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.
Keywords: stationary Gaussian random fields, circulant embedding method, Matérn covariances, fast Fourier transform
CC-BY-NC-ND 4.0
@article{SMAI-JCM_2022__8__327_0,
author = {G\'eraldine Pichot and Simon Legrand and Michel Kern and Nathanael Tepakbong-Tematio},
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},
volume = {8},
year = {2022},
doi = {10.5802/smai-jcm.89},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.89/}
}
TY - JOUR AU - Géraldine Pichot AU - Simon Legrand AU - Michel Kern AU - Nathanael Tepakbong-Tematio TI - Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields JO - The SMAI Journal of computational mathematics PY - 2022 SP - 327 EP - 347 VL - 8 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.89/ DO - 10.5802/smai-jcm.89 LA - en ID - SMAI-JCM_2022__8__327_0 ER -
%0 Journal Article %A Géraldine Pichot %A Simon Legrand %A Michel Kern %A Nathanael Tepakbong-Tematio %T Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields %J The SMAI Journal of computational mathematics %D 2022 %P 327-347 %V 8 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.89/ %R 10.5802/smai-jcm.89 %G en %F SMAI-JCM_2022__8__327_0
Géraldine Pichot; Simon Legrand; Michel Kern; Nathanael Tepakbong-Tematio. Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields. The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 327-347. doi : 10.5802/smai-jcm.89. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.89/
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