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

Chuanfu Xiao, Kejun Tang, Zhitao Zhu

TL;DR

The paper addresses the challenge of representing and computing inverses of large-scale structured matrices arising from PDE discretizations in a TT format. It introduces a computable sufficient condition guaranteeing that the inverse admits a low-rank TT representation and develops a diagonalization-based TT inversion method using Hadamard inverses and an Expanding operator. Theoretical results show the Hadamard inverse of the diagonalized tensor exhibits a displacement structure, yielding exponential decay of singular values under verifiable conditions, and the method applies to Poisson, Boltzmann-BGK, and Fokker-Planck discretizations. Numerical experiments across 3D Poisson, Boltzmann-BGK, and high-dimensional Fokker-Planck problems validate the theory and demonstrate scalable accuracy and memory efficiency for large degrees of freedom.

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.

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

TL;DR

The paper addresses the challenge of representing and computing inverses of large-scale structured matrices arising from PDE discretizations in a TT format. It introduces a computable sufficient condition guaranteeing that the inverse admits a low-rank TT representation and develops a diagonalization-based TT inversion method using Hadamard inverses and an Expanding operator. Theoretical results show the Hadamard inverse of the diagonalized tensor exhibits a displacement structure, yielding exponential decay of singular values under verifiable conditions, and the method applies to Poisson, Boltzmann-BGK, and Fokker-Planck discretizations. Numerical experiments across 3D Poisson, Boltzmann-BGK, and high-dimensional Fokker-Planck problems validate the theory and demonstrate scalable accuracy and memory efficiency for large degrees of freedom.

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.
Paper Structure (26 sections, 8 theorems, 106 equations, 6 figures, 3 tables, 2 algorithms)

This paper contains 26 sections, 8 theorems, 106 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Applications.
  3. Related Work.
  4. Summary of Contributions
  5. Organization
  6. Preliminarlies
  7. Basic tensor operations
  8. TT decomposition
  9. Operations on TT formats
  10. TT-based matrix inversion method
  11. Diagonalization
  12. Low-rank TT representation
  13. Hadamard inverse of TT tensors
  14. Explicit TT approximation of the invese matrix
  15. Overall computational procedure of the TT-based matrix inversion method
  16. ...and 11 more sections

Key Result

Theorem 4.2

If the matrices $\left\{\bm{M}^{(k)}:k=1,2,\ldots,d\right\}$ are invertible, then $\bm{\mathcal{L}}^{\odot-1}$ has $\left(\bm{A}^{(1)},\bm{A}^{(2)},\ldots,\bm{A}^{(d)}\right)$-displacement structure of $\bm{\mathcal{F}}=\bm{f}^{(1)}\otimes\bm{f}^{(2)}\otimes\cdots\otimes\bm{f}^{(d)}$, where for all $k=1,2,\ldots,d$.

Figures (6)

  • Figure 1: Distribution of the singular values of the matricization of the tensor $\bm{\mathcal{L}}^{\odot-1}$ with varying $n$ (the number of grids in each dimension).
  • Figure 2: Solutions of the baseline and TT-based matrix inversion methods in the hyper-plane $(x_1,x_2,0)$ with $n=512$.
  • Figure 3: Numerical results of the TT-based matrix inversion method for solving the 2D2V Boltzmann-BGK equation with different time step sizes. Left: Relative errors with respect to $n$ at $t=1.0$s. Right: Running times (unit: seconds) and averaged TT ranks.
  • Figure 4: Macroscopic density $\rho(\bm{x},t)$, velocity $\bm{U}(\bm{x},t)$, and temperature $T(\bm{x},t)$ of the TT-based matrix inversion method with the time step size $\delta t=0.0025$ at $t=0.5$s and $1.0$s, respectively.
  • Figure 5: Numerical results of the TT-based matrix inversion method for solving the Fokker-Planck equation of different dimensions. Left: Relative errors. Right: Running times (unit: seconds) and averaged TT ranks.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 4.1
  • Theorem 4.2
  • Lemma 4.3
  • Definition 4.4
  • Lemma 4.5
  • Theorem 4.6
  • Corollary 4.7: Estimation of TT ranks
  • ...and 4 more