Multidimensional cubatures with super-power convergence

In many applications, multidimensional integrals over the unit hypercube arise, which are calculated using Monte Carlo methods. The convergence of the best of them turns out to be quite slow. In this paper, fundamentally new cubatures with super-power convergence based on the improved Korobov grids and special variable substitution are proposed. A posteriori error estimates are constructed, which are practically indistinguishable from the actual accuracy. Examples of calculations illustrating the advantages of the proposed methods are given.