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The Trio Identity for Quasi-Monte Carlo Error

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Monte Carlo and Quasi-Monte Carlo Methods (MCQMC 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 241))

Abstract

Monte Carlo methods approximate integrals by sample averages of integrand values. The error of Monte Carlo methods may be expressed as a trio identity: the product of the variation of the integrand, the discrepancy of the sampling measure, and the confounding. The trio identity has different versions, depending on whether the integrand is deterministic or Bayesian and whether the sampling measure is deterministic or random. Although the variation and the discrepancy are common in the literature, the confounding is relatively unknown and under-appreciated. Theory and examples are used to show how the cubature error may be reduced by employing the low discrepancy sampling that defines quasi-Monte Carlo methods. The error may also be reduced by rewriting the integral in terms of a different integrand. Finally, the confounding explains why the cubature error might decay at a rate different from that of the discrepancy.

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Acknowledgements

The author would like to thank the organizers of MCQMC 2016 for an exceptional conference. The author is indebted to his colleagues in the MCQMC community for all that he has learned from them. In particular, the author thanks Xiao-Li Meng for introducing the trio identity and for discussions related to its development. The author also thanks Lluís Antoni Jiménez Rugama for helpful comments in preparing this tutorial. This work is partially supported by the National Science Foundation grant DMS-1522687.

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Authors and Affiliations

  1. Department of Applied Mathematics, Illinois Institute of Technology, 10 W. 32nd Street, RE 208, Chicago, IL, 60616, USA

    Fred J. Hickernell

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  1. Fred J. Hickernell
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Correspondence to Fred J. Hickernell .

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Editors and Affiliations

  1. Department of Statistics, Stanford University, Stanford, California, USA

    Art B. Owen

  2. Department of Management Science and Engineering, Stanford University, Stanford, California, USA

    Peter W. Glynn

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Hickernell, F.J. (2018). The Trio Identity for Quasi-Monte Carlo Error. In: Owen, A., Glynn, P. (eds) Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2016. Springer Proceedings in Mathematics & Statistics, vol 241. Springer, Cham. https://doi.org/10.1007/978-3-319-91436-7_1

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