## 3 Numerical Experiments
### Sensitivity Studies with Respect to Chemical Reaction Rates
Firstly, it should be specified that the quantity \(f_{0}\) is represented by a 6-dimensional integral, while the remaining quantities under consideration are represented by double-dimensional integrals, following the principles of _correlated sampling_. It is one of the approaches suggested in [34] to overcome loss of accuracy when \(\mathbf{V}_{\mathrm{I}_{1}\ldots\mathrm{I}_{v}}<<f_{0}^{2}\). It is valid for our both case studies. The partial and total variance estimations, based on the _correlated sampling_ approach, are presented in the following manner:  
\[\mathrm{V}_{\mathrm{y}}=\int\,f(\mathrm{x})\,\left[f(\mathrm{y}, \mathrm{z}^{\prime})\mathrm{d}\mathrm{x}\mathrm{d}\mathrm{z}^{\prime}-f( \mathrm{x}^{\prime})\right]\mathrm{d}\mathrm{x}\mathrm{d}\mathrm{x}^{\prime},\] \[\mathrm{V}=\int f(\mathrm{x})[f(\mathrm{x})-f(\mathrm{x}^{ \prime})]\,\,\mathrm{d}\mathrm{x}\mathrm{d}\mathrm{x}^{\prime}.\]  
The main conclusions about the properties of the algorithms are as follows: (a) the relative error decreases as the number of points increases; (b) all algorithms are efficient and converge at the expected rate of convergence; (c) the SS-MCA is the most efficient in terms of computational complexity for computing \(f_{0}\) and the total variance, but its efficiency decreases essentially in the case of computing Sobol' global sensitivity indices due to loss of accuracy; (d) the SS-MCA overcomes the disadvantages of the MCA-MSS and MCA-MSS-S in terms of computational cost and ease of implementation, but it is not sufficiently efficient for computational problems where accuracy is compromised; (e) these algorithms result in very high relative errors (see Table 4) in cases with small sensitivity measures, indicating that their efficiency depends not only on the integrand dimension but also significantly on the magnitude of the estimated quantity; (d) the MCA-MSS-S has the longest computational time, but it is the most robust, stable, and reliable. Analyzing the results for the reactions under consideration, one can conclude that the rates of reactions \(\#\#1,3\), and \(22\) have a very significant impact on ozone concentrations. The influence of the rates of reactions \(\#\#7\) and \(27\) is smaller but still notable. Conversely, the impact of the rate of reaction \(\#28\) can be considered negligible. Note that ozone does not necessarily participate in all of these reactions; rather, important ozone precursors are involved. A more detailed analysis can be found in [2]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & **Sobol’ QMCA** & **MCA-MSS** & \multicolumn{2}{c}{**MCA-MSS-S**} & **SS-MCA** \\ \cline{2-5}  & **Rel. Error** & \(\kappa\) & **Rel. Error** & \(\kappa\) & **Rel. Error** & **Rel. Error** \\ \hline \(2^{5}\) & \(4\times 10^{-3}\) & \(2\times 10^{-1}\) & \(3\times 10^{-3}\) & & & \\ \hline \(2\times 2^{5}\) & & & & \(2\times 10^{-2}\) & \(5\times 10^{-4}\) & \\ \hline \(2^{7}\) & \(2\times 10^{-3}\) & \(4\times 10^{-}02\) & \(1\times 10^{-3}\) & & & \(3\times 10^{-3}\) \\ \hline \(2\times 2^{7}\) & & & & \(1\times 10^{-}02\) & \(2\times 10^{-3}\) & \\ \hline \(2^{9}\) & \(9\times 10^{-5}\) & \(5\times 10^{-3}\) & \(2\times 10^{-5}\) & & & \\ \hline \(2\times 2^{9}\) & & & & \(9\times 10^{-4}\) & \(3\times 10^{-4}\) & \(2\times 10^{-5}\) \\ \hline \(2^{11}\) & \(6\times 10^{-5}\) & \(4\times 10^{-3}\) & \(2\times 10^{-5}\) & & & \\ \hline \(2\times 2^{11}\) & & & & \(7\times 10^{-4}\) & \(1\times 10^{-4}\) & \\ \hline \(2^{13}\) & \(2\times 10^{-6}\) & \(2\times 10^{-3}\) & \(5\times 10^{-5}\) & & & \(3\times 10^{-6}\) \\ \hline \(2\times 2^{13}\) & & & & \(2\times 10^{-4}\) & \(5\times 10^{-6}\) & \\ \hline \end{tabular}
\end{table} 
Table 1: Relative errors for the approximate evaluation of \(f_{0}\approx 0.27\) 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** & \multicolumn{2}{c}{**MCA-MSS-S**} & **SS-MCA** \\ \cline{2-5}  & **Rel. Error** & \(\kappa\) & **Rel.