The Rational SPDE Approach for Gaussian Random Fields With General Smoothness

Abstract

A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form Lβu =W, where W is Gaussian white noise, L is a second-order differential operator, and β > 0 is a parameter that determines the smoothness of u. However, this approach has been limited to the case 2β ∈ N, which excludes several important models and makes it necessary to keep β fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension d ∈ N is applicable for any β > d/4, and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function x−β to approximate u. For the resulting approximation, an explicit rate of convergence to u in mean-square sense is derived. Furthermore, we show that our method has the same computational benefits as in the restricted case 2β ∈ N. Several numerical experiments and a statistical application are used to illustrate the accuracy of the method, and to show that it facilitates likelihood-based inference for all model parameters including β. Supplementary materials for this article are available online.