| Quasi-monte Carlo Method |
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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''. : where Ī''s'' is the ''s''-dimensional unit cube, Ī''s'' = 1 × ... × 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 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 : 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 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 SEE ALSO REFERENCES
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