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QuasiMonte Carlo methodIn numerical analysis, a quasiMonte Carlo method is a method for the computation of an integral (or some other problem) that is based on lowdiscrepancy sequences. This is in contrast to a regular Monte Carlo method, which is based on sequences of pseudorandom numbers. Additional recommended knowledgeMonte Carlo and quasiMonte 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}. where Ī^{s} is the sdimensional unit cube, Ī^{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 subsequence of pseudorandom numbers. In a quasiMonte Carlo method, the set is a subsequence of a lowdiscrepancy 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 KoksmaHlawka inequality. The discrepancy of sequences typically used for the quasiMonte Carlo method is bounded by a constant times 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 by the law of the iterated logarithm. Thus it would appear that the accuracy of the quasiMonte 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 quasiMonte Carlo is less than expected theoretically. Still, in the examples studied by Morokoff and Caflisch, the quasiMonte 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 quasiMonte Carlo method is greater if the integrand is smooth, and the number of dimensions s of the integral is small. A technique, coined randomized quasiMonte Carlo, that mixes quasiMonte Carlo with traditional Monte Carlo, extends the benefits of quasiMonte Carlo to medium to large s. Application areasSee alsoReferences

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "QuasiMonte_Carlo_method". A list of authors is available in Wikipedia. 