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Time integration of symmetric and anti-symmetric low-rank matrices and Tucker tensors

Gianluca Ceruti, Christian Lubich

TL;DR

It is shown that the (anti-)symmetric low-rank integrators retain favourable properties of the projector-splitting integrators: given low-Rank time-dependent matrices and tensors are reproduced exactly, and the error behaviour is robust to the presence of small singular values, in contrast to standard integration methods applied to the differential equations of dynamical low- rank approximation.

Abstract

A numerical integrator is presented that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric time-dependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation. A related algorithm is given for the approximation of symmetric or anti-symmetric time-dependent tensors by symmetric or anti-symmetric Tucker tensors of low multilinear rank. The proposed symmetric or anti-symmetric low-rank integrator is different from recently proposed projector-splitting integrators for dynamical low-rank approximation, which do not preserve symmetry or anti-symmetry. However, it is shown that the (anti-)symmetric low-rank integrators retain favourable properties of the projector-splitting integrators: low-rank time-dependent matrices and tensors are reproduced exactly, and the error behaviour is robust to the presence of small singular values, in contrast to standard integration methods applied to the differential equations of dynamical low-rank approximation. Numerical experiments illustrate the behaviour of the proposed integrators.

Time integration of symmetric and anti-symmetric low-rank matrices and Tucker tensors

TL;DR

It is shown that the (anti-)symmetric low-rank integrators retain favourable properties of the projector-splitting integrators: given low-Rank time-dependent matrices and tensors are reproduced exactly, and the error behaviour is robust to the presence of small singular values, in contrast to standard integration methods applied to the differential equations of dynamical low- rank approximation.

Abstract

A numerical integrator is presented that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric time-dependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation. A related algorithm is given for the approximation of symmetric or anti-symmetric time-dependent tensors by symmetric or anti-symmetric Tucker tensors of low multilinear rank. The proposed symmetric or anti-symmetric low-rank integrator is different from recently proposed projector-splitting integrators for dynamical low-rank approximation, which do not preserve symmetry or anti-symmetry. However, it is shown that the (anti-)symmetric low-rank integrators retain favourable properties of the projector-splitting integrators: low-rank time-dependent matrices and tensors are reproduced exactly, and the error behaviour is robust to the presence of small singular values, in contrast to standard integration methods applied to the differential equations of dynamical low-rank approximation. Numerical experiments illustrate the behaviour of the proposed integrators.

Paper Structure

This paper contains 16 sections, 10 theorems, 82 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. General matrices: recap of the projector-splitting integrator for dynamical low-rank approximation
  3. Symmetric and skew-symmetric matrices: a structure-preserving integrator for dynamical low-rank approximation
  4. (Skew-)symmetry preserving integrator
  5. Exactness property of the (skew-)symmetric integrator
  6. Robustness to small singular values
  7. General tensors: recap of the projector-splitting integrator for the dynamical low-rank approximation by Tucker tensors
  8. Symmetric and anti-symmetric tensors: a structure-preserving integrator for dynamical low-rank approximation
  9. (Anti-)symmetry preserving Tucker integrator
  10. Exactness property of the (anti-)symmetric Tucker integrator
  11. Robustness to small singular values
  12. Numerical Experiments
  13. Addition of symmetric tensors: a computationally inexpensive retraction
  14. Robustness with respect to small singular values
  15. Ground state of a fermionic multi-particle system
  16. ...and 1 more sections

Key Result

theorem 1

Let ${\mathbf A}(t) \in \mathbb{R}^{m \times n}$ be of rank $r$ for $t_0 \leq t \leq t_1$, so that ${\mathbf A}(t)$ has a factorization USV, ${\mathbf A}(t)={\mathbf U}(t){\mathbf S}(t){\mathbf V}(t)^\top$. Moreover, assume that the $r\times r$ matrix ${\mathbf V}(t_1)^\top {\mathbf V}(t_0)$ is inve

Figures (4)

  • Figure 1: Comparison of the explicit Runge Kutta method (left) and the proposed symmetry-preserving integrator (right) for different approximation ranks and step sizes in the case of a given time-dependent symmetric matrix.
  • Figure 2: First twelve singular values of the reference solution at time $T=0.1$ and approximation errors for different ranks at different step-sizes for the Lyapunov matrix differential equation.
  • Figure 3: Evolution to the fermionic ground state energy computed with rank 5.
  • Figure 4: Energy evolution computed with rank 5.

Theorems & Definitions (17)

  • theorem 1: Exactness property, LubichOseledets
  • theorem 2: Robust error bound, KieriLubichWalach
  • theorem 3: Exactness property
  • proof
  • theorem 4: Robust error bound
  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma: Local Error
  • proof
  • ...and 7 more