发表机构
University of Washington（华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对动态低秩近似对小奇异值下时间积分器的精度需求，构造二阶和三阶鲁棒投影分裂龙格-库塔方法，证明其误差与小奇异值无关，适用于秩为r的计算场景。
AI 中文摘要
动态低秩近似需要在存在小奇异值时仍保持精度的时间积分器。我们构造了二阶和三阶的鲁棒投影分裂龙格-库塔方法，其核心特征是共用基阶段构造：每个内部阶段和终点均通过将Lubich与Oseledets提出的实用投影分裂算法应用于龙格-库塔增量得到，且始终基于时间步开始时的因子和行空间。在一致有界性、利普希茨、光滑性和法向分量假设下，当投影分裂分解的秩为r时，中点法的局部误差为$C(h^3+h\varepsilon_r)$、全局误差为$C(\delta+\varepsilon_r+h^2)$，三阶方法的局部误差为$C(h^4+h\varepsilon_r)$、全局误差为$C(\delta+\varepsilon_r+h^3)$，其中$\varepsilon_r$为向量场法向分量的界，$\delta$为初始误差，常数与小奇异值无关，所有阶段和输出的秩均为r，计算过程中始终保留原始基宽度。
英文摘要
We develop a general framework for constructing robust high-order projector-splitting integrators for dynamical low-rank approximation. For a prescribed matrix increment, we show that the standard K-S-L projector-splitting step is equivalent to a reduced K-L step, thereby eliminating the explicit backward S-step. We then establish a central relaxed exactness property of the standard projector-splitting integrator: the rank-$r$ approximation inherits the accuracy of the numerical matrix increment, with an error bound independent of small singular values. The resulting schemes evolve fixed rank-$r$ factors and require neither basis augmentation nor rank truncation. As concrete examples, we combine the framework with selected second- and third-order Runge--Kutta methods to obtain robust high-order projector-splitting integrators. Numerical experiments confirm the predicted uniform convergence rates with respect to small singular values.
