Research/Etc_Research2007/01/08 10:41
Monte Carlo Method
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Any method which solves a problem by generating suitable random numbers and observing that fraction of the numbers obeying some property or properties. The method is useful for obtaining numerical solutions to problems which are too complicated to solve analytically. It was named by S. Ulam Eric Weisstein's World of Biography, who in 1946 became the first mathematician to dignify this approach with a name, in honor of a relative having a propensity to gamble (Hoffman 1998, p. 239). Nicolas Metropolis Eric Weisstein's World of Biography also made important contributions to the development of such methods.

The most common application of the Monte Carlo method is Monte Carlo integration.

SEE ALSO: Markov Chain, Monte Carlo Integration, Quasi-Monte Carlo Method, Stochastic Geometry, Uniform Distribution Theory. [Pages Linking Here]

REFERENCES:

Gamerman, D. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Boca Raton, FL: CRC Press, 1997.

Gilks, W. R.; Richardson, S.; and Spiegelhalter, D. J. (Eds.). Markov Chain Monte Carlo in Practice. Boca Raton, FL: Chapman & Hall, 1996.

Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 238-239, 1998.

Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences. New York: Wiley, 1974.

Manno, I. Introduction to the Monte Carlo Method. Budapest, Hungary: Akadémiai Kiadó, 1999.

Metropolis, N. and Ulam, S. "The Monte Carlo Method." J. Amer. Stat. Assoc. 44, 335-341, 1949.

Metropolis, N. "The Beginning of the Monte Carlo Method." Los Alamos Science, No. 15, p. 125. http://jackman.stanford.edu/mcmc/metropolis1.pdf.

Mikhailov, G. A. Parametric Estimates by the Monte Carlo Method. Utrecht, Netherlands: VSP, 1999.

Niederreiter, H. and Spanier, J. (Eds.). Monte Carlo and Quasi-Monte Carlo Methods 1998, Proceedings of a Conference held at the Claremont Graduate University, Claremont, California, USA, June 22-26, 1998. Berlin: Springer-Verlag, 2000.

Sobol, I. M. A Primer for the Monte Carlo Method. Boca Raton, FL: CRC Press, 1994.



Monte Carlo Integration
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In order to integrate a function over a complicated domain D, Monte Carlo integration picks random points over some simple domain D^' which is a superset of D, checks whether each point is within D, and estimates the area of D (volume, n-dimensional content, etc.) as the area of D^' multiplied by the fraction of points falling within D^'. Monte Carlo integration is implemented in Mathematica as NIntegrate[f, ..., Method->MonteCarlo].

Picking N randomly distributed points x_1, x_2, ..., x_N in a multidimensional volume V to determine the integral of a function f in this volume gives a result

intfdV approx V<f>+/-Vsqrt((<f^2>-<f>^2)/N), (1)

where

<f> = 1/Nsum_(i==1)^(N)f(x_i) (2)
<f^2> = 1/Nsum_(i==1)^(N)f^2(x_i) (3)

(Press et al. 1992, p. 295).

SEE ALSO: Monte Carlo Method, Numerical Integration, Quasi-Monte Carlo Integration. [Pages Linking Here]

REFERENCES:

Hammersley, J. M. "Monte Carlo Methods for Solving Multivariable Problems." Ann. New York Acad. Sci. 86, 844-874, 1960.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Simple Monte Carlo Integration" and "Adaptive and Recursive Monte Carlo Methods." §7.6 and 7.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 295-299 and 306-319, 1992.

Ueberhuber, C. W. "Monte Carlo Techniques." §12.4.4 in Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, pp. 124-125 and 132-138, 1997.

Weinzierl, S. "Introduction to Monte Carlo Methods." 23 Jun 2000. http://arxiv.org/abs/hep-ph/0006269/.

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