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Journal Articles SMAI Journal of Computational Mathematics Year : 2022

Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields

Abstract

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 that makes it possible 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 Mat\'ern 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.
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Dates and versions

hal-03190252 , version 1 (06-04-2021)
hal-03190252 , version 2 (24-09-2021)
hal-03190252 , version 3 (04-03-2022)
hal-03190252 , version 4 (04-03-2022)
hal-03190252 , version 5 (04-03-2022)
hal-03190252 , version 6 (15-11-2022)

Identifiers

Cite

Géraldine Pichot, Simon Legrand, Michel Kern, Nathanael Tepakbong-Tematio. Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields. SMAI Journal of Computational Mathematics, 2022, 8, pp.21. ⟨10.5802/smai-jcm.89⟩. ⟨hal-03190252v6⟩
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