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Multilevel hybrid Chernoff tau-leap

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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.

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Acknowledgments

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).

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Authors and Affiliations

  1. Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

    Alvaro Moraes, Raúl Tempone & Pedro Vilanova

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  1. Alvaro Moraes
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  2. Raúl Tempone
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  3. Pedro Vilanova
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Corresponding author

Correspondence to Alvaro Moraes.

Additional information

Communicated by Jan Hesthaven.

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Moraes, A., Tempone, R. & Vilanova, P. Multilevel hybrid Chernoff tau-leap. Bit Numer Math 56, 189–239 (2016). https://doi.org/10.1007/s10543-015-0556-y

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