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 models and makes it necessary to keep $\beta$ fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension $d\in \mathbb{N}$ is applicable for any $\beta >d/4$, and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function $x^{-\beta }$ 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 $2\beta \in \mathbb{N}$. 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 $\beta$.

28 pages, 4 figures

References in corpus (3)

• Numerical solution of fractional elliptic stochastic PDEs with spatial white noise
• Fractional Brownian motion, the Matern process, and stochastic modeling of turbulent dispersion
• Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise

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