Fast simulation for Gaussian random fields on compact Riemannian manifolds

Abstract

We develop a fully discrete approximation scheme to fast simulate centered Gaussian random fields (GRFs) over compact Riemannian manifolds. These GRFs can be either anisotropic or isotropic ones. The scheme is proposed by the truncation of a series expansion depending solely on the decay of the angular power spectrum, which is independent of the chosen space and time discretization. The pth moment convergence for the scheme is proved and the error bound with its associated rate of convergence is determined. Three simulation case studies are conducted by applying the Laplacian operator to find the required surface harmonics for a torus endowed with a Riemannian metric and by applying the ellipsoidal Beltrami and Laplacian operators to find the required surface ellipsoidal harmonics respectively for anisotropic and isotropic GRFs over ellipsoids endowed with two different Riemannian metrics.