Quasi-Monte Carlo Integration

Quasi-Monte Carlo integration is a method of numerical integration that operates in the same way as Monte Carlo integration, but instead uses sequences of quasirandom numbers to compute the integral. Quasirandom numbers are generated algorithmically by computer, and are similar to pseudorandom numbers while having the additional important property of being deterministically chosen based on equidistributed sequences (Ueberhuber 1997, p. 125) in order to minimize errors.

Monte Carlo methods are connected with computer simulation, and there is a distinction between simulation (where the system investigated and the mathematical model are both stochastic in nature, as in the simulation of a supermarket), and Monte Carlo simulation (where the modeled system is deterministic and the model used is stochastic) as in the case of Monte Carlo integration (Neelamkaville 1987, p. 3).

A quasi-Monte Carlo method known as the Halton-Hammersley-Wozniakowski algorithm is implemented in the Wolfram Language as NIntegrate[f, ..., Method ->QuasiMonteCarlo].

REFERENCES:

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

Hammersley, J. M. and Handscomb, D. C. Monte Carlo Methods. New York: Wiley, p. 25, 1964.

Neelamkavil, F. Computer Simulation and Modelling. New York: Wiley, pp. 3-4, 1987.

Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin:Springer-Verlag, pp. 124-125, 1997.

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

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1085, 2002.

Wozniakowski, H. "Average Case Complexity of Multivariate Integration." Bull. Amer. Math. Soc. 24, 185-194, 1991.