# Unified Analysis of Periodization-Based Sampling Methods for Mat\'ern Covariances

## TL;DR

This paper compares periodization methods for Matérn covariances, revealing that smooth truncation offers significant advantages over nonsmooth approaches in terms of eigenvalue decay and required domain size.

## Contribution

It provides a complete analysis of eigenvalue decay and domain size requirements for both nonsmooth and smooth periodization methods for Matérn covariances.

## Key findings

- Smooth truncation outperforms nonsmooth periodization in eigenvalue decay.
- The required torus size depends on the smoothness index and correlation length.
- Numerical results confirm theoretical advantages of smooth truncation.

## Abstract

The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational methods, such as the classical circulant embedding technique using the fast Fourier transform for generating samples on uniform grids. For the family of Mat\'ern covariances with smoothness index $\nu$ and correlation length $\lambda$, we analyse the nonsmooth periodization (corresponding to classical circulant embedding) and an alternative procedure using a smooth truncation of the covariance function. We solve two open problems: the first concerning the $\nu$-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of $\nu$, $\lambda$ of the torus when using a smooth periodization. In doing this we arrive at a complete characterisation of the performance of these two approaches. Both our theoretical estimates and the numerical tests provided here show substantial advantages of smooth truncation.