Gaussian random fields on Riemannian manifolds: Sampling and error analysis

Many applications in spatial and spatio-temporal statistics require data to be modeled by Gaussian processes on non-Euclidean domains, or with non-stationary properties. Using such models generally comes at the price of a drastic increase in operational costs (computational and storage-wise), rendering them hard to apply to large datasets. In this talk, we propose a solution to this problem, which relies on the definition of a class of random fields on Riemannian manifolds. These fields extend ongoing work that has been done to leverage a characterization of the random fields classically used in Geostatistics as solutions of stochastic partial differential equations. The discretization of these generalized random fields, undertaken using a finite element approach, then provides an explicit characterization that is leveraged to solve the scalability problem. In this talk, we present how this approach is used to tackle the simulation and prediction of Gaussian fields on surfaces with given covariance properties (non-stationary and/or spatio-temporal), we present results on the strong error induced by the finite element approximations.