Papers › Analysis of circulant embedding methods for sampling stationary random fields
Analysis of circulant embedding methods for sampling stationary random fields
Ivan G. Graham, Frances Y. Kuo, Dirk Nuyens, Rob Scheichl, Ian H. Sloan
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In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix appearing in the circulant embedding method is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Mat\'ern covariance, and prove (for fixed correlation length) that, as h₀→0, positive definiteness is guaranteed when the random field is sampled on a cube of size order (1 + ν^(1/2) logh₀⁻¹) times larger than the size of the physical domain. (Here h₀ is the mesh spacing of the regular grid and ν the Mat\'ern smoothness parameter.) We show that the sampling cube can become smaller as the correlation length decreases when h₀ and ν are fixed. Our results are confirmed by numerical experiments. We prove several results about the decay of the eigenvalues of the circulant matrix. These lead to the conjecture, verified by numerical experiment, that they decay with the same rate as the Karhunen--Lo\`{e}ve eigenvalues of the covariance operator.
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