In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Ralston-Hermite (RH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.

翻译：在约束于光滑流形的优化问题求解算法中，回缩是确保迭代点始终位于流形上的成熟工具。最近研究表明，回缩同样适用于流形上的其他计算任务（如插值问题）。本文考虑将回缩应用于固定秩矩阵流形上微分方程的数值积分，该问题与动态低秩逼近技术密切相关。事实上，任何回缩均可导出数值积分器，反之，特定动态低秩逼近技术亦与回缩存在直接关联。作为后者的实例，我们引入一种名为KLS回缩的新型回缩方法，该方法源自动态低秩逼近中的非常规积分器。本文同时阐释如何利用回缩重构现有动态低秩逼近技术并设计新方法。特别地，本研究提出两种适用于一般流形上微分方程的新型数值积分格式：加速前向欧拉法和拉尔斯顿-埃尔米特法。两种方法均以回缩为工具对流形上的曲线进行近似。理论证明，这两种方法的局部截断误差均达到三阶精度。基于经典动态低秩逼近例子的数值实验揭示了这些新方法的优势与局限性。