598 RWTH Publication No: 956764        2020       
TITLE Unified Analysis of Periodization-Based Sampling Methods for Matérn Covariances
AUTHORS Markus Bachmayr, Ivan G. Graham, Van Kien Nguyen, Robert Scheichl
ABSTRACT 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 Matérn covariances with smoothness index ν and correlation length λ, we analyse 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 ν-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of ν, λ of the torus when using a smooth periodization. In doing this we arrive at a complete characterisation of the performance of these two approaches. Both our theoretical estimates and the numerical tests provided here show substantial advantages of smooth truncation.
KEYWORDS stationary Gaussian random fields, circulant embedding, periodization, Matérn covariances
DOI 10.1137/19M1269877
PUBLICATION SIAM Journal on Numerical AnalysisVol. 58, Iss. 5 (2020)