## 4 Results Obtained with the Modified Nelder-Mead Method
The comparison is made for different LED types, including forward currents above 100 mA and the full range of forward currents, including those below 100 mA. Consequently, the simulation results obtained using the parameter sets identified by the previous method and those identified by the new, modified Nelder-Mead-based parameter extraction process agree very well in this current range. Considering the entire set of LED operating points, including forward currents below 100 mA, the simulation results obtained from the two different kinds of parameter sets show greater differences. Besides the stronger dependence of the accuracy of the model calculations on the accuracy of the parameter sets at lower forward currents, it is worth mentioning that the previous and the new parameter extraction processes use different objective functions. The new method uses the relative square difference as the cost function, ensuring acceptable accuracy across the entire range of operating forward currents. The statistics for the maximum relative errors obtained by both parameter extraction processes are summarized in Table 3, where the statistics includes the relative errors of the model and the measurements for all operating points that are included in the extraction. Consequently, the simulation results obtained using parameter sets identified by the previous method and those identified by the new, modified Nelder-Mead-based parameter extraction process agree very well in this current range. The statistics show that the multi-domain LED model with parameter sets obtained through the new extraction methodology accurately describes the operation of various high-power LEDs, including color and phosphor-converted white types, with minimal errors. Specifically:  
* For the forward voltage, errors are below 0.5%. * For simulated radiant fluxes, errors are below 5%. * For the luminous flux, errors are below 8%. These error levels are comparable to the industrial average uncertainties associated with measuring these LED properties. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & **Avg. of REs,** & **Variance of REs,** & **Avg. of REs,** & **Variance of REs,** & **Avg. of REs,** & **Variance of REs** \\  & **Electric** & **Electric** & **Optic** & **Optic** & **Luminous** & **Luminous** \\  & **[\%]** & **[10\({}^{-4}\)]** & **[\%]** & **[10\({}^{-4}\)]** & **[\%]** & **[10\({}^{-4}\)]** \\ \hline Brute force \(I_{F}\)\(\geq\)0.1 A & 0.11 & 0.002 & 0.88 & 0.427 & 5.19 & 12.48 \\ Nelder–Mead \(I_{F}\)\(\geq\)0.1 A & 0.11 & 0.001 & 0.43 & 0.029 & 0.62 & 0.053 \\ Brute force, all currents & 1.55 & 0.289 & 12.48 & 18.84 & 10.68 & 14.60 \\ Nelder–Mead, all currents & 0.25 & 0.005 & 2.69 & 1.40 & 3.89 & 1.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Relative errors (REs) statistics for 52 different sets of LED samples. Figure 7: Maximum relative errors for the luminous flux of the multi-domain LED model with parameter sets estimated using the prior brute-force method, compared to the modified Nelder–Mead-based parameter extraction process. The comparison is made for different LED types, including forward currents above 100 mA and the full range of forward currents, including those below 100 mA. The second benchmark evaluates how closely the measured LED characteristics are approximated by the multi-domain LED model equipped with the extracted parameter set. In these evaluations, we have to account for the effects of measurement uncertainties and noise, as their impact is absorbed into the overall error of the parameter fit. Since these measurement-related issues are consistent across any extraction process, and the theoretical accuracy of the multi-domain model is independent of the extraction method, the calculated error of the fit can be considered indicative solely of the quality of the parameter extraction process itself.