Unified Analysis of Periodization-Based Sampling Methods for Matérn Covariances

Markus Bachmayr, Ivan G. Graham, Van Kien Nguyen, Robert Scheichl

Research output: Contribution to journalArticlepeer-review

6 Citations (SciVal)


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 Mat\'ern covariances with smoothness index $\nu$ and correlation length $\lambda$, 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 $\nu$-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of $\nu$, $\lambda$ 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.
Original languageEnglish
Pages (from-to)2953–2980
Number of pages28
JournalSIAM Journal on Numerical Analysis (SINUM)
Issue number5
Early online date20 Oct 2020
Publication statusPublished - 2020

Bibliographical note

22 pages, 2 figures


  • math.NA
  • cs.NA
  • math.PR
  • math.ST
  • stat.TH
  • 60G15, 60G60, 42B05, 65T40


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