The Rational SPDE Approach for Gaussian Random Fields With General Smoothness

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, <i>L</i> is a second-order differential operator, and β&gt;0 is a parameter that determines the smoothness of <i>u</i>. However, this approach has been limited to the case 2β∈N, which excludes several important models and makes it necessary to keep <i>β</i> fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension d∈N is applicable for any β&gt;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 <i>u</i>. For the resulting approximation, an explicit rate of convergence to <i>u</i> 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 <i>β</i>. Supplementary materials for this article are available online.