AbstractA 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/\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}.
However, the multiplier {(\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}.
However, our numerical experiments show that using quasi-random points of Sobol sequences with {N=2^{m}} with natural m makes the integration error approximately proportional to {1/N}.
In our numerical experiments, {d\leq 15}, and we used {N\leq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011.
In this code, {d\leq 2^{14}} and {N\leq 2^{63}}.
Generalized Simpson type inequalities involving Riemann-Liouville fractional integrals and their applications