How to initialize the Circulant Embedding method to speed up the generation of stationary Gaussian Random Fields?

Géraldine Pichot (1) , Simon Legrand (2) , Michel Kern (1) , Nathanael Tepakbong-Tematio (3)

1 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
2 SED - Service Expérimentation et Développement [Paris]
3 ISAE-SUPAERO - Institut Supérieur de l'Aéronautique et de l'Espace

The Circulant Embedding Method (CEM) is a well known technique to generate stationary Gaussian Random Fields (GRF). The main idea is to embed the covariance matrix in a larger nested block circulant matrix, whose factorization can be rapidly computed thanks to the fast Fourier transform (FFT) algorithm. The CEM requires the extended matrix to be at least positive semidefinite which is proven to be the case if the enclosing domain is sufficiently large, as proven by Theorem 2.3 in [Graham et al, SIAM Journal on Numerical Analysis, 2018] for cubic domains. In this paper, we generalize this theorem to the case of rectangular parallelepipeds. Then we propose a new initialization stage of the CEM algorithm which allows to quickly jump to a domain size close to the one needed for the CEM algorithm to work. These domain size estimates are based on fitting functions. Examples of fitting functions are given for the Matérn family of covariances. These functions are inspired by our numerical simulations and by the theoretical work from [Graham et al, SIAM Journal on Numerical Analysis, 2018]. The parameters estimation of the fitting functions is done numerically. Several numerical tests are performed to show the efficiency of the proposed algorithms, for both isotropic and anisotropic Matérn covariances.

Domaines: Analyse numérique [math.NA]

Géraldine Pichot, Simon Legrand, Michel Kern, Nathanael Tepakbong-Tematio. How to initialize the Circulant Embedding method to speed up the generation of stationary Gaussian Random Fields?. 2021. ⟨hal-03190252v2⟩