In this paper, we propose a new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of the set of fixed rank matrices. We first introduce a geometric description of the set of fixed rank matrices which relies on a natural parametrization of matrices. More precisely, it is endowed with the structure of analytic principal bundle, with an explicit description of local charts. For matrix differential equations, we introduce a first order numerical integrator working in local coordinates. The resulting algorithm can be interpreted as a particular splitting of the projection operator onto the tangent space of the low-rank matrix manifold. It is proven to be exact in some particular case. Numerical experiments confirm this result and illustrate the robustness of the proposed algorithm.

在本文中，我们提出了一种新的分裂算法，用于由一组固定秩矩阵的纤维丛结构驱动的动态低秩逼近。我们首先介绍一组固定秩矩阵的几何描述，它依赖于矩阵的自然参数化。更准确地说，它具有解析主丛的结构，具有局部图的显式描述。对于矩阵微分方程，我们引入了在局部坐标中工作的一阶数值积分器。由此产生的算法可以解释为投影算子在低秩矩阵流形的切线空间上的特定分裂。它被证明在某些特定情况下是准确的。数值实验证实了这一结果，并说明了所提出算法的鲁棒性。