In this paper, explicit stable integrators based on symplectic and contact geometries are proposed for a non-autonomous ordinarily differential equation (ODE) found in improving convergence rate of Nesterov's accelerated gradient method. Symplectic geometry is known to be suitable for describing Hamiltonian mechanics, and contact geometry is known as an odd-dimensional counterpart of symplectic geometry. Moreover, a procedure, called symplectization, is a known way to construct a symplectic manifold from a contact manifold, yielding Hamiltonian systems from contact ones. It is found in this paper that a previously investigated non-autonomous ODE can be written as a contact Hamiltonian system. Then, by symplectization of a non-autonomous contact Hamiltonian vector field expressing the non-autonomous ODE, novel symplectic integrators are derived. Because the proposed symplectic integrators preserve hidden symplectic and contact structures in the ODE, they should be more stable than the Runge-Kutta method. Numerical experiments demonstrate that, as expected, the second-order symplectic integrator is stable and high convergence rates are achieved.

中文翻译：

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Nesterov型加速方法的快速辛积分器

在本文中，针对在提高 Nesterov 加速梯度方法的收敛速度中发现的非自治常微分方程 (ODE)，提出了基于辛几何和接触几何的显式稳定积分器。已知辛几何适合描述哈密顿力学，接触几何被称为辛几何的奇维对应物。此外，一个称为辛化的过程是一种从接触流形构造辛流形的已知方法，从接触流形产生哈密顿系统。在本文中发现，先前研究的非自治常微分方程可以写成接触哈密顿系统。然后，通过表达非自治常微分方程的非自治接触哈密顿向量场的辛化，推导出新的辛积分器。因为所提出的辛积分器在 ODE 中保留了隐藏的辛和接触结构，所以它们应该比 Runge-Kutta 方法更稳定。数值实验表明，正如预期的那样，二阶辛积分器是稳定的，并且达到了高收敛速度。