论文标题
一种新的分裂算法，用于由基质歧管的纤维束结构促进的动态低级别近似
A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds
论文作者
论文摘要
在本文中，我们提出了一种新的分裂算法，用于由固定级别矩阵集的纤维束结构激励的动态低级别近似。我们首先介绍了固定级别矩阵集的几何描述，该矩阵依赖于矩阵的自然参数化。更确切地说，它具有分析主捆绑包的结构，并具有对本地图表的明确描述。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集成器。所得算法可以解释为投影算子在低级矩阵歧管的切线空间上的特定分裂。事实证明，在某些特殊情况下是准确的。数值实验证实了这一结果，并说明了所提出算法的鲁棒性。
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.
