Due to its reduced memory and computational demands, dynamical low-rank approximation (DLRA) has sparked significant interest in multiple research communities. A central challenge in DLRA is the development of time integrators that are robust to the curvature of the manifold of low-rank matrices. Recently, a parallel robust time integrator that permits dynamic rank adaptation and enables a fully parallel update of all low-rank factors was introduced. Despite its favorable computational efficiency, the construction as a first-order approximation to the augmented basis-update & Galerkin integrator restricts the parallel integrator's accuracy to order one. In this work, an extension to higher order is proposed by a careful basis augmentation before solving the matrix differential equations of the factorized solution. A robust error bound with an improved dependence on normal components of the vector field together with a norm preservation property up to small terms is derived. These analytic results are complemented and demonstrated through a series of numerical experiments.

翻译：因其降低的内存和计算需求，动态低秩近似（DLRA）已在多个研究领域引发广泛关注。DLRA的核心挑战在于开发对低秩矩阵流形曲率具有鲁棒性的时间积分器。近期，一种支持动态秩自适应并能实现所有低秩因子完全并行更新的鲁棒并行时间积分器被提出。尽管该积分器具有显著的计算效率优势，但其作为增广基更新与伽辽金积分器的一阶近似构造，将并行积分器的精度限制为一阶。本研究通过在对分解解进行矩阵微分方程求解前进行谨慎的基增广，提出了一种扩展至高阶的方法。我们推导了关于向量场法向分量具有改进依赖性的鲁棒误差界，以及保留至小项的范数保持性质。通过一系列数值实验，这些分析结果得到了补充与验证。