# A Semi-Parallel Active Learning Method Based on Kriging for Structural Reliability Analysis
## 4 Numerical Examples
### Example 2: Six-Hump Camel Back Function  
The six-hump camel back function is used for the second example [58]. The design space of the SC function in this paper is smaller compared to the original function, as shown in Equation (22). The design variables are uniform distribution in the interval \(-2\leq x_{1}\leq 2,-0.5\leq x_{2}\leq 1\). This function has six minimums, one of which is global minimum, which occurs at \(x=[-0.09029,\ 0.7126]\) with \(G(x)=-0.0316\), as indicated by the asterisk in Figure 5. To simulate a small probability of failure, the function is shifted up by one unit.  
\[G(x)=4x_{1}^{2}-2.1x_{1}^{4}+\frac{1}{3}x_{1}^{6}+x_{1}x_{2}-4x_{2}^{2}+4x_{2} ^{4}+1 \tag{22}\]train the surrogate models around the threshold value, AK adds seven points in Figure 6c, and SPAK adds six points in Figure 6d. It is important to emphasize that SPAK adds the training points semi-parallel, the triangle training points are generated based on the U function in sequential, while the plus training points based on the \(U_{A}\) and the asterisk points based on the U function are in parallel. Therefore, SPAK only iterates 14 times with 26 training points. It has significantly fewer iterations than the AK.  
Furthermore, to eliminate the contingency of the results caused by the initial samples, 20 groups of random samples are selected for the experiment, and the average values of 20 results are finally taken for comparisons. Table 2 shows the results among random Kriging, AK, EGO, SPAK, and brute MCS. The true failure probability is calculated as 0.002971 based on MCS with \(1\times 10^{6}\) sampling evaluations. The results obtained by the four methods are quite different. The random Kriging method has the maximal relative error 11.63%. Obviously, this method is no longer applicable to the problem of small failure probability. The EGO has error 3.036%, better than the random Kriging, and only needs 35.45 number of function evaluations (NoF) and 20.45 NoI. The NoF based on AK and SPAK is close to the former, 38.85, and the latter, 39.1; however, SPAK has fewer NoI than AK, which only has 14.2 iterations. Iteration determines the number of cycles. There is no doubt that SPAK will be more efficient. In terms of accuracy, SPAK is also the best, with a relative error 0.0471%. The absolute error \(y_{res}\) and \(\sigma\) of SPAK around the threshold value is reduced, which makes SPAK more accurate. The surrogate model is closer to the real performance function.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**NoF**} & \multirow{2}{*}{**NoI**} & \multirow{2}{*}{**P\({}_{\text{f}}\)**} & \multirow{2}{*}{**Relative err**} & \multirow{2}{*}{\(\begin{subarray}{c}y_{res}\\ \text{(\%)}\end{subarray}\)} & \multirow{2}{*}{\(\begin{subarray}{c}\sigma\\ \text{(\text{\text{x}}_{1}\in[-0.5,0.5]}\end{subarray}\)} & \multirow{2}{*}{\(\begin{subarray}{c}\text{(\text{\text{x}}_{1}\in[-0.5,0.5]}\end{subarray}\)} \\ & & & \(\begin{subarray}{c}\text{\text{\text{x}}_{2}\in[0.5,1]}\end{subarray}\) & \(\begin{subarray}{c}\text{\text{x}_{2}\in[0.5,1]}\end{subarray}\) \\ \hline MCS & \(1\times 10^{6}\) & / & 0.002971 & / & / & / \\ \hline Kriging & 60 & / & 0.002625 & 11.63 & 0.27934 & 0.06587 \\ \hline EGO & 35.45 & 20.45 & 0.002881 & 3.036 & 0.34872 & 0.09747 \\ \hline AK & 38.85 & 23.85 & 0.002955 & 0.5301 & 0.14960 & 0.03135 \\ \hline SPAK & 39.1 & 14.2 & 0.002972 & 0.0471 & 0.10461 & 0.02629 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of different methods for Example 2.  
Figure 5: Contour map and 3D map of SC function.