**1. Introduction**

One of the main problem of the statistical modeling method (the Monte Carlo method) is the problem of quality for pseudorandom numbers. In the paper, we consider a task of multidimensional integrals calculation by the statistical modeling method and give sufficient conditions for the stability of this task to quality of pseudorandom numbers. Included results of various numerical experiences with the mt19937 pseudorandom number generator. In our work, we discuss important issues of algorithms optimization in the statistical modeling. In particular, we apply the new approach to the following: a task of finding of integral functionals from solution of boundary-value problems for both the linear [1] or nonlinear [2] elliptic equations (the estimations are given near to a boundary).

The paper is organized as follows: In Section 2, we give the sufficient conditions of stability. Calculation of an integral of very large dimensions is discussed in Section 3. Rare events effect is the subject of Section 4. In Section 5, we describe calculation of integral moments. In Section 6, we apply our approach to calculate an integral of the regression function. In Section 7, we give the conclusion of our studies.
