DISCRETIZED DYNAMICAL LOW-RANK APPROXIMATION IN THE PRESENCE OF SMALL SINGULAR VALUES
DISCRETIZED DYNAMICAL LOW-RANK APPROXIMATION IN THE PRESENCE OF SMALL SINGULAR VALUES
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DOI：
10.1137/15m1026791
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发表时间：
2016-01-01
影响因子：
2.9
通讯作者：
Walach, Hanna
中科院分区：
文献类型：
--
作者：
Kieri, Emil;Lubich, Christian;Walach, Hanna
本文的主题是对大型时变矩阵和张量的低秩近似。这些矩阵和张量要么是显式给出的，要么是矩阵和张量微分方程的未知解。基于对低秩流形切空间上的正交投影进行分裂，近年来提出了一种新的时间积分器，用于获得低秩矩阵和低秩张量列的近似.根据标准理论，Lie{Trotter和斯特朗投影分裂方法分别具有一阶和二阶精度，但是当低秩近似具有小奇异值时，通常的误差界被打破。当解的奇异值衰减而没有明显的间隙时，或者当解的有效秩被高估时，就会发生这种情况。另一方面，当给定的时间相关矩阵或张量已经具有规定的秩时，积分器是精确的。我们提供了一个错误的分析，统一这些属性。我们表明，在精确解是一个低秩矩阵或张量列的ε-扰动的情况下，投影分裂积分器的误差是有利的有界的ε-和步长，独立于奇异值的小。这样的结果不适用于任何标准积分器。数值实验验证了理论的正确性。
Low-rank approximations to large time-dependent matrices and tensors are the subject of this paper. These matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations. Based on splitting the orthogonal projection onto the tangent space of the low-rank manifold, novel time integrators for obtaining approximations by low-rank matrices and low-rank tensor trains were recently proposed. By standard theory, the Lie{Trotter and Strang projector-splitting methods are first and second order accurate, respectively, but the usual error bounds break down when the low-rank approximation has small singular values. This happens when the singular values of the solution decay without a distinct gap or when the effective rank of the solution is overestimated. On the other hand, the integrators are exact when given time-dependent matrices or tensors are already of the prescribed rank. We provide an error analysis which unifies these properties. We show that in cases where the exact solution is an epsilon-perturbation of a low-rank matrix or tensor train, the error of the projector-splitting integrator is favorably bounded in terms of epsilon and the stepsize, independently of the smallness of the singular values. Such a result does not hold for any standard integrator. Numerical experiments illustrate the theory.