## 3 Numerical Simulations
The goal of the next experiment is to investigate the effectiveness of the proposed method as function of \(M\) and \(2S\) ranging from 2 to 18. In Figures 5-7, the equi-probability curves of the rate of success of the recovered excitations are plotted considering strong non-convex norm (\(p=0.1\)) and moderate non-convex norm (\(p=0.5\)) with the number of random starting points \(Q=10\) and \(Q=5\), respectively. In the same figure the CDF curve of the standard \(\ell_{1}\) norm minimization is also plotted in order to make a comparison. The estimation of the excitations is considered to be exact if MSE is lower than \(-30\) dB and the rate of success (i.e., the number of trials terminated with an MSE lower than \(-30\) dB over the total number of trials) is evaluated for each \((M,2S)\) considering 100 trials. These figures demonstrate that, for a fixed number of failures \(S\), the rate of success increases from 0.25 (purple curve) to 0.95 (yellow curve) varying the number of measurements \(M\) in a small range, indicating the presence of a narrow phase transition behavior also in proposed method. In addition, these figures also clearly show that the proposed algorithm requires a number of measurements less than the \(\ell_{1}\) norm to obtain a desired \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Norm** & \(Q=1\) & \(Q=5\) & \(Q=10\) \\ \hline
0.1 & 0.0135 & 1.1849 & 2.3709 \\
0.5 & 0.0134 & 0.9591 & 1.9187 \\
0.9 & 0.0332 & 1.0699 & 2.1371 \\
1.0 & 0.2462 & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of computational time for linear array using different methods (unit second). Figure 4: CDF of MSE of the retrieved excitations using proposed method in the case \(Q=10\) and the standard \(\ell_{1}\) norm minimization; solid line: \(\ell_{1}\) norm minimization; dashed line: proposed method. probability of success rate. For example, supposing that a probability of success rate of 0.95 or higher, and considering the presence of no more than five failures, namely \(2S=10\), by properly using random perturbation technique, a number of measurements \(M=12\) is required for both \(\ell_{0.1}\) and \(\ell_{0.5}\) norm minimization with a tolerable computational burden, while \(\ell_{1}\) norm minimization requires at least \(M=14\) measurements, confirming again the better performance of the proposed method. A large number of simulations on different kinds of arrays confirm the above observations. As another example, let us consider an uniform planar array (UPA) composed of \(N_{x}\times N_{y}=9\times 9\) point-like isotropic radiating elements, placed on \(d_{x}\times d_{y}=0.5\lambda\) mesh. The far-field data are measured in \(M=25\) points using random sampling strategy and the AUT is affected by \(S=8\) faulty elements. The type of faulty sensors and the measurement noise have the same configurations as reported in the  
Figure 5: Rate of success of the retrieved excitations versus \(M\) and \(2S\) using \(\ell_{1}\) norm minimization; 32 elements ULA; SNR = 35 dB. Figure 6: Rate of success of the retrieved excitations versus \(M\) and \(2S\) using proposed method in the case \(p=0.1\), \(Q=10\); 32 elements ULA; SNR = 35 dB. Similar conclusions could be made according to the simulation results for planar array. The plots show that a value of \(Q\) equal or higher than 5 is able to overcome the moderate non-convex feature of \(\ell_{0.5}\) norm and provide better performance compared to the standard \(\ell_{1}\) norm minimization also in this scenario. While for the strong non-convex norm \(\ell_{0.1}\), a larger number of random starting points \(Q=20\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Norm** & \(Q=1\) & \(Q=5\) & \(Q=20\) \\ \hline
0.1 & 0.0242 & 2.2760 & 9.1220 \\
0.5 & 0.0260 & 1.4287 & 5.7080 \\
0.9 & 0.0793 & 1.7199 & 6.8736 \\
1.0 & 0.2940 & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of computational time for planar array using different methods (unit: second).