## 1 Introduction: Problem Setting
One of the first steps is metamodeling or approximation, which serves as a vital connection between generating experimental data and applying mathematical techniques for sensitivity analysis. Accurate approximation of data is essential for the overall reliability of the resulting sensitivity indices; therefore, identifying an effective method for approximating discrete functions is crucial. Detailed information on the approximation stage of the mathematical model under consideration can be found in [2]. The subsequent important step in performing sensitivity analysis involves selecting an appropriate technique (e.g., local approaches like one-at-a-time experiments, screening methods, variance-based methods, or derivative-based global sensitivity approaches [3]). This generally requires robust sampling techniques, such as Latin hypercube sampling, importance sampling, stratified sampling, and low-discrepancy sequences. A strong contender for reliably analyzing models with nonlinearities is the variance-based approach [4]. Its core principle is to estimate how variations in an input parameter or group of inputs contribute to the variance of the model output. To measure this, the total sensitivity index (TSI) is employed (see Section 1.5). We consider this measure more suitable and reliable than others when multi-component analysis is required. Mathematically, it is defined as a multidimensional integral:  
\[I=\int_{\Omega}g(\mathrm{x})p(\mathrm{x})\,\mathrm{dx},\ \ \Omega\subset \mathbf{R}^{d}, \tag{2}\]  
where \(g(\mathrm{x})\) is a square-integrable function in \(\Omega\) and \(p(\mathrm{x})\geq 0\) is a probability density function such that \(\int_{\Omega}p(\mathrm{x})\,\mathrm{dx}=1\). The advantages and reasons to choose a variance-based approach for SA in the current work are provided in Section 1.4. A detailed description of the [PERSON]' approach (a well-known and efficient variance-based approach for SA) is presented in Section 1.5. Following Equation (2), one can conclude that the last crucial step of providing reliable SA is to choose an efficient Monte Carlo (stochastic) algorithm for multidimensional integration to compute sensitivity indices of inputs in a reliable way. From an environmental security perspective, it is crucial to examine the impact of variations in chemical reaction rates and emission levels on the results generated by the UNI-DEM. Conducting such an analysis can yield valuable insights that serve as a solid foundation for making well-informed assumptions, reasonable simplifications, or identifying parameters whose accuracy needs enhancement. This is because the model's outputs can be highly sensitive to fluctuations in these parameters. The primary objective of our research is to enhance the reliability of the model's outputs by employing efficient Monte Carlo algorithms (MCAs) that are optimal for a specific category of integrands. In certain scenarios, an issue termed \"loss of accuracy\" arises when attempting to compute small sensitivity indices. This study introduces and evaluates specialized Monte Carlo techniques designed to mitigate these challenges, thereby improving the overall accuracy and dependability of the model's results. The main tasks of the present work are to
* Study the sensitivity of the concentration levels of key pollutants (e.g., \(O_{3}\)) due to variations in chemical rates and emission levels. Apply unimprovable Monte Carlo algorithms for numerical integration to perform Sobol' variance-based sensitivity analysis. * [PERSON]' approach carried out by a Gray code implementation and sets of direction numbers proposed by [PERSON] and [PERSON] [5];
- [PERSON]'s scrambling [6] taken from the collection of NAG C Library [7];
- eFAST approach carried out via SIMLAB [8];
- Plain Monte Carlo algorithm [9].