## 1 Introduction: Problem Setting
### Concept of Sensitivity Analysis  
The process of sensitivity analysis (SA) is of crucial importance for large-scale mathematical models. This process includes the following three steps:  
* first, one should define the probability distributions for the input parameters under consideration;
* second, samples should be generated according to the defined probability distributions using a proper sampling strategy;
* third, an efficient approach is necessary for sensitivity analysis to be applied to study the output variance according to the variation in the inputs. Moreover, there are additional stages during the process of providing sensitivity analysis to a particular mathematical model: (i) approximation, which is an important link between the generation of experimental data and the mathematical technology for sensitivity analysis, and (ii) use of a proper probability approach for computing specific sensitivity measures. A variety of sensitivity analysis techniques are documented [29]. Most current methods for conducting SA hinge on particular assumptions about the model's behavior (such as linearity, monotonicity, and additivity of the relationship between model inputs and outputs). These assumptions commonly apply to a wide array of mathematical models. Nonetheless, some models incorporate substantial nonlinearities and/or stiffness, thereby invalidating assumptions about linearity and additivity. This issue is particularly pronounced when working with nonlinear systems of partial differential equations. Our research focuses on the UNI-DEM, a mathematical model that simulates the transport of air pollutants and other substances across extensive geographical regions. The system of partial differential equations captures key physical phenomena like advection, diffusion, deposition, and both chemical and photochemical processes affecting the studied species. Additionally, emissions and fluctuating meteorological conditions are considered. Nonlinearity in the equations predominantly stems from modeling chemical reactions [1]. If the model outputs are sensitive to a particular process, it suggests that the process may require refinement or more accurate modeling. Our aim is to enhance the reliability of the model predictions and identify processes needing further scrutiny as well as input parameters that demand higher-precision measurements. Thorough sensitivity analysis is crucial for uncovering potential areas where model simplification can occur. As a result, the development and exploration of more precise and robust sensitivity analysis methodologies are essential. Among quantitative methods, variance-based techniques are frequently employed [4]. A literature review based on high-impact-factor journals from Science and Nature (Thomson Reuters, Journal citation reports, April 2015) and all Elsevier journals using Scopus bibliometric search tools for publications from 2005 to 2014 is presented in [30]. This study yields the following conclusions:  
* There is a progressively increasing proportion of global sensitivity analysis approaches, although local techniques remain predominant. * Regression- and variance-based techniques are the most favored. * Medicine and chemistry are the leading scientific domains in applying global sensitivity analysis. Two prominent variance-based methods were utilized: the [PERSON]' approach and the extended Fourier Amplitude Sensitivity Test (eFAST). These methods were implemented using Monte Carlo algorithms (MCAs) or the SIMLAB software tool for sensitivity analysis [8]. In the eFAST method developed to estimate the total effects, the variance of the model output \(y\) (a \(d\)-dimensional integral) is expressed as a single-dimensional integral with respect to a scalar variable \(s\). This transformation involves representing each input variable \(x_{i}\) as \(x_{i}=G_{i}(sin(\omega_{i}\ s))\), where \(G_{i}\) represents a suitable set of transformations and \(\omega_{i}\) are integer frequencies (refer to [31] for detailed information).