# Meta-Heuristics Optimization of Mirrors for Gravitational Wave Detectors: Cryogenic Case
## 5 Numerical Experiments
### Aseevation of Convergence  
In order to better describe the results of the convergence of the various optimization methods used, we introduce the common Pareto Front. This common Pareto front is obtained by combining (i.e., joining together) the lists of the different Pareto fronts found with the methods indicated above and extracting a single Pareto front made of all the non-dominated points [49]. In Figure 2, we show the common Pareto front for the above methods; the parameters used in the simulations are reported in the caption of the figure itself (the same as the end of Table 1); the number of the layers is \(N_{L}=23\), and the red dot shows the position of the quarter wavelength design (which terminates the tradeoff curve). Some interesting comments have already been made on Figure 2; even in the case of cryogenic temperature (where the parameters fulfill the relation \(\eta_{H}<\eta_{L}\)), the Pareto front has several bumps (see closeup in Figure 2) and is therefore not convex. Compared to the case studied in the literature by some of us (i.e., \(\eta_{H}>\eta_{L}\)), a discontinuity appears in the closeup region. This discontinuity present at high transmittance arises from the abrupt transition in the optimal mirror design that occurs when going from one layer of the \(H\)-material to two layers of the \(H\) and \(L\) materials (in the cryogenic case it has a highly specific noise coefficient). \begin{table}
\begin{tabular}{l l l} \hline
**Property** & \(H\)**Material** & \(L\)**Material** \\ \hline refractive index & 2.1 & 1.45 \\ \hline loss angle & \(10^{-4}\) & \(7.8\times 10^{-4}\) \\ \hline extinction & \(4\times 10^{-8}\) & \(8.4\times 10^{-11}\) \\ \hline \(\eta\) coeff. & 1 & 7 \\ \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of the \(H\)- and \(L\)-type materials, used in the simulations at cryogenic temperature (we suppose that the laser operating wavelength is \(\lambda_{0}=1550\) nm). The substrate has the same refractive index as the \(L\)-type material \(n_{s}=n_{L}\). In Figure 3, the absolute errors of the various trade off curves with respect to the common Pareto front, each computed with its algorithm, are shown. The vertical green line shows the last point computed by the heuristics, indicating the termination of the numerical Pareto curve. This line is not shown when it coincides with the quarter wavelength coating design (the endpoint of the Pareto front, the red dot in Figure 2). Note the empty spaces (see the dashed light blue area) in the panel of the Borg MOEA and MOEA/D methods, before the last point on the Pareto curve. In that range, the absolute error is comparable to numerical accuracy, which is why it was not reported. The tendency of the Pareto fronts to concentrate on a region of the constraint space (often called focus in the jargon) is one of the methodological problems studied in the literature [50]. In some recent papers on the subject, some authors have stated that they can exploit this tendency to reduce the computational load and speed up the convergence to a specific region of constraint space of greater interest [51]. For the heuristics in Figure 3 indicated with NSGA2 and NSGA3 labels, we show, in red, the error curves of the methods when the evolution time is halved. These curves give us quantitative information about error reduction and confirm that the optimization methods converge. Similar curves have been computed for all algorithms (although not reported) showing the same direction of error reduction. In the panels related to MOEA/D and GDE3, we show, in red, the error curves for the Pareto fronts computed, adding a further constraint on the transmittance (i.e., \(\tau_{c}<0.01\)) and leaving the computation time unchanged. With this expedient, we see that is possible to slightly improve the performance of these two methods in the low transmittance (high thermal noise) region. The improvement of the  
Figure 2: The figure shows the common Pareto Front for the GDE3, NSGA2, NSGA3, MOEA/D, Borg MOEA, and MOPSO methods.