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.

翻译：在本文中,我们提出了一套由固定级矩阵的纤维捆绑结构驱动的动态低级近似新分解算法。我们首先对一组依赖自然矩阵的平衡化的固定级矩阵进行几何描述。更准确地说,它具有分析式主矩阵结构,对本地图表有明确描述。对于矩阵差异方程式,我们引入了第一个按顺序排列的以本地坐标工作的数字集成器。由此产生的算法可以被解释为投影操作员与低级矩阵多层相近空间的特殊分割。在某些特定情况下,它被证明是准确的。数字实验证实了这一结果,并说明了拟议算法的稳健性。