Generating Realisations of Stationary Gaussian Random Fields by Circulant Embedding.

Random fields are families of random variables, indexed by a d-dimensional parameter x with d> 1. They are important in many applications and are used, for example, to model properties of biological tissue, velocity fields in turbulent flows and permeability coefficients of rocks. Mark 24 of the NAG Fortran library includes new routines for generating realisations of stationary Gaussian random fields using the method of circulant embedding. This short note illustrates the main ideas behind circulant embedding and how to use the routines g05zr and g05zs in the NAG toolbox for MATLAB. The routines g05zm, g05zn and g05zp can also be used to generate realisations of stationary Gaussian stochastic processes (the d = 1 case).1 Random Fields For a two-dimensional domain D ⊂ R2, a (real-valued) random field {Z(x) : x ∈ D}, also written Z(x, ω), is a set of real-valued random variables on a probability space (Ω,F,P). That is, for each x ∈ D, Z(x) : Ω → R is a random variable. The random field is second-order if Z(x) has finite variance for each x ∈ D and for such fields we can define the mean function µ(x) = E [Z(x)] and the covariance function C(x,y) = Cov(Z(x), Z(y)) = E Z(x) − µ(x) Z(y) − µ(y), x,y ∈ D. Important cases are stationary random fields, where µ(x) is constant and the covariance de-pends on x − y and isotropic random fields, where the covariance depends on ‖x − y‖2. For a fixed ω ∈ Ω, the associated realisation of a random field is the deterministic function f: D → R defined by f(x): = Z(x, ω) for x ∈ D. Thus, a realisation represents one possibility for the quantity Z as a function of x. Examples are given in Figures 2 and 3, later. If Z(x, ω) represents an input for a mathematical model such as a system of PDEs, then we often need to generate multiple realisations of Z(x, ω) so that statistical analysis of the solution can be performed (e.g., using Monte Carlo methods)