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Monte Carlo Methods

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Nuclear Computational Science

Abstract

Monte Carlo methods comprise a large and still growing collection of methods of repetitive simulation designed to obtain approximate solutions of various problems by playing games of chance. Often these methods are motivated by randomness inherent in the problem being studied (as, e.g., when simulating the random walks of “particles” undergoing diffusive transport), but this is not an essential feature of Monte Carlo methods. As long ago as the eighteenth century, the distinguished French naturalist Compte de Buffon [1] described an experiment that is by now well known: a thin needle of length l is dropped repeatedly on a plane surface that has been ruled with parallel lines at a fixed distance d apart. Then, as Laplace suggested many years later [2], an empirical estimate of the probability P of an intersection obtained by dropping a needle at random a large number, N, of times and observing the number, n, of intersections provides a practical means for estimating π

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Notes

  1. 1.

    Of course, even pseudorandom sequences are deterministic, a fact that has stirred some debate about whether a probabilistic analysis made any sense at all for pseudorandomly implemented Monte Carlo. This, in turn, was one of the motives for developing a mathematically rigorous analysis divorced from probability theory and based only on a notion of uniformity arising out of number-theoretic considerations.

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Acknowledgments

The author gratefully acknowledges the support of the Laser Microbeam and Medical Program NIH P-41-RR-01192, and grants UCOP 41730 and NSF/DMS 0712853 during the preparation of this chapter.

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  1. Beckman Laser Institute, University of California, Irvine, California, USA

    Jerome Spanier

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Spanier, J. (2010). Monte Carlo Methods. In: Nuclear Computational Science. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3411-3_3

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