## 3 Numerical Simulations
### The Sparse Measurement Point Arrangement Based on the Theory of ARM
Figure 9: The sound prediction errors of CMESM on the prediction plane when 90 uniformly arranged measurement points are used at the frequency of 71 Hz. (**a**) Free field (\(L1=55\), upper: \(p_{E}\); lower: \(\varphi_{E}\) ) and (**b**) \(h_{U}=0.2\) m (\(L1=62\), upper: \(p_{E}\); lower: \(\varphi_{E}\) ). pressure amplitude could be quite low after adjusting \(\varepsilon\), the prediction error of the sound pressure phase could still be awful as the ill-condition problem of sensing matrices may be inevitable when the measurement points are uniformly arranged and excessively sparse. Though regularization methods are introduced to solve the ill-condition problem of the sensing matrices, the result of sound prediction is still limited. Meanwhile, when the value of \(\varepsilon\) is quite low, the solutions of \(\beta\) may not exist as the solutions of underdetermined equations could hardly have extravagantly high accuracy. Increasing the value of \(\varepsilon\) could help in obtaining a solution of \(\beta\), but the accuracy of the ARMs could not be guaranteed. The quality of the prediction could be improved via increasing the quantity of measurement points [31]. Therefore, increasing the measurement point quantity to 135, the variation of \(\gamma\) with the increase of ARM order is shown in Figure 10. Several values of \(\gamma\) at different orders are given in Table 4.  
Compared with Table 3, it could be found in Table 4 that, when the quantity of measurement points increases to 135, the values of \(\gamma\) could also be quite low at high orders, which is beneficial to find the balance between computation convergence and prediction robustness. Repeating the simulation in Section 3.1, the prediction errors are shown in Figure 11. It could be seen that, with more measurement points, the prediction results become much better because it is easy to obtain optimal solutions of \(\beta\) that satisfy the sparsity and the penalty terms. Meanwhile, it is not necessary to change the quantity of measurement points when the reflection boundary exists, which breaks through the limitation of the SVM. However, the expansion of the measurement system scale will definitely increase the engineering cost. [PERSON] and [PERSON] indicated that, with a fixed scale of the measurement system, a non-uniform system could have better performance in sound prediction than a uniform system [28; 30]. However, the optimization method was not introduced in those works. Therefore, the EFI could be applied for optimizing the measurement system. With the help \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Case\textbackslash{}Order** & **10** & **30** & **50** & **70** & **90** & **110** & **130** & **168** \\ \hline Free Field & 2.67 & 4.75 & 7.87 & 20.02 & 44.96 & 853.99 & \(7.48\times 10^{5}\) & \(1.68\times 10^{10}\) \\ Half-space & 2.78 & 5.02 & 8.99 & 21.05 & 46.99 & 482.14 & \(2.76\times 10^{4}\) & \(7.44\times 10^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Different ARM truncation numbers in CMESM for 135 measurement points. Figure 10: The variation of \(\gamma\) with the increase of ARM order when 135 uniformly arranged measurement points are used. of the EFI, the measurement points contributing most to the ARMs are picked out from the candidate set.