Discrepancy behaviour in the non-asymptotic regime Original Research Article

We have computed true star-discrepancies D∗(N) for Halton and Niederreiter point sequences in dimensions s=2,3,4, and 6 with N up to 250000, 10000, 2000 and 300, respectively. For comparison, we also calculated some L2 discrepancies T∗. The mean behaviour of D∗(N) can well be approximated by power laws with an exponent between about −0.7 and −0.9, which slowly decreases with N. This behaviour is far from the generally assumed asymptotic one ∼C(s)(logN)s/N. The factor between the true and the asymptotic behaviour increases strongly with s and reaches many orders of magnitude for large s. The ratios of D∗(N) for different low discrepancy sequences are not proportional to the presumed asymptotic pre-factors C(s). Especially for the range of bases investigated, Niederreiter sequences have about the same D∗ as Halton ones in a range of N, which we conjecture to be at least of the order of 1010.