# Quasi-Monte Carlo method

In

numerical analysis , a**quasi-Monte Carlo method**is a method for the computation of anintegral (or some other problem) that is based onlow-discrepancy sequence s. This is in contrast to a regularMonte Carlo method , which is based on sequences ofpseudorandom numbers.Monte Carlo and quasi-Monte Carlo methods are stated in a similar way.The problem is to approximate the integral of a function "f" as the average of the function evaluated at a set of points "x"

_{1}, ..., "x"_{"N"}.:$int\_\{ar\; I^s\}\; f(u),du\; approx\; frac\{1\}\{N\},sum\_\{i=1\}^N\; f(x\_i),$

where

^{"s"}is the "s"-dimensional unit cube,^{"s"}= [0, 1] × ... × [0, 1] . (Thus each "x"_{"i"}is a vector of "s" elements.)In a Monte Carlo method,the set "x"_{1}, ..., "x"_{"N"}is a subsequenceof pseudorandom numbers.In a quasi-Monte Carlo method,the set is a subsequence of a low-discrepancy sequence.The approximation error of a method of the above type is bounded by a term proportional to the discrepancy of the set "x"

_{1}, ..., "x"_{"N"}, by the Koksma-Hlawka inequality.The discrepancy of sequences typically used for the quasi-Monte Carlo method is bounded by a constant times:$frac\{(log\; N)^s\}\{N\}.$

In comparison, with probability one, the expected discrepancy of a uniform random sequence (as used in the Monte Carlo method) has an order of convergence

:$sqrt\{frac\{log\; log\; N\}\{2N$

by the

law of the iterated logarithm .Thus it would appear that the accuracy of the quasi-Monte Carlo method increases faster than that of the Monte Carlo method. However, Morokoff and Caflisch cite examples of problems in which the advantage of the quasi-Monte Carlo is less than expected theoretically. Still, in the examples studied by Morokoff and Caflisch, the quasi-Monte Carlo method did yield a more accurate result than the Monte Carlo method with the same number of points.

Morokoff and Caflisch remark that the advantage of the quasi-Monte Carlo method is greater if the integrand is smooth, and the number of dimensions "s" of the integral is small. A technique, coined randomized quasi-Monte Carlo, that mixes quasi-Monte Carlo with traditional Monte Carlo, extends the benefits of quasi-Monte Carlo to medium to large "s".

**Application areas***

Monte Carlo methods in finance **See also***

Monte Carlo method **References*** Michael Drmota and Robert F. Tichy, "Sequences, discrepancies and applications", Lecture Notes in Math.,

**1651**, Springer, Berlin, 1997, ISBN 3-540-62606-9

* Harald Niederreiter. "Random Number Generation and Quasi-Monte Carlo Methods." Society for Industrial and Applied Mathematics, 1992. ISBN 0-89871-295-5

* Harald G. Niederreiter, "Quasi-Monte Carlo methods and pseudo-random numbers", Bull. Amer. Math. Soc.**84**(1978), no. 6, 957--1041

* William J. Morokoff and Russel E. Caflisch, "Quasi-random sequences and their discrepancies", SIAM J. Sci. Comput.**15**(1994), no. 6, 1251--1279 "(AtCiteSeer : [*http://citeseer.ist.psu.edu/morokoff94quasirandom.html*] )"

* William J. Morokoff and Russel E. Caflisch, "Quasi-Monte Carlo integration", J. Comput. Phys.**122**(1995), no. 2, 218--230. "(AtCiteSeer : [*http://citeseer.ist.psu.edu/morokoff95quasimonte.html*] )"

* Oto Strauch and Štefan Porubský, "Distribution of Sequences: A Sampler", Peter Lang Publishing House, Frankfurt am Main 2005, ISBN 3-631-54013-2

* R. E. Caflisch, "Monte Carlo and quasi-Monte Carlo methods", Acta Numerica vol. 7, Cambridge University Press, 1998, pp. 1-49.**External links*** [

*http://www.puc-rio.br/marco.ind/quasi_mc.html A very intuitive and comprehensive introduction to Quasi-Monte Carlo methods*]

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