## 4 Case Study on the Multiple-Update-Infill Sampling Methods
### Preliminary Description of the Case Studies
\begin{table}
\begin{tabular}{l l} \hline \hline Input: & Number of initial training samples \(n_{0}\); Validation samples \(\mathbf{X}_{val}\); \\  & Number of final training samples \(n_{final}\) \\ \hline \multicolumn{2}{l}{1. Generate validation samples over input space} \\ \hline \multicolumn{2}{l}{(1) Generate \(\mathbf{X}_{val}\) using Mm LHD (Equation (10))} \\ \multicolumn{2}{l}{(2) Compute outputs \(\mathbf{Y}_{val}\) using the computational model of \(\mathbf{X}_{val}\)} \\ \hline \multicolumn{2}{l}{2. Perform sequential sampling method} \\ \hline \multicolumn{2}{l}{For \(i=1,\cdots,10\)} \\ \multicolumn{2}{l}{(1) Set current infill stage to \(n_{s}=1\)} \\ \multicolumn{2}{l}{(2) Generate \(\mathbf{X}_{i}^{(n_{s})}\) using Mm LHD (Equation (10)) with \(n_{0}\)} \\ \multicolumn{2}{l}{(3) Compute outputs \(\hat{\mathbf{Y}}_{i}^{(n_{s})}\) using the computational model of \(\mathbf{X}_{i}^{(n_{s})}\)} \\ \multicolumn{2}{l}{(4) Construct kriging model (\(KRG_{i}^{(n_{s})}\)) using \(\mathbf{X}_{i}^{(n_{s})}\) and \(\mathbf{Y}_{i}^{(n_{s})}\)} \\ \multicolumn{2}{l}{(5) Set \# of training sample to \(n_{train}=n_{0}\)} \\ \multicolumn{2}{l}{Until (\(n_{train}=n_{final}\))} \\ \multicolumn{2}{l}{(6) Predict outputs of \(\mathbf{X}_{val}\) (\(\hat{\mathbf{Y}}_{val}\)) using \(KRG_{i}^{(n_{s})}\)} \\ \multicolumn{2}{l}{(7) Evaluate and save performance measures using \(\mathbf{Y}_{val}\) and \(\hat{\mathbf{Y}}_{val}\)} \\ \multicolumn{2}{l}{(8) Search the infill sample(s) (\(\mathbf{X}_{infill}\)) based on \(KRG_{i}^{(n_{s})}\)} \\ \multicolumn{2}{l}{(9) Compute outputs \(\mathbf{Y}_{infill}\) using computational model of \(\mathbf{X}_{infill}\)} \\ \multicolumn{2}{l}{(10) Add \(\mathbf{X}_{infill}\) and \(\mathbf{Y}_{infill}\) to \(\mathbf{X}_{i}^{(n_{s})}\) and \(\mathbf{Y}_{i}^{(n_{s})}\)} \\ \multicolumn{2}{l}{(11) Update kriging model (\(KRG_{i}^{(n_{s})}\)) using \(\mathbf{X}_{i}^{(n_{s})}\) and \(\mathbf{Y}_{i}^{(n_{s})}\)} \\ \multicolumn{2}{l}{(12) Set \(n_{train}=n_{train}+N\big{(}\mathbf{X}_{infill}\big{)}\) and \(n_{s}=n_{s}+1\)} \\ \multicolumn{2}{l}{End} \\ \multicolumn{2}{l}{(13) Save final results: \(KRG_{i}^{(final)},\mathbf{X}_{i}^{(final)}\), and history of the performance measure} \\ End \\ \hline \hline \end{tabular} Note: N(\(X_{infill}\)) denotes the number of the infill sample(s).  
\end{table} 
Table 1: The flowchart of the multiple-update-infill sampling methods in the case studies. \[\rho^{2}=\frac{\sum_{i=2}^{k}\sum_{j=1}^{i-1}\rho_{ij}^{2}}{k(k-1)/2} \tag{29}\]  
where \(\rho_{ij}^{2}\) is the linear correlation between columns \(i\) and \(j\). The last performance measure is \(T_{norm}\). Since single-update-infill sampling (that is, \(MSE\)) requires more infill stages than others, this can be used as the reference of the computational time. This measure is the normalized time by the average of the computational time based on the MSE:  
\[T_{norm,i}=\frac{T_{i}}{1/N\sum_{k=1}^{10}T_{k}^{MSE}} \tag{30}\]  
where \(T_{i}\) is the computational time of the SS method, and \(T_{k}^{MSE}\) denotes the computation time for the \(k\)th SS method based on MSE.