BIT Numerical Mathematics

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

Multilevel hybrid Chernoff tau-leap

Article

Abstract

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.

Keywords

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 

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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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