## 4 Case Study on the Multiple-Update-Infill Sampling Methods
### Results and Discussion
In other words, some samples are clustered to certain regions in the input space. This may result in the numerical instability in the construction of the surrogate model. **Computational efficiency:** The computational cost of each SS method is shown in Figures 14-16. As expected, the values of \(T_{norm}\) became smaller in accordance with the increase of the infill samples at each infill stage. The optimization scheme (firefly algorithm) was implemented to search the infill samples, and numerous evaluations of the current surrogate model were required. The computational cost for the optimization scheme would be negligible in applications with computationally intensive models. **General discussion:**\(MSE+MED_{max}+MED_{min}\) showed the best performance in terms of convergence, accuracy, sampling, and computational efficiency, while \(EI_{max}+EI_{min}\) showed the worst performance. Since \(EI_{max}\) and \(EI_{min}\) were originally developed to find the global optima, they tend to locate more infill samples in the regions near the optima. As the surrogate model becomes more accurate, infill criteria make the infill samples near the global optima for exploitation. On the contrary, \(MED_{max}\) and \(MED_{min}\) can allocate the infill samples that are far apart from the existing samples. Since more multiple infill criteria can save computational costs as discussed above, \(MSE+MED_{max}+MED_{min}\) can accelerate the learning capability of the unknown response surface and can avoid the numerical instability caused by samples being too close to each other. It is worth noting that MED does not require a specific surrogate model. For example, many infill criteria (for example, MSE and EI) can be applied to only the kriging model. On the contrary, the multiple-update-infill sampling method using the MED criteria (that is \(MED_{max}+MED_{min}\)) can be applied to any surrogate models (penalized spline regression model [33], radial-basis function [34], and multivariate adaptive regression spline (MARS) [22]). **Limitations:** The proposed method has a limitation for some applications. Since the kriging model in this study is based on the stationary correlation specification, it cannot suitably represent strong non-stationary response surfaces or noisy response surfaces (that is, high-frequency components due to approximation and truncation errors). Under such response surfaces, the kriging model with stationary correlation specification may result in numerical instabilities and poor predictive performances by violating the stationary assumption. To deal with such response surfaces, treed GPs [35] or a non-stationary covariance-based kriging method [36] are needed to ensure the predictive performance of the kriging model. Figure 11: The convergence rates of root mean square errors (\(RMSE\)) in Case Study 1: (**a**) mean; (**b**) mean (zoom-in); (**c**) standard deviation; (**d**) standard deviation (zoom-in). Figure 12: \(Cont.\)  
Figure 12: The convergence rates of \(RMSE\) in Case Study 2: (**a**) mean; (**b**) mean (zoom-in); (**c**) standard deviation; (**d**) standard deviation (zoom-in). Figure 13: The convergence rates of \(RMSE\) in Case Study 3: (**a**) mean; (**b**) mean (zoom-in); (**c**) standard deviation; (**d**) standard deviation (zoom-in). Figure 14: The performance evaluations in Case Study 1: (**a**) \(d_{min}\) versus \(RMSE\); (**b**) \(RMSE\) versus \(\rho^{2}\); (**c**) \(d_{min}\) versus \(\rho^{2}\); (**d**) \(RMSE\) versus \(T_{norm}\); (**e**) \(d_{min}\) versus \(T_{norm}\). Figure 15: The performance evaluations in Case Study 2; (**a**) \(d_{min}\) versus \(RMSE\); (**b**) \(RMSE\) versus \(\rho^{2}\); (**c**) \(d_{min}\) versus \(\rho^{2}\); (**d**) \(RMSE\) versus \(T_{norm}\); (**e**) \(d_{min}\) versus \(T_{norm}\).