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: 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. Numerical experiments illustrate the behaviour of the proposed integrators.
Highlights
In this paper we propose and analyse an algorithm that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric timedependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation
Motivation for this work comes from Lyapunov and Riccati differential equations, which have large symmetric matrices as solutions, which can often be well approximated by low-rank matrices [19]
The constants are independent of singular values of the exact or approximate solution. It is further shown in [10, Section 2.6.3] that an inexact solution of the matrix differential equations in the projector-splitting integrator leads to an additional error that is bounded in terms of the local errors in the inexact substeps, again with constants that do not depend on small singular values
Summary
In this paper we propose and analyse an algorithm that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric timedependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation. We will show that the (anti-)symmetry-preserving integrators proposed here retain the robustness with respect to small singular values of the projector-splitting algorithms This relies on an exactness property, namely that explicitly given timedependent matrices and tensors of the approximation rank are reproduced exactly by the integrator. The second remarkable property is the robustness of the algorithm to the presence of small singular values of the solution or its approximation This is in contrast to standard integrators applied to (2) or the equivalent differential equations for the factors U(t), S(t), V(t), which contain a factor S(t)−1 on the right-hand sides [12, Prop. While the projector-splitting integrator for dynamical low-rank approximation described in the previous section has favourable properties, it does not preserve symmetry or skew-symmetry of the solution A(t) to (1).
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