## 3 Numerical Experiments
### Sensitivity Studies with Respect to Emission Levels
The results also highlight that the computational efficiency of the algorithms depends on both the integrand's dimension and the magnitude of the estimated quantity. The order of relative error varies for different quantities of interest (as seen in the _Reference value_ column), even for the same sample size. Notably, the MCA-MSS-S tends to yield smaller relative errors, although, in most cases, the Sobol' QMCA and MCA-MSS deliver similar results. The primary objective here is to examine the computational efficiency of the stochastic algorithms under consideration for evaluating sensitivity measures, such as multidimensional integrals (total variance) or ratios of multidimensional integrals (Sobol' global sensitivity indices). Nonetheless, some conclusions can be drawn regarding the sensitivity of mean monthly ammonia concentrations in Milan to the input emissions of anthropogenic pollutants based on the preliminary reference or estimated values (see Table 7) of the quantities of interest. The most influential emissions affecting ammonia concentrations are ammonia emissions themselves, which account for approximately 89% in the southern part of Milan. Sulfur dioxide emissions also impact ammonia concentrations, although to a lesser degree--around 11%. A more detailed analysis is available in [44]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & **Sobol’** **QMCA** & **MCA-MSS** & **MCA-MSS-S** & **SS-MCA** \\ \cline{2-6}  & **Rel. Error** & \(\kappa\) & **Rel. Error** & \(\kappa\) & **Rel. Error** & **Rel. Error** \\ \hline \(2^{5}\) & \(2\times 10^{-2}\) & \(1\times 10^{-2}\) & \(1\times 10^{-2}\) & & \(5\times 10^{-3}\) \\ \hline \(2\times 2^{5}\) & & & & \(1\times 10^{-2}\) & \(1\times 10^{-2}\) & \\ \hline \(2^{7}\) & \(3\times 10^{-2}\) & \(8\times 10^{-3}\) & \(3\times 10^{-2}\) & & \(7\times 10^{-3}\) \\ \hline \(2\times 2^{7}\) & & & & \(8\times 10^{-3}\) & \(7\times 10^{-3}\) & \\ \hline \(2^{9}\) & \(5\times 10^{-3}\) & \(2\times 10^{-3}\) & \(1\times 10^{-3}\) & & \(9\times 10^{-4}\) \\ \hline \(2\times 2^{9}\) & & & & \(7\times 10^{-4}\) & \(2\times 10^{-2}\) & \\ \hline \(2^{11}\) & \(7\times 10^{-4}\) & \(2\times 10^{-3}\) & \(4\times 10^{-4}\) & & \(3\times 10^{-4}\) \\ \hline \(2\times 2^{11}\) & & & & \(6\times 10^{-4}\) & \(1\times 10^{-4}\) & \\ \hline \(2^{13}\) & \(6\times 10^{-4}\) & \(5\times 10^{-4}\) & \(2\times 10^{-4}\) & & \(2\times 10^{-4}\) \\ \hline \(2\times 2^{13}\) & & & & \(1\times 10^{-4}\) & \(6\times 10^{-5}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Relative errors for the approximate evaluation of the total variance \(\mathbf{V}\approx 0.0002\) for various numbers of points \(n\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & **Sobol’** **QMCA** & **MCA-MSS** & **MCA-MSS-S** & **SS-MCA** \\ \cline{2-6}  & **Rel. Error** & \(\kappa\) & **Rel. Error** & \(\kappa\) & **Rel. Error** & **Rel. Error** \\ \hline \(2^{5}\) & \(2\times 10^{-2}\) & \(1\times 10^{-1}\) & \(7\times 10^{-3}\) & & \(3\times 10^{-5}\) \\ \hline \(2\times 2^{5}\) & & & & \(1\times 10^{-2}\) & \(1\times 10^{-2}\) & \\ \hline \(2^{7}\) & \(3\times 10^{-3}\) & \(1\times 10^{-2}\) & \(3\times 10^{-4}\) & & \(1\times 10^{-4}\) \\ \hline \(2\times 2^{7}\) & & & & \(4\times 10^{-3}\) & \(4\times 10^{-3}\) & \\ \