## 6 Results
Following the logic of our study, we investigate the relationship between speed of convergence and each of the three factors of the simulation, namely the number of anomalies in image, the initial guess error, and the choice of the learning rate. To verify the impact of number of anomalies on the speed of convergence, the Kruskal-Wallis test is applied on each algorithm speed of convergence sample data, as grouped by the number of inclusions. Kruskal-Wallis test results are presented in Figure 5, and we can conclude (by failing to reject the Kruskal-Wallis null hypothesis) that the number  
Figure 3: (**a**) Densities of number of iteration for each optimizer and (**b**) the empirical cumulative density functions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Groups** & **Point Estimation** & **Standard Error** & **Normal** & **Percentile** & **Pivotal** \\ \hline AmsGrad vs. Nadam & 4 & 0.36 & \((3.29,4.71)\) & \((3,4)\) & \((4,5)\) \\ \hline Adam vs. Nadam & 12 & 0.5 & \((11.04,12.96)\) & \((11,12)\) & \((12,13)\) \\ \hline AmsGrad vs. Adam & 15 & 0.44 & \((14.13,15.87)\) & \((15,16)\) & \((14,15)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The 95% non-parametric confidence intervals based on the bootstrap method. of anomalies present in the image is not significantly affecting the speed of convergence (\(p>0.05\)) for different optimizers. To fulfill our investigation, we discuss the impact of initial guess error and learning rate parameter over number of iterations as shown in Figure 6a and Figure 6b, respectively. The Spearman's coefficient of correlation is used due to its robustness against outliers which appears in data. Scatter plots in Figure 6a,b show the relationship between the initial guess error and the learning rate parameter on the speed of convergence, respectively. [PERSON]'s coefficient of correlation R and \(p\)-value are mentioned at the top of each graph. From Figure 6b, we notice that, when the learning rate ranges in [0.001, 0.2], [PERSON] and Adam algorithms take more iterations than the AmsGrad algorithm. In addition, we note that the AmsGrad algorithm presents some robustness toward the learning rate in this range and presents some outliers in the range [0.001, 0.2]. According to the [PERSON]'s correlation coefficient, we observe a very strong correlation (\(R=-1\)) between learning rate parameter and number of  
Figure 4: Kruskal–Wallis paired test for the number of iterations between different groups of optimizers with the resulting \(p\)-value of the test for each group. Figure 5: Kruskal–Wallis test on number of anomalies for different optimizers with the resulting \(p\)-value of the test for each group. iterations for Adam and Nadam optimizers and a negligible correlation for the case of AmsGrad optimizer and presents some outliers when the learning rate ranges in [0.2, 0.3]. On the other hand, Figure 6a shows the relationship between the initial guess error and number of iterations taken by each optimizer to reach convergence of cost functional. We note that the error has the same impact on [PERSON] and Nadam algorithms, when comparing their \(p\)-value and coefficient of correlation. However, we observe that the AmsGrad is more efficient than the other two optimizers even if the error is far from the real image. To assess the quality performance in reconstructed images between these optimizers, we performed statistical tests for differences of means on PSNR and SSIM as measured for reconstructed images, between the optimizers. These two scores are defined as follows:  
\[PSNR=10\log_{10}(\frac{max^{2}(\mu_{a}^{true})}{\frac{1}{N}\sum_{i=1}^{N}(\mu_ {a}^{recon}(i)-\mu_{a}^{true}(i))^{2}}),\]  
\[ \begin{split} SSIM(\mu_{a}^{true},\mu_{a}^{recon})& =[l(\mu_{a}^{true},\mu_{a}^{recon})]^{x}\\ &+[c(\mu_{a}^{true},\mu_{a}^{recon})]^{y}+[s(\mu_{a}^{true},\mu_ {a}^{recon}]^{z},\end{split} \]  
\[