  Sensitivity analysis (SA) is a procedure for studying how sensitive are the
output results of large-scale mathematical models to some uncertainties of the
input data. The models are described as a system of partial differential
equations. Often such systems contain a large number of input parameters.
Obviously, it is important to know how sensitive is the solution to some
uncontrolled variations or uncertainties in the input parameters of the model.
Algorithms based on analysis of variances technique (ANOVA) for calculating
numerical indicators of sensitivity and computationally efficient Monte Carlo
integration techniques have recently been developed by the authors. They have
been successfully applied to sensitivity studies of air pollution levels
calculated by the Unified Danish Eulerian Model (UNI-DEM) with respect to
several important input parameters. In this paper a comprehensive theoretical
and experimental study of the Monte Carlo algorithm based on
\textit{symmetrised shaking} of Sobol sequences has been done. It has been
proven that this algorithm has an optimal rate of convergence for functions
with continuous and bounded second derivatives in terms of probability and mean
square error. Extensive numerical experiments with Monte Carlo, quasi-Monte
Carlo (QMC) and scrambled quasi-Monte Carlo algorithms based on Sobol sequences
are performed to support the theoretical studies and to analyze applicability
of the algorithms to various classes of problems. The numerical tests show that
the Monte Carlo algorithm based on \textit{symmetrised shaking} of Sobol
sequences gives reliable results for multidimensional integration problems
under consideration.
