### Abstract

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.

Language | English |
---|---|

Pages | 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 |

### Fingerprint

### Keywords

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

### Cite this

*SIAM Journal on Numerical Analysis*,

*52*(6), 3106-3127. https://doi.org/10.1137/130940761

}

*SIAM Journal on Numerical Analysis*, vol. 52, no. 6, pp. 3106-3127. https://doi.org/10.1137/130940761

**Complexity of multilevel Monte Carlo tau-Leaping.** / Anderson, David F.; Higham, Desmond J.; Sun, Yu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Complexity of multilevel Monte Carlo tau-Leaping

AU - Anderson, David F.

AU - Higham, Desmond J.

AU - Sun, Yu

PY - 2014

Y1 - 2014

N2 - 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 overseveral 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 considerhere. We also present computational results that illustrate the new analysis.

AB - 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 overseveral 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 considerhere. We also present computational results that illustrate the new analysis.

KW - computational complexity

KW - coupling

KW - continuous time Markov chain

KW - tau-leaping

KW - variance

KW - multilevel Monte Carlo

U2 - 10.1137/130940761

DO - 10.1137/130940761

M3 - Article

VL - 52

SP - 3106

EP - 3127

JO - SIAM Journal on Numerical Analysis

T2 - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 6

ER -