### Abstract

Tau-leaping is a popular discretization method for generating approximate paths of continuous time, discrete space Markov chains, notably for biochemical reaction systems. To compute expected values in this context, an appropriate multilevel Monte Carlo form of tau-leaping has been shown to improve efficiency dramatically. In this work we derive new analytic results concerning the computational complexity of multilevel Monte Carlo tau-leaping that are significantly sharper than previous ones. We avoid taking asymptotic limits and focus on a practical setting where the system size is large enough for many events to take place along a path, so that exact simulation of paths is expensive, making tau-leaping an attractive option. We use a general scaling of the system components that allows for the reaction rate constants and the abundances of species to vary over

several orders of magnitude, and we exploit the random time change representation developed by Kurtz. The key feature of the analysis that allows for the sharper bounds is that when comparing relevant pairs of processes we analyze the variance of their difference directly rather than bounding via the second moment. Use of the second moment is natural in the setting of a diffusion equation, where multilevel Monte Carlo was first developed and where strong convergence results for numerical methods are readily available, but is not optimal for the Poisson-driven jump systems that we consider

here. We also present computational results that illustrate the new analysis.

several orders of magnitude, and we exploit the random time change representation developed by Kurtz. The key feature of the analysis that allows for the sharper bounds is that when comparing relevant pairs of processes we analyze the variance of their difference directly rather than bounding via the second moment. Use of the second moment is natural in the setting of a diffusion equation, where multilevel Monte Carlo was first developed and where strong convergence results for numerical methods are readily available, but is not optimal for the Poisson-driven jump systems that we consider

here. We also present computational results that illustrate the new analysis.

Original language | English |
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Pages (from-to) | 3106-3127 |

Number of pages | 22 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 52 |

Issue number | 6 |

Early online date | 18 Dec 2014 |

DOIs | |

Publication status | Published - 2014 |

### Keywords

- computational complexity
- coupling
- continuous time Markov chain
- tau-leaping
- variance
- multilevel Monte Carlo

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## Cite this

Anderson, D. F., Higham, D. J., & Sun, Y. (2014). Complexity of multilevel Monte Carlo tau-leaping.

*SIAM Journal on Numerical Analysis*,*52*(6), 3106-3127. https://doi.org/10.1137/130940761