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Randomized Quasi-Monte Carlo: An Introduction for Practitioners

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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))

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Abstract

We survey basic ideas and results on randomized quasi-Monte Carlo (RQMC) methods, discuss their practical aspects, and give numerical illustrations. RQMC can improve accuracy compared with standard Monte Carlo (MC) when estimating an integral interpreted as a mathematical expectation. RQMC estimators are unbiased and their variance converges at a faster rate (under certain conditions) than MC estimators, as a function of the sample size. Variants of RQMC also work for the simulation of Markov chains, for function approximation and optimization, for solving partial differential equations, etc. In this introductory survey, we look at how RQMC point sets and sequences are constructed, how we measure their uniformity, why they can work for high-dimensional integrals, and how can they work when simulating Markov chains over a large number of steps.

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Acknowledgements

This work has been supported by a Canada Research Chair, an Inria International Chair, a NSERC Discovery Grant, to the author. It was presented at the MCQMC conference with the help of SAMSI funding. David Munger made Figs. 9 and 10. Several comments from the Guest Editor Art Owen helped improving the paper.

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  1. Département d’Informatique et de Recherche Opérationnelle, Université de Montréal, Canada and Inria, Rennes, France

    Pierre L’Ecuyer

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  1. Pierre L’Ecuyer
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Correspondence to Pierre L’Ecuyer .

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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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L’Ecuyer, P. (2018). Randomized Quasi-Monte Carlo: An Introduction for Practitioners. 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_2

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