Title: 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 $L^{\beta}u = \mathcal{W}$, where
$\mathcal{W}$ is Gaussian white noise, $L$ is a second-order differential
operator, and $\beta>0$ is a parameter that determines the smoothness of $u$.
However, this approach has been limited to the case $2\beta\in\mathbb{N}$,
which excludes several important covariance models and makes it necessary to
keep $\beta$ fixed during inference.
We introduce a new method, the rational SPDE approach, which is applicable
for any $\beta>0$ and therefore remedies the mentioned limitation. The
presented scheme combines a finite element discretization in space with a
rational approximation of the function $x^{-\beta}$ to approximate $u$. For the
resulting approximation, an explicit rate of strong convergence to $u$ is
derived and we show that the method has the same computational benefits as in
the restricted case $2\beta\in\mathbb{N}$ 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 $\beta$.