Cubatures with Superpower Convergence Law

In many applications, multidimensional integrals over the unit hypercube arise and are calculated using Monte Carlo (MC) methods. The convergence of the best among them turns out to be quite slow. In this paper, fundamentally new cubatures with superpower convergence based on 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.