Generation of Gaussian random fields on the sphere

Abstract

In order to approximate solutions to a fractional elliptic SPDE used to generate random fields on the sphere with Matérn covariance, a sinc quadrature approach combined with a surface finite element method is used. The right hand side of the equation is replaced with trace class noise on the sphere. Error bounds in L2( ; L2(S2))-norm are proved using energy estimates and an Aubin–Nitsche duality type argument. P-almost surely asymptotic error estimates are shown. Some properties of the exact solution to both trace class right hand side and white noise right hand side are discussed and the relation between the mean square differentiability and the parameters of the equation are established. Some properties of white noise on the sphere are considered. The method is implemented in FEniCS and parts of the algorithm is verified which agree with the theoretical results.