Tensor train-Karhunen-Loève expansion: New theoretical and algorithmic frameworks for representing general non-Gaussian random fields

Abstract

The goals of this work are two-fold: firstly, to propose a new theoretical framework for representing random fields on a large class of multidimensional physical domain in the tensor train format; secondly, to develop a new algorithmic framework for accurately computing the modes and describing the correlation structure of the latent factors beyond second order within moderate time. Central to the algorithmic framework is the tensor train decomposition of cumulant functions. This decomposition is accurately computed with a novel rank-revealing algorithm. For computing the modes, compared with existing Galerkin-type and collocation-type methods, the proposed computational procedure totally removes the need of selecting the basis functions or collocation points and the quadrature points, which not only greatly enhances adaptivity, but also avoids solving large-scale eigenvalue problems. Moreover, by computing with higher-order cumulant functions, the new theoretical and algorithmic frameworks show great potential for representing general non-Gaussian non-homogeneous random fields. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the efficiency and accuracy of the proposed algorithmic framework.