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

  • Yunfeng Cai (a1), Tiejun Li (a1), Jiushu Shao (a2) and Zhiming Wang (a1)

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.

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Corresponding author

*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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[1] Bilger, R.W., Conditional moment closure for turbulent reacting flow, Phys. Fluids A, 5 (1993), pp. 436444.
[2] Boffi, D., Finite element approximation of eigenvalue problems, Acta Numer., 19 (2010), pp. 1120.
[3] Bourgault, Y., Broizat, D. and Jabin, P.-E., Convergence rate for the method of moments with linear closure relations, arXiv:1206.4831v1.
[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. 547571.
[5] Cai, Z., Fan, Y. and Li, R., Globally hyperbolic regularization of grad's moment system, Commun. Pure Appl. Math., 67 (2014), pp. 464518.
[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. 29682973.
[7] W. E, , Khanin, K., Mazel, A. and Sinai, Y., Invariant measures for burgers equation with stochastic forcing, Ann. Math., 151 (2000), pp. 877960.
[8] Frankel, D. and Smit, B., Understanding Molecular Simulation, 2nd edition, Academic Press, San Diego, 2001.
[9] Frisch, U., Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1996.
[10] Ghanem, R. and Spanos, P., Stochastic Finite Element: A Spectral Approach, Springer-Verlag, New York, 1991.
[11] Gillespie, D. T., Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem., 58 (2007), pp. 3555.
[12] Gillespie, C. S., Moment-closure approximations for mass-action models, IET Sys. Bio., 3 (2009), pp. 5258.
[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. 687706.
[14] Lee, C., Kim, K. and Kim, P., A moment closure method for stochastic reaction networks, J. Chem. Phys., 130 (2009), 134107.
[15] Leggett, A. J. et al., Dynamics of the dissipative two-state system, Rev. Mod. Phys., 59 (1987), pp. 185.
[16] McAdams, H. H. and Arkin, A., Stochastic mechanisms in gene expression, Proc. Natl. Acad. Sci., 94 (1997), pp. 814819.
[17] Mori, H., Transport, Collective Motion, and Brownian Motion, Prog. Theor. Phys., 33 (1965), pp. 423455.
[18] Orszag, S. A. and Bissonnette, L. R., Dynamical properties of truncated wiener hermite expansions, Phys. Fluids, 10 (1967), pp. 26032613.
[19] Schmiedl, T. and Seifert, U., Stochastic thermodynamics of chemical reaction networks, J. Chem. Phys., 126 (2007), 044101.
[20] Shao, J., Decoupling quantum dissipation interaction via stochastic fields, J. Chem. Phys., 120(11) (2004), pp. 50535056.
[21] Szegö, G., Orthogonal Polynomials, 4th ed., Amer. Math. Soc., Rhode Island, 1975.
[22] Xiu, D. and Karniadakis, G. E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619644.
[23] Xiu, D. and Hesthaven, J. S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 11181139.
[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. 216221.
[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.

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

  • Yunfeng Cai (a1), Tiejun Li (a1), Jiushu Shao (a2) and Zhiming Wang (a1)

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