Chapter

Modeling Uncertainty

Volume 46 of the series International Series in Operations Research & Management Science pp 419-474

Recent Advances in Randomized Quasi-Monte Carlo Methods

  • Pierre L’EcuyerAffiliated withDépartement d’Informatique et de Recherche Opérationnelle, Université de Montréal
  • , Christiane LemieuxAffiliated withDepartment of Mathematics and Statistics, University of Calgary

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Abstract

We survey some of the recent developments on quasi-Monte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a high-dimensional integral. We review several QMC constructions and different randomizations that have been proposed to provide unbiased estimators and for error estimation. Randomizing QMC methods allows us to view them as variance reduction techniques. New and old results on this topic are used to explain how these methods can improve over the MC method in practice. We also discuss how this methodology can be coupled with clever transformations of the integrand in order to reduce the variance further. Additional topics included in this survey are the description of figures of merit used to measure the quality of the constructions underlying these methods, and other related techniques for multidimensional integration.