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 Projected Ralston--Hermite (PRH) 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：在受限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保持在流形上的成熟工具。近期研究表明，回缩对流形上的其他计算任务（包括插值任务）同样具有实用价值。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，这与动态低秩逼近技术密切相关。事实上，任何回缩都能导出数值积分器，反之亦然，某些动态低秩逼近技术也与回缩存在直接关联。作为后者的实例，我们提出一种名为KLS回缩的新型回缩方法，该方法源于动态低秩逼近的非传统积分器。本文还阐述了如何利用回缩恢复已知的动态低秩逼近技术并设计新方法。特别地，本研究提出了两种适用于一般流形上微分方程的新型数值积分方案：加速前向欧拉方法和投影Ralston-Hermite方法。两种方法均以回缩作为逼近流形上曲线的工具，并被证明具有三阶局部截断误差。基于经典动态低秩逼近算例的数值实验揭示了这些新方法的优势与不足。