Multidimensional Sensitivity Analysis of Large-scale Mathematical Models

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.