  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
geometrical domain in the tensor train format; secondly, to develop a new
algorithm framework for accurately computing the modes and the second and
third-order cumulant tensors within moderate time. The core of the new
theoretical framework is the tensor train decomposition of cumulant functions.
This decomposition is accurately computed with a novel rank-revealing
algorithm. 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 third-order cumulant functions, the new
theoretical and algorithm 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 algorithm framework.
