The SPDE approach for Gaussian random fields with general smoothness
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A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) L^βu = 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 covariance models such as the exponential covariance on R^2. We demonstrate how this restriction can be avoided by combining a finite element discretization in space with a rational approximation of the function x^-β to approximate the solution u. For the resulting approximation, an explicit rate of strong convergence is derived and we show that the method has the same computational benefits as in the restricted case 2β∈N when used for statistical inference and prediction. Several numerical experiments are performed to illustrate the accuracy of the method, and to show how it can be used for likelihood-based inference for all model parameters including β.
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