Quasi-Monte Carlo method

In numerical analysis, a quasi-Monte Carlo method is a method for the computation of an integral (or some other problem) that is based on low-discrepancy sequences. This is in contrast to a regular Monte Carlo method, which is based on sequences of pseudorandom 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 x1, ..., xN.


 * $$ \int_{\bar I^s} f(u)\,du \approx \frac{1}{N}\,\sum_{i=1}^N f(x_i), $$

where &#298;s is the s-dimensional unit cube, &#298;s = [0, 1] &times; ... &times; [0, 1]. (Thus each xi is a vector of s elements.) In a Monte Carlo method, the set x1, ..., xN 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 x1, ..., xN, 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