## 4 Experimental Result and Discussion
### Line Search Algorithm Comparison and Correctness Verification  
We applied our parallel implementation to the earthquake source parameter estimation procedure using the L-BFGS-B. In doing so, we evaluated four different solutions introduced in Section 3.3, which are Armijo, MT, BWW, and cuLBFGSB, to derive the best-fit source parameters. Five thousand iterations of the Monte Carlo method were performed for each candidate. We defined the lower and upper bound of source parameters based on the study of [PERSON] [52], as summarized in Table 7. Starting points of the Monte Carlo iterations were randomly sampled from a uniform distribution in the boundaries, and termination conditions of the L-BFGS-B were set to \(maxIter=1000\), \(ftol=10^{-3}\), \(m=8\), and \(gtol=10^{-3}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{
\begin{tabular}{c} **Programming** \\ **Model** \\ **(  
We used the RMSE, computation time, and 95% confidence interval obtained from the Monte Carlo iterations as the indicators to evaluate and compare the performance and correctness of candidates. Table 8 lists the statistics of the RMSE and the computation time measured for the candidate implementations. As shown in Table 8, Armijo and cuLBFGSB outperform MT and BWW in the mean and the standard deviation values of RMSE. Although Armijo, MT, and cuLBFGSB show a similar mean computation time, Armijo has the lowest mean and standard deviation values of the computation time. In addition, the summary of the best-fit parameters minimizing the RMSE and their 95% confidence interval is given in Table 9. The best-fit parameters derived from candidates are almost identical, but Armijo has a much smaller confidence interval than the other candidates, with a 42\(\sim\)84% interval size reduction. This result remarks that the best-fit parameters determined by Armijo are the most reasonable. To estimate the uncertainties of parameters in detail, we illustrated the histogram of each parameter in Figure 11. All parameters from Armijo feature a unimodal histogram with a small standard deviation. However, most source parameters estimated from MT show multiple peaks, and the _Dip_ derived from BWW peaks around both the lower and upper bound. Furthermore, cuLBFGSB shows that most results of the _Length_ and the _Width_ converge on the lower bound, and the _Slip_ does not converge on the unique solution either with a high standard deviation. It seems that the highly biased results of cuLBFGSB were caused by its simplified line search algorithm. This result confirms again that Armijo has the best agreement with the earthquake source parameter estimation. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Constraints**} & \multirow{2}{*}{_E (km)_} & \multirow{2}{*}{_N (km)_} & \multirow{2}{*}{_Depth_} & \multirow{2}{*}{_Strike_} & \multirow{2}{*}{_Dip (deg.)_} & \multirow{2}{*}{_Length_} & \multirow{2}{*}{_Width_} & \multirow{2}{*}{_Rake_} & \multirow{2}{*}{_Slip (cm)_} \\  & & & **(km)** & & & & & \\ \hline Lower bound & 4.8 & \(-12.0\) & 3.0 & 110.0 & 33.0 & 4.0 & 4.0 & 80.0 & 10.0 \\ Upper bound & 10.0 & \(-6.0\) & 5.7 & 235.0 & 55.0 & 6.5 & 6.7 & 150.0 & 30.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Bound constraints of source parameters. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**RMSE (cm)**} & \multicolumn{4}{c}{**Computation Time (s)**} \\  & **Min.** & **Max.** & **Mean.** & **Std.** & **Min.** & **Max.** & **Mean.** & **Std.** \\ \hline Armijo & 0.4741 & 0.5103 & 0.4784 & 0.0030 & 0.2465 & 1.9259 & 1.0198 & 0.2481 \\ MT & 0.4745 & 0.8243 & 0.5291 & 0.0589 & 0.2309 & 3.6965 & 1.1638 & 0.5253 \\ BWW & 0.4748 & 0.8164 & 0.5273 & 0.0613 & 0.1737 & 11.4612 & 1.8269 & 0.9649 \\ cuLBFGSB & 0.4739 & 0.9450 & 0.4860 & 0.0258 & 0.1163 & 2.5101 & 1.1922 & 0.3514 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: RMSE and the computation time statistics of four candidates.