kolley et al [25], Leonardi et al [29], Mishra and Schwab [33], Nobile et al [36], Tang and Zhou [38] and the references therein.…”