Comments on "On the use of low discrepancy sequences in Monte Carlo methods"

Quasi-random (or low discrepancy) sequences are sequences for which the convergence to the uniform distribution on [0; 1)s occurs rapidly. Such sequences are used in quasi-Monte Carlo methods for which the convergence speed, with respect to the N first terms of the sequence, is in O(N 1(lnN)s), where s is the mathematical dimension of the problem considered. The disadvantage of these methods is that error bounds, even if they exist theoretically, are inefficient in practice. Nevertheless, to take advantage of these methods for what concerns their convergence speed, we use them as a variance reduction technique, which lead to great improvements with respect to standard Monte Carlo methods. We consider in this paper two different approaches which combine Monte Carlo and quasi-Monte Carlo methods. The first one can use every low discrepancy sequence and the second one, called Owen’s method, uses only Niederreiter sequences. We prove that the first approach has the same convergence speed than Owen’s one, for which it is known, and we perform numerical comparisons for every choice of low discrepancy sequence. Key-words: Monte Carlo, variance reduction, quasi-Monte Carlo, low discrepancy sequences.