## 3 Numerical Experiments
### Sensitivity Studies with Respect to Chemical Reaction Rates  
Firstly, the results regarding the sensitivity of ozone concentration in Genova to variations in the rates of certain chemical reactions are presented. These reactions include #1, 3, 7, and 22 (time-dependent) as well as #27 and #28 (time-independent) from the condensed CBM-IV scheme ( [1]). The simplified chemical equations for these reactions are as follows:  
\[ \begin{array}{lclcl}[\mathbf{\#1}]&NO_{2}+h\
u&\Longrightarrow&NO+O;&[\mathbf{\#2}]& HO_{2}+NO\Longrightarrow&OH+NO_{2};\\ [\mathbf{\#3}]&O_{3}+NO&\Longrightarrow&NO_{2};&[\mathbf{\#27}]&HO_{2}+HO_{2} \Longrightarrow&H_{2}\mathrm{O}_{2};\\ [\mathbf{\#7}]&NO_{2}+O_{3}&\Longrightarrow&NO_{3};&[\mathbf{\#28}]&OH+CO \Longrightarrow&HO_{2}.\end{array} \]  
The domain under consideration is the 6-dimensional hypercubic domain \([0.6,1.4]^{6}\)). Polynomials of second degree were used again as an approximation tool [2]. It means that the integrands in our case studies are smooth functions and, in general, do not lead to computational difficulties, but some of the quantities that should be estimated are small in value and a loss of accuracy occurs. That is why a more accurate analysis and numerical approaches should be performed during the numerical experiments. Following the definitions of Sobol' global sensitivity measures, it is clear that the precise computing of the total variance is crucial. In this context, there is a discussion [38] about the better estimation of \(f_{0}^{2}=\left(\int_{\mathrm{L}^{d}}f(\mathrm{x})\mathrm{d}\mathrm{x}\right)^ {2}\) inside the expression of the total variance. The first formula is  
\[f_{0}^{2}\approx\frac{1}{n}\sum_{i=1}^{n}f(\mathrm{x}_{i,1},\ldots,\mathrm{x} _{i,d})\;f(\mathrm{x}_{i,1}^{\prime},\ldots,\mathrm{x}_{i,d}^{\prime})\]and the second one is  
\[f_{0}^{2}\approx\left\{\frac{1}{n}\sum_{i=1}^{n}f(\mathrm{x}_{i,1},\ldots, \mathrm{x}_{i,d})\right\}^{2},\]  
where \(\mathrm{x}\) and \(\mathrm{x}^{\prime}\) are two independent sample vectors. During the current numerical tests, a strong influence of the chosen formula on the accuracy of the SS-MCA was observed. This held true for both case studies with dimensions (specifically, 8 and 12 in the double-dimension case). The second formula yielded reliable results for numerically estimating total variance, aligning with the recommendation in [38], whereas the first formula produced results significantly deviating from the corresponding reference values. The first formula performs better in the case of estimation of sensitivity indices of a fixed order according to the recommendation in [38]). While the other algorithms (Sobol' QMCA, MCA-MSS, and MCA-MSS-S) were not significantly influenced by the choice in formulas, the same formulas used for the SS-MCA were applied to estimate the corresponding quantities. This approach was taken to ensure a consistent comparison of relative errors across all the algorithms under consideration. The results for the relative errors in the approximate evaluation of the quantities \(f_{0}\), total variance, first-order and total sensitivity indices, and second-order sensitivity indices, using various stochastic approaches for numerical integration, are presented in Table 1, Table 2, Table 3, and Table 4, respectively. In the tables, only those sensitivity indices larger than \(1\times 10^{-4}\) were estimated by the algorithms under consideration. The other case corresponds to negligibly small or unimportant main or interaction effects of inputs. One of the best available random number generators, the SIMD-oriented Fast Mersenne Twister (SFMT) [42; 43], a 128-bit pseudo-random number generator with a period of \(2^{19937}-1\), was used to generate the required random points. The calculations with pseudo-random numbers generated by the SFMT were averaged based on 10 algorithm runs.