Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 58, Issue 5, Pages 2953-2980Publisher
SIAM PUBLICATIONS
DOI: 10.1137/19M1269877
Keywords
stationary Gaussian random fields; circulant embedding; periodization; Matern covariances
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Funding
- Hausdorff Center of Mathematics, University of Bonn
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The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational methods, such as the classical circulant embedding technique using the fast Fourier transform for generating samples on uniform grids. For the family of Matern covariances with smoothness index v and correlation length A, we analyze the nonsmooth periodization (corresponding to classical circulant embedding) and an alternative procedure using a smooth truncation of the covariance function. We solve two open problems: the first concerning the v-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of v, A of the torus when using a smooth periodization. In doing this we arrive at a complete characterization of the performance of these two approaches. Both our theoretical estimates and the numerical tests provided here show substantial advantages of smooth truncation.
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