## 4 Numerical Experiments
### Observations  
Regardless of the number of chosen layers, similar observations in each experiment can be deduced. Accurate results are achieved for \(L_{i}\) if the number of excitation frequencies is at least as large as the number of searched layer thicknesses. Choosing only a single frequency (Tables 2-4) for the one-layered case (Experiment 1, \(n=1\)) shows that frequency adaption is needed such that the data carry sufficient information for the solver to reconstruct the thickness for every initial value. Here, Table 4 represents the overall best results for the lowest frequency. Adding a second frequency (Table 5) does not seem to improve the results and may appear unnecessary, but since it enriches the data, it stabilizes the minimization process in such a way that the frequency adaption can be omitted. Choosing only a single frequency (Table 6) for the two-layered case (Experiment 2, \(n=2\)) is insufficient, although it can generate good results when the initial value is close to the exact solution \(\mathbf{L_{\text{exact}}}=(5,5)^{T}\times 10^{-5}\). Again, adding more frequencies to the process for Experiment 2, one is able to reconstruct \(L_{i}\) for the majority of the initial values.  
Obviously, the average relative error \(\overline{\text{error}}(\omega_{1},\dots,\omega_{m})\) is not an appropriate indicator for the approximation accuracy of \(L_{i}\); see e.g., Table 2 for the initial value \(L_{\text{initial}}=5.6316\times 10^{-5}\), where the relative average error is actually negligible, but the calculated thickness shows a relative error of approximately 25 percent. This is a typical behavior of nonlinear inverse problems, suggesting that minimizing the objective functional (Equation (25)), even with exact data, can lead to inexact solutions. Only in a few cases is the algorithm unable to find an appropriate minimizer  
\[\mathbf{L_{\text{calculated}}}=\left(L_{\text{1 calculated}},\dots,L_{\text{ calculated}}\right)^{T} \tag{32}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \(\boldsymbol{L_{\text{initial}}}\) **[m]** & \(\boldsymbol{L_{\text{1 calculated}}}\) **[m]** & \(\boldsymbol{L_{\text{2 calculated}}}\) **[m]** & \(\boldsymbol{L_{\text{3 calculated}}}\) **[m]** & \(\overline{\text{error}}(\boldsymbol{\omega_{1}},\boldsymbol{\omega_{2}}, \boldsymbol{\omega_{3}},\boldsymbol{\omega_{4}})\) \\ \hline  
at all, such that the residual is \(\approx 0\); see e.g., Table 9 for the intial value \(L_{\text{initial}}=1.4211\times 10^{-5}\), but the average relative error is approximately 90 percent. Here, the algorithm stops in a local minimum since the initial value has not been chosen appropriately. Comparing results for multiple initial values is therefore crucial in order to evaluate the calculated \(L_{i}\). With increasing \(n\), the structure of the objective functional gets more complex. This often leads to strings of initial values getting trapped in undesired local minima, which complicates the evaluation of the results. For example, the results of the last six initial values in Table 9 give the impression that \(\mathbf{L}_{\text{calculated}}\approx(4.2,23.7,15.2)^{T}\times 10^{-5}\) has to be accepted as a solution, whereas the exact solution \((5,5,5)^{T}\times 10^{-5}\) is only represented four times in the middle section of the same table. Adding a fourth frequency (Table 10) resolves this ambiguity and leads to the exact result.