Error trends in Quasi-Monte Carlo integration

Several test functions, whose variation could be calculated, were integrated with up to 10 10 trials using different low-discrepancy sequences in dimensions 3, 6, 12, and 24. The integration errors divided by the variation of the functions were compared with exact and asymptotic discrepancies. These errors follow an approximate power law, whose constant is essentially given by the variance of the integrand, and whose power depends on its effective dimension. Included were also some calculations with scrambled Niederreiter sequences, and with Niederreiter–Xing sequences. A notable result is that the pre-factors of the asymptotic discrepancy function D ∗(N) , which are so often used as a quality measure of the sequences, have little relevance in practical ranges of N.