Sparse approximations of fractional Mat\'ern fields

TL;DR

This paper introduces a fast, sparse approximation method for fractional Matérn fields using spectral compactness and Taylor expansion, enabling efficient computation with Gaussian Markov random fields.

Contribution

It presents a novel approximation approach for Matérn fields based on spectral and Taylor expansion techniques, leading to sparse matrix systems for efficient solutions.

Findings

The method achieves accurate approximations of Matérn fields.

Sparse Cholesky decomposition enables efficient computation.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

We consider a fast approximation method for a solution of a certain stochastic non-local pseudodifferential equation. This equation defines a Mat\'ern 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.