A note on the relationship between PDE-based precision operators and Mat\'ern covariances

TL;DR

This paper explores the connection between PDE-based precision operators and Matérn covariance functions in Gaussian random fields, focusing on how their parameters relate to variance and correlation length.

Contribution

It clarifies the relationship between PDE coefficients and Matérn covariance parameters, providing insights into their interplay in Gaussian random fields.

Findings

Derived explicit relationships between PDE coefficients and Matérn parameters

Clarified how variance and correlation length are influenced by PDE-based precision operators

Provided theoretical insights into Gaussian field modeling using PDEs

Abstract

The purpose of this technical note is to summarize the relationship between the marginal variance and correlation length of a Gaussian random field with Mat\'ern covariance and the coefficients of the corresponding partial-differential-equation (PDE)-based precision operator.