Analysis of circulant embedding methods for sampling stationary random fields

A standard problem in uncertainty quantification and in computational statistics is the sampling of stationary Gaussian random fields with given covariance in a d-dimensional (physical) domain. In many applications it is sufficient to perform the sampling on a regular grid on a cube enclosing the physical domain, in which case the corresponding covariance matrix is nested block Toeplitz. After extension to a nested block circulant matrix, this can be diagonalized by FFT—the “circulant embedding method.” Provided the circulant matrix is positive definite, this provides a finite expansion of the field in terms of a deterministic basis, with coefficients given by i.i.d. standard normals. In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Matérn covariance, and prove (for fixed correlation length) that, as h ₀ ? 0, positive definiteness is guaranteed when the random field is sampled on a cube of size order (1 + ? ¹/ ² log h ^- ₀ ¹ ) times larger than the size of the physical domain. (Here h ₀ is the mesh spacing of the regular grid and ? the Matérn smoothness parameter.) We show that the sampling cube can become smaller as the correlation length decreases when h ₀ and ? 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–Loève eigenvalues of the covariance operator. The method analyzed here complements the numerical experiments for uncertainty quantification in porous media problems in an earlier paper by the same authors in J. Comput. Phys., 230 (2011), pp. 3668–3694.