Abstract
This paper shows that it is relatively easy to incorporate adaptive timesteps into multilevel Monte Carlo simulations without violating the telescoping sum on which multilevel Monte Carlo is based. The numerical approach is presented for both SDEs and continuous-time Markov processes. Numerical experiments are given for each, with the full code available for those who are interested in seeing the implementation details.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Anderson, D., Higham, D.: Multi-level Monte Carlo for continuous time Markov chains with applications in biochemical kinetics. SIAM Multiscale Model. Simul. 10(1), 146–179 (2012)
Anderson, D., Higham, D., Sun, Y.: Complexity of multilevel Monte Carlo tau-leaping. SIAM J. Numer. Anal. 52(6), 3106–3127 (2014)
Barrett, J., Süli, E.: Existence of global weak solutions to some regularized kinetic models for dilute polymers. SIAM Multiscale Model. Simul. 6(2), 506–546 (2007)
Giles, M.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)
Giles, M.: Matlab code for multilevel Monte Carlo computations. http://people.maths.ox.ac.uk/gilesm/acta/ (2014)
Giles, M.: Multilevel Monte Carlo methods. Acta Numer. 24, 259–328 (2015)
Gillespie, D.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115(4), 1716–1733 (2001)
Heinrich, S.: Multilevel Monte Carlo methods. In: Multigrid Methods. Lecture Notes in Computer Science, vol. 2179, pp. 58–67. Springer, Heidelberg (2001)
Hoel, H., von Schwerin, E., Szepessy, A., Tempone, R.: Adaptive multilevel Monte Carlo simulation. In: Engquist, B., Runborg, O., Tsai, Y.H. (eds.) Numerical Analysis of Multiscale Computations, vol. 82, pp. 217–234. Lecture Notes in Computational Science and Engineering. Springer, Heidelberg (2012)
Hoel, H., von Schwerin, E., Szepessy, A., Tempone, R.: Implementation and analysis of an adaptive multilevel Monte Carlo algorithm. Monte Carlo Methods Appl. 20(1), 1–41 (2014)
Hutzenthaler, M., Jentzen, A., Kloeden, P.: Divergence of the multilevel Monte Carlo method. Ann. Appl. Prob. 23(5), 1913–1966 (2013)
Lester, C., Yates, C., Giles, M., Baker, R.: An adaptive multi-level simulation algorithm for stochastic biological systems. J. Chem. Phys. 142(2) (2015)
Moraes, A., Tempone, R., Vilanova, P.: A multilevel adaptive reaction-splitting simulation method for stochastic reaction networks. Preprint arXiv:1406.1989 (2014)
Moraes, A., Tempone, R., Vilanova, P.: Multilevel hybrid Chernoff tau-leap. SIAM J. Multiscale Model. Simul. 12(2), 581–615 (2014)
Müller-Gronbach, T.: Strong approximation of systems of stochastic differential equations. Habilitation thesis, TU Darmstadt (2002)
Tian, T., Burrage, K.: Binomial leap methods for simulating stochastic chemical kinetics. J. Chem. Phys. 121(10), 356 (2004)
Acknowledgments
MBG’s research was funded in part by EPSRC grant EP/H05183X/1, and CL and JW were funded in part by a CCoE grant from NVIDIA. In compliance with EPSRC’s open access initiative, the data in this paper, and the MATLAB codes which generated it, are available from doi:10.5287/bodleian:s4655j04n. This work has benefitted from extensive discussions with Ruth Baker, Endre Süli, Kit Yates and Shenghan Ye.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Giles, M.B., Lester, C., Whittle, J. (2016). Non-nested Adaptive Timesteps in Multilevel Monte Carlo Computations. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-33507-0_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33505-6
Online ISBN: 978-3-319-33507-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)