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.

中文翻译：

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对称和反对称低秩矩阵和 Tucker 张量的时间积分

提出了一个数值积分器，用于计算大型对称或偏对称时间相关矩阵的对称或偏对称低秩近似，这些矩阵要么是明确给出的，要么是矩阵微分方程的未知解。给出了一种相关算法，用于通过低多重线性秩的对称或反对称 Tucker 张量逼近对称或反对称时间相关张量。所提出的对称或反对称低秩积分器不同于最近提出的用于动态低秩逼近的投影分割积分器，后者不保持对称性或反对称性。然而，结果表明（反对）对称低阶积分器保留了投影仪分裂积分器的有利特性：与应用于动态低秩逼近微分方程的标准积分方法相比，给定的低秩时间相关矩阵和张量被精确再现，并且误差行为对于小奇异值的存在具有鲁棒性。数值实验说明了所提出的积分器的行为。