Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields
The SMAI Journal of computational mathematics, Tome 8 (2022), pp. 327-347.

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 [9] 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 Matérn family of covariances. These functions are inspired by our numerical simulations and by the theoretical work from [9]. 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.

Publié le :
DOI : 10.5802/smai-jcm.89
Classification : 60G60, 65C10, 65C05, 86A32
Mots clés : stationary Gaussian random fields, circulant embedding method, Matérn covariances, fast Fourier transform
Pichot, Géraldine 1 ; Legrand, Simon 2 ; Kern, Michel 1 ; Tepakbong-Tematio, Nathanael 3

1 Inria, 2 rue Simone Iff, 75589 Paris, France and Université Paris-Est, CERMICS (ENPC), 6 et 8 av. Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France
2 Inria, 2 rue Simone Iff, 75589 Paris, France
3 ISAE-SUPAERO, 10, avenue Édouard-Belin, BP 54032, 31055 Toulouse Cedex 4, France
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     title = {Initialization of the {Circulant} {Embedding} method to speed up the generation of {Gaussian} random fields},
     journal = {The SMAI Journal of computational mathematics},
     pages = {327--347},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Pichot, Géraldine; Legrand, Simon; Kern, Michel; Tepakbong-Tematio, Nathanael. Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields. The SMAI Journal of computational mathematics, Tome 8 (2022), pp. 327-347. doi : 10.5802/smai-jcm.89. http://www.numdam.org/articles/10.5802/smai-jcm.89/

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