Simulation of Random Fields via Local Average Subdivision

TLDR

The method addresses the need to model engineering properties as local averages and to condition realizations for known data or variable resolution within sub‑regions. The paper presents a fast, accurate technique for generating realizations of homogeneous Gaussian scalar random processes in one, two, or three dimensions. It constructs discrete processes by averaging a homogeneous random function over incremental domains at multiple discretization levels, illustrated with Ornstein–Uhlenbeck and fractional Gaussian noise processes. The technique enables conditioning and local refinement, improving finite element modeling of random phenomena.

Abstract

A fast and accurate method of generating realizations of a homogeneous Gaussian scalar random process in one, two, or three dimensions is presented. The resulting discrete process represents local averages of a homogeneous random function defined by its mean and covariance function, the averaging being performed over incremental domains formed by different levels of discretization of the field. The approach is motivated first by the need to represent engineering properties as local averages (since many properties are not well defined at a point and show significant scale effects), and second to be able to condition the realization easily to incorporate known data or change resolution within sub‐regions. The ability to condition the realization or increase the resolution in certain regions is an important contribution to finite element modeling of random phenomena. The Ornstein‐Uhlenbeck and fractional Gaussian noise processes are used as illustrations.