BIT Numerical Mathematics

, Volume 56, Issue 1, pp 189–239 | Cite as

Multilevel hybrid Chernoff tau-leap



In this work, we extend the hybrid Chernoff tau-leap method to the multilevel Monte Carlo (MLMC) setting. Inspired by the work of Anderson and Higham on the tau-leap MLMC method with uniform time steps, we develop a novel algorithm that is able to couple two hybrid Chernoff tau-leap paths at different levels. Using dual-weighted residual expansion techniques, we also develop a new way to estimate the variance of the difference of two consecutive levels and the bias. This is crucial because the computational work required to stabilize the coefficient of variation of the sample estimators of both quantities is often unaffordable for the deepest levels of the MLMC hierarchy. Our method bounds the global computational error to be below a prescribed tolerance, TOL, within a given confidence level. This is achieved with nearly optimal computational work. Indeed, the computational complexity of our method is of order \(\mathcal {O}\left( \textit{TOL}^{-2}\right) \), the same as with an exact method, but with a smaller constant. Our numerical examples show substantial gains with respect to the previous single-level approach and the Stochastic Simulation Algorithm.


Stochastic reaction networks Continuous time Markov chains Multilevel Monte Carlo Hybrid simulation methods Chernoff tau-leap  Dual-weighted estimation Strong error estimation Global error control Computational Complexity 

Mathematics Subject Classification

60J75 60J27 65G20 92C40 



The authors would like to thank two anonymous reviewers for their constructive comments that helped us to improve our manuscript. We also would like to thank Prof. Mike Giles for very enlightening discussions. The authors are members of the KAUST SRI Center for Uncertainty Quantification in the Computer, Electrical and Mathematical Sciences and Engineering Division at King Abdullah University of Science and Technology (KAUST).


  1. 1.
    Anderson, D.F.: A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J. Chem. Phys. 127(21), 214107 (2007)CrossRefGoogle Scholar
  2. 2.
    Anderson, D.F., Higham, D.J.: Multilevel Monte Carlo for continuous Markov chains, with applications in biochemical kinetics. Multiscale Model. Simul. 10(1), 146–179 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Anderson, D.F., Higham, D.J., Sun, Y.: Complexity of multilevel Monte Carlo tau-leaping. arXiv:1310.2676v1 (2013)
  4. 4.
    Aparicio, J.P., Solari, H.: Population dynamics: Poisson approximation and its relation to the Langevin processs. Phys. Rev. Lett. 86(18), 4183–4186 (2001)CrossRefGoogle Scholar
  5. 5.
    Bierig, C., Chernov, A.: Convergence analysis of multilevel variance estimators in multilevel Monte Carlo Methods and application for random obstacle problems. Preprint 1309, Institute for Numerical Simulation, University of Bonn (2013)Google Scholar
  6. 6.
    Collier, N., Haji-Ali, A.-L., Nobile, F., von Schwerin, E., Tempone, R.: A continuation multilevel Monte Carlo algorithm. Mathematics Institute of Computational Science and Engineering, Technical report Nr. 10.2014, EPFL (2014)Google Scholar
  7. 7.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence (Wiley Series in Probability and Statistics), vol. 9, 2nd edn. Wiley-Interscience, New York (2005)Google Scholar
  8. 8.
    Gibson, M.A., Bruck, J.: Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 104(9), 1876–1889 (2000)CrossRefGoogle Scholar
  9. 9.
    Giles, M.: Multi-level Monte Carlo path simulation. Oper. Res. 53(3), 607–617 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gillespie, D.T.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115, 1716–1733 (2001)CrossRefGoogle Scholar
  12. 12.
    Heinrich, S.: Multilevel Monte Carlo Methods. Large-Scale Scientific Computing. Lecture Notes in Computer Science, vol. 2179, pp. 58–67. Springer, Berlin Heidelberg (2001)CrossRefMATHGoogle Scholar
  13. 13.
    Karlsson, J., Katsoulakis, M., Szepessy, A., Tempone, R.: Automatic weak global error control for the tau-leap method, pp. 1–22. arXiv:1004.2948v3 (2010)
  14. 14.
    Karlsson, J., Tempone, R.: Towards automatic global error control: computable weak error expansion for the tau-leap method. Monte Carlo Methods Appl. 17(3), 233–278 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kurtz, T.G.: Representation and approximation of counting processes. In: Advances in Filtering and Optimal Stochastic Control, LNCIS vol. 42, pp. 177–191. Springer, Berlin (1982)Google Scholar
  16. 16.
    Li, T.: Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model. Simul. 6(2), 417–436 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming (International Series in Operations Research and Management Science). Springer, Berlin (2010)Google Scholar
  18. 18.
    Moraes, A., Tempone, R., Vilanova, P.: Hybrid Chernoff tau-leap. Multiscale Model. Simul. 12(2), 581–615 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Shapiro, S.S., Wilk, M.B.: An analysis of variance test for normality (complete samples). Biometrika 52(3/4), 591–611 (1965)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Speight, A.: A multilevel approach to control variates. J. Comput. Finance 12, 1–25 (2009)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Computer, Electrical and Mathematical Sciences and Engineering DivisionKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

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