# A Semi-Parallel Active Learning Method Based on Kriging for Structural Reliability Analysis
## 6 Conclusions  
Currently, the reliability analysis system is becoming more complex, and the efficiency and accuracy of reliability analysis are also increasing. This paper presents a new algorithm called SPAK that uses Kriging models for small failure probabilities. There are two innovations:  
1. The method proposes a novel learning function that considers the relationship between the training points and samples. It is called the \(U_{\text{A}}\) function. Additionally, it is more advanced than the U function, and while it is based on the U function, it is also different from it. The learning function pays attention to the correlation function, predictions and variance, so it overcomes the shortcoming that the training points are always around the area near the threshold. Based on the \(U_{\text{A}}\) function, the other essential areas have been highlighted as supplementary. Therefore, the accuracy of the surrogate model based on SPAK is improved. 2. To enhance the efficiency of iteration, the method proposes the semi-parallel learning process. It will add one or two training points intelligently, which is only based on one stopping criterion. This learning process reduces the number of iterations and enhances efficiency. The new learning function and intelligent semi-parallel added points are the most significant advantages compared to other parallel learning methods and sequential learning methods. To test this new method, three numerical examples and a nonlinear and a multidimensional application are studied. We obtain the following results:  
1. The failure probability based on SPAK is more accurate than the Kriging, AK and EGO methods in Example 1 to 3. Especially for Examples 2 and 3, the relative error of the failure probability based on SPAK is only one tenth of the other methods. It verifies the accuracy of this method for the failure probability analysis of the engineering application because the real value of failure probability cannot be obtained for practical engineering problems. 2. Due to the semi-parallel learning mechanism, the number of iterations required to construct the surrogate model is significantly reduced. In the application of the reliability analysis of the aquarium, evaluation efficiency is increased by at least 14.5% and the iteration efficiency increased by 35.7%. It shows that the semi-parallel learning method has more advantages in computational efficiency, especially in iterative computation. 3. Due to SPAK being an approximate model construction method, it focuses on the analysis of the accuracy around the model threshold (\(G=0\)). Therefore, this method \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **NoF** & **NoI** & **P\({}_{\text{f}}\)** & **P\({}_{\text{r}}\)** \\ \hline Kriging & 300 & / & \(5.784\times 10^{-5}\) & 0.99994216 \\ \hline EGO & 77 & 57 & \(3.510\times 10^{-4}\) & 0.99964900 \\ \hline AK & 62 & 42 & \(5.522\times 10^{-5}\) & 0.99994478 \\ \hline SPAK & 53 & 27 & \(5.453\times 10^{-5}\) & 0.99994547 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of different methods of the aquarium. is more suitable for dealing with critical value problems, such as reliability analysis, but may not be the most suitable for solving extreme value problems. Based on this work, it can be concluded that the proposed algorithm is valuable for constructing reliability the surrogate model. Further, we can combine IS sampling, subset and other methods with SPAK to improve its application value for more rare failure probability analyses. This research group will also carry out relevant research in the follow-up work. Conceptualization, C.L. and J.G. ; Methodology, Z.L. and X.L. ; Formal analysis, Z.L. ; Investigation, Y.Q. ; Resources, X.L. and Y.Q. ; Writing--original draft, Z.L. ; Writing--review & editing, J.G. ; Funding acquisition, C.L. All authors have read and agreed to the published version of the manuscript.