Monte Carlo Integration

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

Picking  randomly distributed points , , ...,  in a multidimensional volume  to determine the integral of a function  in this volume gives a result

(1)

where

			(2)
			(3)

(Press et al. 1992, p. 295).

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.