QMC integration errors and quasi-asymptotics

A crude Monte Carlo (MC) method allows to calculate integrals over a d-dimensional cube. As the number N of integration nodes becomes large, the rate of probable error of the MC method decreases as O⁢(1/N){O(1/\sqrt{N})}. The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of d-dimensional functions contains a multiplier 1/N{1/N}. However, the multiplier (ln⁡N)d{(\ln N)^{d}} is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor 1/N{1/N}. However, our numerical experiments show that using quasi-random points of Sobol sequences with N=2m{N=2^{m}} with natural m makes the integration error approximately proportional to 1/N{1/N}. In our numerical experiments, d≤15{d\leq 15}, and we used N≤240{N\leq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, d≤214{d\leq 2^{14}} and N≤263{N\leq 2^{63}}.