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Title: Tau leaping of stiff stochastic chemical systems via local central limit approximation

Abstract

Stiffness manifests in stochastic dynamic systems in a more complex manner than in deterministic systems; it is not only important for a time-stepping method to remain stable but it is also important for the method to capture the asymptotic variances accurately. In the context of stochastic chemical systems, time stepping methods are known as tau leaping. Well known existing tau leaping methods have shortcomings in this regard. The implicit tau method is far more stable than the trapezoidal tau method but underestimates the asymptotic variance. On the other hand, the trapezoidal tau method which estimates the asymptotic variance exactly for linear systems suffers from the fact that the transients of the method do not decay fast enough in the context of very stiff systems. We propose a tau leaping method that possesses the same stability properties as the implicit method while it also captures the asymptotic variance with reasonable accuracy at least for the test system S{sub 1}↔S{sub 2}. The proposed method uses a central limit approximation (CLA) locally over the tau leaping interval and is referred to as the LCLA-τ. The CLA predicts the mean and covariance as solutions of certain differential equations (ODEs) and for efficiency we solvemore » these using a single time step of a suitable low order method. We perform a mean/covariance stability analysis of various possible low order schemes to determine the best scheme. Numerical experiments presented show that LCLA-τ performs favorably for stiff systems and that the LCLA-τ is also able to capture bimodal distributions unlike the CLA itself. The proposed LCLA-τ method uses a split implicit step to compute the mean update. We also prove that any tau leaping method employing a split implicit step converges in the fluid limit to the implicit Euler method as applied to the fluid limit differential equation.« less

Authors:
 [1];  [1]
  1. Department of Mathematics and Statistics, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 (United States)
Publication Date:
OSTI Identifier:
22233582
Resource Type:
Journal Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 242; Other Information: Copyright (c) 2013 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ACCURACY; APPROXIMATIONS; ASYMPTOTIC SOLUTIONS; DIFFERENTIAL EQUATIONS; EFFICIENCY; FLEXIBILITY; FLUIDS; KINETICS; STABILITY; STOCHASTIC PROCESSES; TRANSIENTS

Citation Formats

Yang, Yushu, and Rathinam, Muruhan. Tau leaping of stiff stochastic chemical systems via local central limit approximation. United States: N. p., 2013. Web. doi:10.1016/J.JCP.2013.02.011.
Yang, Yushu, & Rathinam, Muruhan. Tau leaping of stiff stochastic chemical systems via local central limit approximation. United States. https://doi.org/10.1016/J.JCP.2013.02.011
Yang, Yushu, and Rathinam, Muruhan. 2013. "Tau leaping of stiff stochastic chemical systems via local central limit approximation". United States. https://doi.org/10.1016/J.JCP.2013.02.011.
@article{osti_22233582,
title = {Tau leaping of stiff stochastic chemical systems via local central limit approximation},
author = {Yang, Yushu and Rathinam, Muruhan},
abstractNote = {Stiffness manifests in stochastic dynamic systems in a more complex manner than in deterministic systems; it is not only important for a time-stepping method to remain stable but it is also important for the method to capture the asymptotic variances accurately. In the context of stochastic chemical systems, time stepping methods are known as tau leaping. Well known existing tau leaping methods have shortcomings in this regard. The implicit tau method is far more stable than the trapezoidal tau method but underestimates the asymptotic variance. On the other hand, the trapezoidal tau method which estimates the asymptotic variance exactly for linear systems suffers from the fact that the transients of the method do not decay fast enough in the context of very stiff systems. We propose a tau leaping method that possesses the same stability properties as the implicit method while it also captures the asymptotic variance with reasonable accuracy at least for the test system S{sub 1}↔S{sub 2}. The proposed method uses a central limit approximation (CLA) locally over the tau leaping interval and is referred to as the LCLA-τ. The CLA predicts the mean and covariance as solutions of certain differential equations (ODEs) and for efficiency we solve these using a single time step of a suitable low order method. We perform a mean/covariance stability analysis of various possible low order schemes to determine the best scheme. Numerical experiments presented show that LCLA-τ performs favorably for stiff systems and that the LCLA-τ is also able to capture bimodal distributions unlike the CLA itself. The proposed LCLA-τ method uses a split implicit step to compute the mean update. We also prove that any tau leaping method employing a split implicit step converges in the fluid limit to the implicit Euler method as applied to the fluid limit differential equation.},
doi = {10.1016/J.JCP.2013.02.011},
url = {https://www.osti.gov/biblio/22233582}, journal = {Journal of Computational Physics},
issn = {0021-9991},
number = ,
volume = 242,
place = {United States},
year = {Sat Jun 01 00:00:00 EDT 2013},
month = {Sat Jun 01 00:00:00 EDT 2013}
}