论文标题
时间自适应拉格朗日变异集成符，以加速优化。
Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds
论文作者
论文摘要
最近在Wibisono等人的规范矢量空间和Riemannian歧管上引入了一个加速优化的差异框架。 （2016）以及Duruisseaux和Leok（2021）。据观察，在数值集成中，谨慎地结合了时间匹配性和符号性，可能会导致计算效率的显着提高。然而，众所周知，当使用可变的时间步长时，符号整合物会失去其近乎能量保存的性能。规避此问题的最常见方法涉及哈密顿侧的繁殖性转化，并在Duruisseaux等人中使用。 （2021）构建有效的显式算法，以进行符号加速优化。但是，哈密顿变分集成剂的当前表述对诸如Riemannian歧管和谎言组等更一般的空间没有内在的意义。相比之下，拉格朗日的变分集成剂在流形上定义明确，因此我们在这里开发了拉格朗日变量积分器中时间适应性的框架，并使用所得的几何积分器来解决规范矢量空间和谎言组的优化问题。
A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in Wibisono et al. (2016) and Duruisseaux and Leok (2021). It was observed that a careful combination of timeadaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincare transformation on the Hamiltonian side, and was used in Duruisseaux et al. (2021) to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on normed vector spaces and Lie groups.
