Generating correlated gaussian random fields by orthogonal polynomial approximations to the square root of the covariance matrix

Abstract

Monte Carlo simulations in geostatistics often require the generation of realizations of multi-dimensional Gaussian processes with prescribed mean and covariance over a specified grid Ω of sampling points in R:d In principle, given a factorization C = AAT of the process’ covariance matrix C, realizations can always be constructed from the product Aϵ where ϵ is a white noise vector with unit variance. However, this approach generally has a high computational cost when compared with techniques such as the spectral or turning bands methods. Nevertheless, matrix factorization has advantages in that it does not require any special structure, such as isotropy or stationarity of the covariance model, o r regularity of the sampling grid Ω. Therefore, in an effort to speed up the approach at the cost of some loss in accuracy, Davis (1987b) proposed approximating the square root A by a low order matrix polynomial in C. This paper explores the construction of such polynomial approximations in more detail The paper f...