Provable Low-Rank Tensor-Train Approximations in the Inverse of Large-Scale Structured Matrices

TL;DR

This paper demonstrates that the inverse of certain large-scale structured matrices from differential operators can be efficiently approximated using low-rank tensor-train formats, enabling faster PDE solutions.

Contribution

It provides a verifiable condition under which the inverse admits a low-rank TT approximation and develops an efficient method for computing such inverses.

Findings

Inverse matrices have low-rank TT formats for PDE-related matrices.

The method efficiently solves PDEs like Poisson, Boltzmann, Fokker-Planck.

Numerical results confirm theoretical predictions and computational advantages.

Abstract

This paper studies the low-rank property of the inverse of a class of large-scale structured matrices in the tensor-train (TT) format, which is typically discretized from differential operators. An interesting question that we are concerned with is: Does the inverse of the large-scale structured matrix still admit the low-rank TT representation with guaranteed accuracy? In this paper, we provide a computationally verifiable sufficient condition such that the inverse matrix can be well approximated in a low-rank TT format. It not only answers what kind of structured matrix whose inverse has the low-rank TT representation but also motivates us to develop an efficient TT-based method to compute the inverse matrix. Furthermore, we prove that the inverse matrix indeed has the low-rank tensor format for a class of large-scale structured matrices induced by differential operators involved in several PDEs, such as the Poisson, Boltzmann, and Fokker-Planck equations. Thus, the proposed algorithm is suitable for solving these PDEs with massive degrees of freedom. Numerical results on the Poisson, Boltzmann, and Fokker-Planck equations validate the correctness of our theory and the advantages of our methodology.