This paper introduces a sophisticated approach to multidimensional sensitivity analysis by utilizing novel stochastic techniques for air pollution modeling within the large-scale, long-range transport framework of the Unified Danish Eulerian Model (UNI-DEM). This model plays a crucial role in assessing the detrimental impacts of elevated air pollution levels, and our research leverages it to tackle essential environmental protection issues. We develop and implement advanced Monte Carlo and quasi-Monte Carlo methods, incorporating specialized lattice sequences, to enhance the computational efficiency of multidimensional numerical integration. The study centers on evaluating the sensitivity of the UNI-DEM model outputs to variations in anthropogenic pollutant emissions and the rates of key chemical reactions. Our algorithms are applied to calculate global Sobol sensitivity indices, assessing the influence of several input parameters on significant air pollutant concentrations across diverse European cities with varying geographical characteristics. The goal of this research is to deepen the understanding of factors influencing air pollution and to support the development of effective strategies for reducing its harmful environmental impacts.