Efficient approximation of high-dimensional exponentials by tensornetworks

TL;DR

This paper introduces a tensor network-based method for efficiently approximating high-dimensional exponential functions, crucial in uncertainty quantification, by solving an associated differential equation with error control.

Contribution

It presents a novel approach combining differential equations and tensor trains to approximate high-dimensional exponentials with reliable error estimates.

Findings

Effective tensor train representation of exponential functions.

Numerical validation on log-normal fields and Bayesian problems.

Outperforms existing low-rank approximation methods.

Abstract

In this work a general approach to compute a compressed representation of the exponential $\exp(h)$ of a high-dimensional function $h$ is presented. Such exponential functions play an important role in several problems in Uncertainty Quantification, e.g. the approximation of log-normal random fields or the evaluation of Bayesian posterior measures. Usually, these high-dimensional objects are intractable numerically and can only be accessed pointwise in sampling methods. In contrast, the proposed method constructs a functional representation of the exponential by exploiting its nature as a solution of an ordinary differential equation. The application of a Petrov--Galerkin scheme to this equation provides a tensor train representation of the solution for which we derive an efficient and reliable a posteriori error estimator. Numerical experiments with a log-normal random field and a Bayesian likelihood illustrate the performance of the approach in comparison to other recent low-rank representations for the respective applications. Although the present work considers only a specific differential equation, the presented method can be applied in a more general setting. We show that the composition of a generic holonomic function and a high-dimensional function corresponds to a differential equation that can be used in our method. Moreover, the differential equation can be modified to adapt the norm in the a posteriori error estimates to the problem at hand.