SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2949-A2980, January 2021.
It is well known that symplectic integrators lose their near energy preservation properties when variable time-steps are used. The most common approach to combining adaptive time-steps and symplectic integrators involves the Poincaré transformation of the original Hamiltonian. In this article, we provide a framework for the construction of variational integrators using the Poincaré transformation. Since the transformed Hamiltonian is typically degenerate, the use of Hamiltonian variational integrators based on Type II or Type III generating functions is required instead of the more traditional Lagrangian variational integrators based on Type I generating functions. Error analysis is provided, and numerical tests based on the Taylor variational integrator approach in [J. M. Schmitt, T. Shingel, and M. Leok, BIT, 58 (2018), pp. 457--488] to time-adaptive variational integration of Kepler's 2-body problem are presented. Finally, we use our adaptive framework together with the variational approach to accelerated optimization presented in [A. Wibisono, A. Wilson, and M. Jordan, Proc. Natl. Acad. Sci. USA, 113 (2016), pp. E7351--E7358] to design efficient variational and nonvariational explicit integrators for symplectic accelerated optimization.

中文翻译：

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自适应哈密顿变分积分器及其在辛加速优化中的应用

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2949-A2980 页，2021 年 1 月。
众所周知，当使用可变时间步长时，辛积分器会失去它们的近能量保存特性。结合自适应时间步长和辛积分器的最常见方法涉及原始哈密顿量的庞加莱变换。在本文中，我们提供了使用庞加莱变换构建变分积分器的框架。由于变换后的哈密顿量通常是退化的，因此需要使用基于类型 II 或类型 III 生成函数的哈密顿变分积分器，而不是基于类型 I 生成函数的更传统的拉格朗日变分积分器。[JM Schmitt, T. Shingel, and M. Leok, BIT, 58 (2018), pp.] 提供了误差分析和基于泰勒变分积分器方法的数值测试。457--488] 提出了开普勒二体问题的时间自适应变分积分。最后，我们将我们的自适应框架与 [A. Wibisono、A. Wilson 和 M. Jordan，Proc。纳特尔。阿卡德。科学。USA, 113 (2016), pp. E7351--E7358] 设计用于辛加速优化的高效变分和非变分显式积分器。