一类高振荡随机哈密顿系统的李代数方法

doi:10.7523/j.issn.2095-6134.2019.01.002

中国科学院大学学报 ›› 2019, Vol. 36 ›› Issue (1): 5-10.DOI: 10.7523/j.issn.2095-6134.2019.01.002

一类高振荡随机哈密顿系统的李代数方法

阮家麟, 王丽瑾

中国科学院大学数学科学学院, 北京 100049

收稿日期:2017-09-26 修回日期:2018-01-04 发布日期:2019-01-15
通讯作者: 王丽瑾,E-mail:ljwang@ucas.ac.cn
基金资助:
Supported by the National Nature Science Foundation of China (11471310,11071251)

A Lie algebraic approach for a class of highly oscillatory stochastic Hamiltonian systems

RUAN Jialin, WANG Lijin

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received:2017-09-26 Revised:2018-01-04 Published:2019-01-15
Supported by:
Supported by the National Nature Science Foundation of China (11471310,11071251)

摘要/Abstract

摘要： 为一类高振荡随机哈密顿系统提出一种李代数数值方法。对一个具体的高振荡随机哈密顿系统，给出两个基于李代数方法的数值格式，并证明它们近似保辛结构。通过数值实验展示这两种格式的根均方收敛阶，以及它们在数值求解该高振荡随机哈密顿系统中的有效性和优越性。

关键词: 随机微分方程数值解, 随机哈密顿系统, 高振荡随机微分方程, 算子分裂方法, 李代数方法

Abstract: In this work, we propose a Lie algebraic approach for numerically solving a class of highly oscillatory stochastic Hamiltonian systems (SHSs). For a concrete highly oscillatory SHS, we construct two numerical schemes based on the Lie algebraic approach, and prove their near preservation of the symplecticity. We also show by numerical tests their root mean-square convergence orders, as well as their effectiveness and merits in solving the highly oscillatory SHS.

Key words: numerical solutions of stochastic differential equations, stochastic Hamiltonian systems, highly oscillatory SDEs, operator splitting methods, Lie algebraic approach

中图分类号:

O241.8

阮家麟, 王丽瑾. 一类高振荡随机哈密顿系统的李代数方法[J]. 中国科学院大学学报, 2019, 36(1): 5-10.

RUAN Jialin, WANG Lijin. A Lie algebraic approach for a class of highly oscillatory stochastic Hamiltonian systems[J]. , 2019, 36(1): 5-10.

参考文献

[1] Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM review, 2001, 43(3):525-546.
[2] Misawa T. A Lie algebraic approach to numerical integration of stochastic differential equations[J]. SIAM Journal on Scientific Computing, 2001, 23(3):866-890.
[3] Feng K, Qin M. Symplectic geometric algorithms for hamiltonian systems[M]. Berlin:Springer, 2010.
[4] Milstein G N, Repin Y M, Tretyakov M V. Symplectic integration of Hamiltonian systems with additive noise[J]. SIAM Journal on Numerical Analysis, 2002,39(6):2066-2088.
[5] Milstein G N, Repin Y M, Tretyakov M V. Numerical methods for stochastic systems preserving symplectic structure[J]. SIAM Journal on Numerical Analysis, 2002, 40(4):1583-1604.
[6] Kloeden P E, Platen E. Numerical solution of stochastic differential equations[M]. Springer-Verlag, 1992.
[7] Milstein G N. Numerical integration of stochastic differential equations[M]. Springer Science & Business Media, 1994.
[8] Milstein G N, Tretyakov M V. Stochastic numerics for mathematical physics[M]. Springer Science & Business Media, 2013.
[9] Chartier P, Makazaga J, Murua A, et al. Multi-revolution composition methods for highly oscillatory differential equations[J]. Numerische Mathematik, 2014, 128(1):167-192.
[10] Vilmart G. Weak second order multirevolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise[J]. Siam Journal on Scientific Computing, 2015, 36(4):A1770-A1796.
[11] Cohen D. On the numerical discretisation of stochastic oscillators[J]. Mathematics & Computers in Simulation, 2012, 82(8):1478-1495.
[12] Cohen D. Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations[J]. Numerische Mathematik, 2012, 121(1):1-29.
[13] Hairer E, Lubich C, Wanner G. Geometric numerical integration:structure-preserving algorithms for ordinary differential equations[J]. Series in Computational Mathematics, 2006, 25(1):805-882.
[14] Kunita H. On the representation of solutions of stochastic differential equations[M]. Séminaire de Probabilités XIV 1978/79. Springer Berlin Heidelberg, 1980:282-304.

一类高振荡随机哈密顿系统的李代数方法

A Lie algebraic approach for a class of highly oscillatory stochastic Hamiltonian systems

PDF (PC)

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 7

编辑推荐

Metrics

本文评价

访问统计

联系我们

[1]	王余超, 王丽瑾. 一类随机泊松系统的解的研究[J]. 中国科学院大学学报, 0, (): 103-103.
[2]	王鹏钧, 王丽瑾. 线性随机泊松系统基于Padé近似的随机泊松积分子[J]. 中国科学院大学学报, 2021, 38(2): 160-170.
[3]	孙丽莹, 王丽瑾. 带有一个噪声的随机微分方程的全隐式格式[J]. 中国科学院大学学报, 2016, 33(3): 306-310.
[4]	王辉, 徐明海. 一种新的径向导热界面参数插值格式[J]. 中国科学院大学学报, 2011, 28(5): 624-629.
[5]	王艳丽赵平福. 一类高振荡常微分方程数值解法的误差分析[J]. 中国科学院大学学报, 2007, 24(2): 154-160.
[6]	肖存英胡雄. 启示性方法用于分析有限差分方程的计算稳定性（英文） [J]. 中国科学院大学学报, 2007, 24(2): 179-185.
[7]	朱伟; 郑继明. 一类多时滞的向量差分方程的渐近稳定性（英文）[J]. 中国科学院大学学报, 2006, 23(5): 593-596.