# Tensor train-Karhunen-Lo\`eve expansion for continuous-indexed random fields using higher-order cumulant functions

## TL;DR

This paper introduces a novel tensor train-based framework for representing and computing with multidimensional non-Gaussian random fields, improving efficiency and adaptivity over traditional methods.

## Contribution

It develops a new theoretical and algorithmic approach using tensor train decomposition of cumulant functions, eliminating the need for basis or quadrature point selection.

## Key findings

- Accurate tensor train decomposition of cumulant functions achieved
- Algorithm outperforms traditional methods in efficiency and adaptivity
- Effective for non-Gaussian, non-homogeneous 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 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.