Advances in Gaussian random field generation: a review

Abstract

Gaussian (normal) distribution is a basic continuous probability distribution in statistics, it plays a substantial role in scientific and engineering problems that related to stochastic phenomena. This paper aims to review state of the art of Gaussian random field generation methods, their applications in scientific and engineering issues of interest, and open-source software/packages for Gaussian random field generation. To this end, first we briefly introduce basic mathematical concepts and theories in Gaussian random field, then seven commonly-used Gaussian random field generation methods are systematically presented. The basic idea, mathematical framework of each generation method are introduced in detail and comparisons of these methods are summarized. Then representative applications of Gaussian random field in various areas, especially of engineering interest in recent two decades, are reviewed. For readers' convenience, four representative example codes are provided, and several relevant up-to-date open-source software and packages that freely available from the Internet are introduced.

Local hydraulic resistance in heterogeneous porous media

“…To generate heterogeneous porous media we make use of the Gaussian Random Fields (GRF), which are increasingly used to represent realistic porous media (Liu et al, 2019).…”

Estimating CPT Parameters at Unsampled Locations Based on Kriging Interpolation Method

“…Where e indexes the unsampled location, i indexes the ith sampled location, Z is the CPT parameter (i.e., q c or f s ), λ is the corresponding weight, and n is the number of samples. The weight λ i could be obtained by solving marix presented in Equations ( 7) and (8).…”

“…Random field theory is often used to analyze time series. Although it has been successfully applied in analyzing soil spatial variability by different researchers [8,9], it is mostly applied to one-dimensional situations. Geostatistics often refers to the Kriging method, which has been widely used in the field of geographic science.…”

“…Description of code(8). Note: regpoly0, regpoly2, and regpoly3 mean zero-order polynomial, first polynomial, and second-order polynomial; correxp, correpg, corrgauss, corrlin, corrspherical, and corrspline mean exponential model, generalized exponential model, Gaussian model, linear model, spherical model, and cubic spline model.…”

A fractional PDE model for turbulent velocity fields near solid walls

“…Gaussian random fields are essentially ubiquitous in contemporary UQ and many convenient features of them are well-known; see, e.g. Liu et al (2019) and references therein.…”