Stepwise covariance matrix decomposition for efficient simulation of multivariate large-scale three-dimensional random fields

Abstract

• A stepwise covariance matrix decomposition method is proposed for multivariate large-scale 3-D random field simulation. • This method is accurate when the 3D correlation function is separable. • The requirement on memory space can be reduced by ten orders of magnitude. • The maximum 3-D random field resolution can be improved from 21 × 21 × 41 to over 501 × 501 × 1001. The spatial variability of geomaterials affects the failure mechanism and reliability of geotechnical structures significantly, and can be modeled rigorously as a three-dimensional (3-D) random field. However, the simulation of multivariate, large-scale and high-resolution 3-D random fields is a challenging task due to extraordinary demands in computational resources. This paper proposes a stepwise covariance matrix decomposition method (CMD) with the aid of separable correlation functions, in which the 3-D random field is generated sequentially along each single dimension with small one-dimensional correlation matrices. The method not only inherits the simplicity of the widely-used general CMD, but also significantly reduces the computational time and required memory space. It only takes a few seconds to construct large-scale and high-resolution 3-D random fields, with the requirement on memory space reduced by more than ten orders of magnitude. The maximum random field resolution is significantly improved from approximately 21 × 21 × 41 using the general CMD to over 501 × 501 × 1001 using the stepwise CMD, which suffices in most engineering applications. The stepwise CMD facilitates 3-D spatial variability modeling in probabilistic site characterization and routine geotechnical reliability analysis.