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
, where
is Gaussian white noise, L is a second-order differential operator, and
is a parameter that determines the smoothness of u. However, this approach has been limited to the case
, 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
is applicable for any
, and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function
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
. 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.