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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) Probabilistic methods, simulation and stochastic differential equations Numerical linear algebra

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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References

[1] Bilger, R.W., Conditional moment closure for turbulent reacting flow, Phys. Fluids A, 5 (1993), pp. 436–444.Google Scholar
[2] Boffi, D., Finite element approximation of eigenvalue problems, Acta Numer., 19 (2010), pp. 1–120.Google Scholar
[3] Bourgault, Y., Broizat, D. and Jabin, P.-E., Convergence rate for the method of moments with linear closure relations, arXiv:1206.4831v1.Google Scholar
[4] Cai, Z., Fan, Y. and Li, R., Globally hyperbolic regularization of grad's moment system in one dimensional space, Commun. Math. Sci., 11 (2012), pp. 547–571.Google Scholar
[5] Cai, Z., Fan, Y. and Li, R., Globally hyperbolic regularization of grad's moment system, Commun. Pure Appl. Math., 67 (2014), pp. 464–518.Google Scholar
[6] Chorin, A. J., Hald, O. H., and Kupferman, R., Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci., 97 (2000), pp. 2968–2973.CrossRefGoogle ScholarPubMed
[7] W. E, , Khanin, K., Mazel, A. and Sinai, Y., Invariant measures for burgers equation with stochastic forcing, Ann. Math., 151 (2000), pp. 877–960.Google Scholar
[8] Frankel, D. and Smit, B., Understanding Molecular Simulation, 2nd edition, Academic Press, San Diego, 2001.Google Scholar
[9] Frisch, U., Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1996.Google Scholar
[10] Ghanem, R. and Spanos, P., Stochastic Finite Element: A Spectral Approach, Springer-Verlag, New York, 1991.Google Scholar
[11] Gillespie, D. T., Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem., 58 (2007), pp. 35–55.Google Scholar
[12] Gillespie, C. S., Moment-closure approximations for mass-action models, IET Sys. Bio., 3 (2009), pp. 52–58.Google Scholar
[13] Hou, T. Y., Luo, W., Rozovskii, B. and Zhou, H., Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics, J. Comp. Phys., 216 (2006), pp. 687–706.Google Scholar
[14] Lee, C., Kim, K. and Kim, P., A moment closure method for stochastic reaction networks, J. Chem. Phys., 130 (2009), 134107.Google Scholar
[15] Leggett, A. J. et al., Dynamics of the dissipative two-state system, Rev. Mod. Phys., 59 (1987), pp. 1–85.Google Scholar
[16] McAdams, H. H. and Arkin, A., Stochastic mechanisms in gene expression, Proc. Natl. Acad. Sci., 94 (1997), pp. 814–819.Google Scholar
[17] Mori, H., Transport, Collective Motion, and Brownian Motion, Prog. Theor. Phys., 33 (1965), pp. 423–455.Google Scholar
[18] Orszag, S. A. and Bissonnette, L. R., Dynamical properties of truncated wiener hermite expansions, Phys. Fluids, 10 (1967), pp. 2603–2613.Google Scholar
[19] Schmiedl, T. and Seifert, U., Stochastic thermodynamics of chemical reaction networks, J. Chem. Phys., 126 (2007), 044101.Google Scholar
[20] Shao, J., Decoupling quantum dissipation interaction via stochastic fields, J. Chem. Phys., 120(11) (2004), pp. 5053–5056.Google Scholar
[21] Szegö, G., Orthogonal Polynomials, 4th ed., Amer. Math. Soc., Rhode Island, 1975.Google Scholar
[22] Xiu, D. and Karniadakis, G. E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619–644.Google Scholar
[23] Xiu, D. and Hesthaven, J. S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 1118–1139.Google Scholar
[24] Yan, Y., Yang, F., Liu, Y., and Shao, J., Hierarchical approach based on stochastic decoupling to dissipative systems, Chem. Phys. Lett., 395 (2004), pp. 216–221.Google Scholar
[25] Zhou, Y. and Shao, J., Solving the spin-boson model of strong dissipation with flexible random-deterministic scheme, J. Chem. Phys., 128 (2008), 034106.Google Scholar