Sparse Approximations of Fractional Matérn Fields

Abstract

We consider a fast approximation method for a solution of a certain stochastic non-local pseudodifferential equation. This equation defines a Matérn class random field. The approximation method is based on the spectral compactness of the solution. We approximate the pseudodifferential operator with a Taylor expansion. By truncating the expansion, we can construct an approximation with Gaussian Markov random fields. We show that the solution of the truncated version can be constructed with an over-determined system of stochastic matrix equations with sparse matrices. We solve the system of equations with a sparse Cholesky decomposition. We consider the convergence of the discrete approximation of the solution to the continuous one. Finally numerical examples are given.

The Rational SPDE Approach for Gaussian Random Fields With General Smoothness

“…This method was used for the comparison in (Heaton et al, 2018), and we will use it as a baseline method when analyzing the accuracy of the rational SPDE approximations in later sections. Another Markov approximation based on the spectral density was proposed by Roininen et al (2018). These Markov approximations may be sufficient in certain applications; however, any approach based on the spectral density or the covariance function is difficult to generalize to models on more general domains than R d , nonstationary models, or non-Gaussian models.…”

Gaussian Markov Random Field Priors in Ionospheric 3-D Multi-Instrument Tomography

“…So, even though the actual covariance matrix is never constructed, the precision matrix can be determined by its covariance properties. The detailed theoretical background for these types of GMRFs is given in [21]- [24]. Here, an example is given, where the target precision −1 pr is determined with a squared exponential covariance function…”

“…Following [24], an anisotropic continuous GMRF with a covariance approximating (11) can be formed as a solution to a stochastic partial differential equation…”

Approximate state-space Gaussian processes via spectral transformation

“…because it defines how many terms we can include to the expansion while still retaining a valid spectral density [10]. To cope with this problem we introduce spectral preconditioning, where we write the spectral density S as…”