Analysis of circulant embedding methods for sampling stationary random fields

TL;DR

This paper proves conditions under which circulant embedding methods guarantee positive definiteness for sampling stationary random fields, especially for Matérn covariance, and confirms findings with numerical experiments.

Contribution

It provides new theoretical guarantees for the positive definiteness of circulant matrices in sampling methods, particularly for Matérn fields, and analyzes eigenvalue decay rates.

Findings

Positive definiteness is guaranteed for sufficiently large embedding cubes.

Eigenvalues of the circulant matrix decay at the same rate as Karhunen–Loève eigenvalues.

Numerical experiments confirm theoretical results.

Abstract

In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix appearing in the circulant embedding method is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Mat\'ern covariance, and prove (for fixed correlation length) that, as $h_0\rightarrow 0$, positive definiteness is guaranteed when the random field is sampled on a cube of size order $(1 + \nu^{1/2} \log h_0^{-1})$ times larger than the size of the physical domain. (Here $h_0$ is the mesh spacing of the regular grid and $\nu$ the Mat\'ern smoothness parameter.) We show that the sampling cube can become smaller as the correlation length decreases when $h_0$ and $\nu$ are fixed. Our results are confirmed by numerical experiments. We prove several results about the decay of the eigenvalues of the circulant matrix. These lead to the conjecture, verified by numerical experiment, that they decay with the same rate as the Karhunen--Lo\`{e}ve eigenvalues of the covariance operator.