Fast symplectic integrator for Nesterov-type acceleration method
Center of Mathematics for Artificial Intelligence and Data Science（人工智能与数据科学数学中心）；Chubu University（中部大学）；The Institute of Statistical Mathematics（统计数学研究所）；RIKEN AIP（理化学研究所先进智能项目中心）
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.
A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems
Qufu Normal University（曲阜师范大学）；University of Tübingen（蒂宾根大学）；Nanjing University（南京大学）
Preservation of stability properties near fixed points of linear hamiltonian systems by symplectic integrators
Harbin Institute of Technology（哈尔滨工业大学）；University of Washington（华盛顿大学）；Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
High order symplectic integrators based on continuous-stage Runge-Kutta Nystrom methods
College of Mathematics and Statistics, Changsha University of Science and Technology（长沙理工大学数学与统计学院）；LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；School of Science, East China Jiaotong University（华东交通大学理学院）
Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka--Volterra model
School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）；School of Science, East China Jiaotong University（华东交通大学理学院）
Exponential Stability Estimate of Symplectic Integrators for Integrable Hamiltonian Systems
Inner Mongolia University（内蒙古大学）；Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）；South China University of Technology（华南理工大学）
Institute of Fundamental Sciences, Massey University（马斯特里赫特大学基础科学研究所）；Mathematical Sciences, Chalmers University of Technology（查尔姆斯理工大学数学科学系）；Mathematics and Mathematical Statistics, Umeå Universitet（于默奥大学数学与数理统计系）
Multi-symplectic Preserving Integrator for the Schrödinger Equation with Wave Operator
Jiangxi Normal University（江西师范大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, CAS（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
High precision Symplectic Integrators for the Solar System
Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC（巴黎天文动力学与系统研究所）；Instituto de Matemática Multidisciplinar, Universitat Politécnica de València（瓦伦西亚理工大学多学科数学研究所）；Institut de Matemàtiques i Aplicacions de Castelló and Departament de Matemàtiques, Universitat Jaume I（卡斯特利翁数学与应用研究所及马德里卡洛斯三世大学数学系）；Konputazio Zientziak eta A.A. saila, Informatika Fakultatea, EHU/UPV（巴斯克大学计算机学院）
Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems
Inner Mongolia University（内蒙古大学）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院华罗庚数学重点实验室）
Collective Lie-Poisson integrators on $\mathbf{R}^{3}$
Institute of Fundamental Sciences, Massey University（梅西大学基础科学研究所）；Department of Mathematical Sciences, Chalmers University of Technology（查尔姆斯理工大学数学科学系）；Department of Mathematics, University of Bergen（卑尔根大学数学系）
Numerical Construction of Quasi-Periodic Solutions Beyond Symplectic Integrators
University of Chinese Academy of Sciences（中国科学院大学）；Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）
Butcher series for Hamiltonian Poisson integrators through symplectic groupoids
Univ Rennes, INRIA (Research team MINGuS), IRMAR (CNRS UMR 6625) and ENS Rennes（雷恩大学、INRIA（MINGuS研究团队）、IRMAR（CNRS UMR 6625）和雷恩高等师范学院）；Göttingen Mathematisches Institut, Georg-August-Universität Göttingen（哥廷根数学研究所，格奥尔格-奥古斯特哥廷根大学）
