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Analysis of the Closure Approximation for a Class of Stochastic Differential Equations

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Numerical linear algebra Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  09 May 2017

Yunfeng Cai ,
Tiejun Li ,
Jiushu Shao  and
Zhiming Wang
Yunfeng Cai*
Affiliation:
Laboratory of Mathematics and Applied Mathematics and School of Mathematical Sciences, Peking University, Beijing 100871, China
Tiejun Li*
Affiliation:
Laboratory of Mathematics and Applied Mathematics and School of Mathematical Sciences, Peking University, Beijing 100871, China
Jiushu Shao*
Affiliation:
College of Chemistry, Beijing Normal University, Beijing 100875, China
Zhiming Wang*
Affiliation:
Laboratory of Mathematics and Applied Mathematics and School of Mathematical Sciences, Peking University, Beijing 100871, China
*
*Corresponding author. Email addresses:yfcai@math.pku.edu.cn (Y. F. Cai), wangzm@pku.edu.cn (Z. M. Wang), tieli@pku.edu.cn (T. J. Li), jiushu@bnu.edu.cn (J. S. Shao)
*Corresponding author. Email addresses:yfcai@math.pku.edu.cn (Y. F. Cai), wangzm@pku.edu.cn (Z. M. Wang), tieli@pku.edu.cn (T. J. Li), jiushu@bnu.edu.cn (J. S. Shao)
*Corresponding author. Email addresses:yfcai@math.pku.edu.cn (Y. F. Cai), wangzm@pku.edu.cn (Z. M. Wang), tieli@pku.edu.cn (T. J. Li), jiushu@bnu.edu.cn (J. S. Shao)
*Corresponding author. Email addresses:yfcai@math.pku.edu.cn (Y. F. Cai), wangzm@pku.edu.cn (Z. M. Wang), tieli@pku.edu.cn (T. J. Li), jiushu@bnu.edu.cn (J. S. Shao)
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Abstract

Motivated by the numerical study of spin-boson dynamics in quantum open systems, we present a convergence analysis of the closure approximation for a class of stochastic differential equations. We show that the naive Monte Carlo simulation of the system by direct temporal discretization is not feasible through variance analysis and numerical experiments. We also show that the Wiener chaos expansion exhibits very slow convergence and high computational cost. Though efficient and accurate, the rationale of the moment closure approach remains mysterious. We rigorously prove that the low moments in the moment closure approximation of the considered model are of exponential convergence to the exact result. It is further extended to more general nonlinear problems and applied to the original spin-boson model with similar structure.

Keywords

Moment closurespin-boson dynamicsconvergence analysisbanded structure

MSC classification

Secondary: 65C30: Stochastic differential and integral equations 65F99: None of the above, but in this section 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.)
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 299 - 330
Copyright
Copyright © Global-Science Press 2017 

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