Multidimensional cubatures on Sobol sequences

Calculation of the multidimensional cubatures in the unit hypercube is a complex problem of numerical methods, and its application value is great. This paper compares various calculation methods: product of regular onedimensional grid formulae, classical Monte Carlo method using pseudorandom points and Sobol sequences. It is suggested to use not any Sobol sequences, but only with magic numbers $N=2^n$. In addition, offset Sobol points are proposed: all coordinates of magic Sobol points are simultaneously increased by $(2N)^{-1}$. Comparisons on the test showed that the latter method is significantly more accurate than all the others.