# A Semi-Parallel Active Learning Method Based on Kriging for Structural Reliability Analysis
## 4 Numerical Examples
### Example 1: Highly Nonlinear Curve Function  
The first mathematical example, designed to find the extreme response of \(G(x)\), is employed to demonstrate the difference of the new training points among the typical Kriging surrogate model methods. The highly nonlinear performance function with one-dimensional space, cited in paper [57], is provided as Equation (20). \[G(x)=1.0417x^{5}-13.25x^{4}+59.792x^{3}-112.75x^{2}+75.167x+5 \tag{20}\]  
where variable \(x\) is within the interval [0, 5]. The global minimum occurs at \(x=4.515\) with \(G(4.515)=-2.5036\), whereas the local minimum is located at \(x=1.957\) with \(G(1.957)=3.9824\), shown in Figure 2.  
The performance function is firstly evaluated at initial samples \(x_{1}\sim x_{5}\): [0, 1, 2, 2.5, 5], and the obtained performance function values are \(G(x_{1})\sim G(x_{5})\): [5, 15, 4.004, 6.630, 10.148]. With these initial sample points, the one-dimensional surrogate model can be built to approximate the limit state function \(G(x)\). In order to clarify the difference of the modeling process among the methods, the random Kriging method without the learning process is built by 15 points. Based on the learning functions and stopping criterions of different methods, the learning processes are shown as Table 1 and Figure 3. The EGO achieves convergence in nine iterations, the AK and SPAK require 10 and six iterations, respectively. It should be noted that SPAK adds two training points at the first, second and fifth iterations; thus, reduces the number of iterations. So, SPAK is the most efficient. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **No. of Iteration** \\ **(NoI)** \\ \end{tabular} & **Kriging** & **EGO** & **AK** & **SPAK** \\ \hline \multirow{4}{*}{0} & 0 & 0 & 0 & 0 \\ \cline{2-5}  & 1 & 1 & 1 & 1 \\ \cline{2-5}  & 2 & 2 & 2 & 2 \\ \cline{2-5}  & 2.500 & 2.500 & 2.5000 & 2.500 \\ \cline{2-5}  & 5 & 5 & 5 & 5 \\ \hline \multirow{2}{*}{1} & 0.4877 & 3.469 & 3.6818 & \begin{tabular}{c} 3.6818 \\ 3.7500 \\ \end{tabular} \\ \hline \multirow{2}{*}{2} & 0.6349 & 4.238 & 3.0951 & \
\begin{tabular}{c} 4.2338 \\ 4.3749 \\ \end{tabular} \\ \hline \multirow{2}{*}{3} & 1.3924 & 2.975 & 1.6927 & 4.6184 \\ \hline \multirow{2}{*}{4} & 2.7344 & 1.541 & 4.2485 & 4.7260 \\ \hline \multirow{2}{*}{5} & 3.1617 & 0.457 & 4.1366 &
\begin{tabular}{c} 4.7512 \\ 3.0909 \\ \end{tabular} \\ \hline \multirow{2}{*}{6} & 4.0736 & 4.588 & 4.4860 & 4.2054 \\ \hline \multirow{2}{*}{7} & 4.5289 & 3.864 & 4.6726 & / \\ \cline{2-5}  & 4.5668 & 2.245 & 4.2030 & / \\ \hline \multirow{2}{*}{9} & 4.7875 & 4.516 & 4.7538 & / \\ \cline{2-5}  & 4.8244 & / & 4.2054 & / \\ \hline \(\mathrm{P_{f}}\) & 0.10996 & 0.10996 & 0.10996 & 0.10996 \\ \hline \(\mathrm{err_{max}}\) & \(4.4123\times 10^{-4}\) & \(26.747\times 10^{-4}\) & \(7.1643\times 10^{-4}\) & \(4.6021\times 10^{-4}\) \\ \hline \(\mathrm{\sigma}\) & \(2.3423\times 10^{-4}\) & \(11.6403\times 10^{-4}\) & \(2.7163\times 10^{-4}\) & \(1.5857\times 10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Results of adding training points for different methods for Example 1. Figure 2: Performance function of mathematical example. In Figure 3, the EGO adds the new points near extreme point with a high probability. It seems more efficient than the AK and SPAK on a global space to identify the extreme value of the performance function. However, for the reliability analysis, under the premise of global modeling, more attention should be paid to the threshold region to determine a critical boundary. Based on the learning function in Equation (15), the AK selects a new point that may be crossed, as shown in Figure 3c. The SPAK considers the correlation between sample points and training points based on Equation (16), the added training points are shown clearly in Figure 3d. Meanwhile, SPAK may add the two new training points in once iterated parallelly; the training points are shown as a plus sign (+) and an asterisk (*) in Figure 3d.