# A Semi-Parallel Active Learning Method Based on Kriging for Structural Reliability Analysis
## 3 Proposed Method
### A Semi-Parallel Active Learning Method Based on Kriging  
Based on the \(U_{\text{A}}(x)\) aforementioned, a new semi-parallel active learning method based on Kriging is proposed to balance the estimation accuracy and the computational cost. In SPAK, the new training points are added semi-parallel to update the surrogate model. The semi-parallel indicates that the added new training points may be one point or two points during one learning process, which depends on the different positions of learning functions. So SPAK is substantially different from the traditional parallel learning and sequential learning method. The flowchart of SPAK is shown in Figure 1 and the procedure is described as follows:  
Step 1. Initialization. Generate a population of \(n_{\text{MC}}\) sample points set \(S=\{x_{i}:i=1,2,\ldots,n_{\text{MC}}\}\) based on PDF in the design space. It presents the candidate training points of the active learning process. The population of \(n_{\text{MC}}\) will be changed greater if the condition of updating is satisfied. Step 2. Define original DoE. In order to build an initial Kriging surrogate model, a DoE contains \(n_{\text{TP}}\) samples that are stochastically selected from \(n_{\text{MC}}\) or the Latin hypercube sampled in the truncated domain as \([n_{x}\pm 5\sigma_{x}]\). Then, the performance function is invoked to calculate the responses of these samples. The size of samples in the original DoE usually is small. In the cited paper [4], the number is larger with the number of random variables increased. Step 3. Construct or refine the Kriging model. The Kriging model is constructed or refined based on the current DoE and responses of these training points. Here, the toolbox of DACE is recommended. Step 4. Predict using the Kriging model. Kriging predictions \(\hat{G}(x_{i})\), variances \(\hat{\sigma_{G}}^{2}(x_{i})\), and correlation coefficient vector \(r(x_{i})\) for \(i=1,2,\ldots,n_{\text{MC}}\) with Equations (6)-(8) are obtained. Then, the probability of failure is obtained based on Equation (1). Step 5. Identify new training points in semi-parallel. In this step, the two learning functions in Equations (15) and (16) are calculated for all \(n_{\text{MC}}\) sample points, simultaneously. The new potential training points are identified, where \(U(x)\) and \(U_{\text{A}}(x)\) reach the minimum. For most situations, learning functions are primarily based on predictions \(\hat{G}(x_{i})\) and variances \(\hat{\sigma_{G}}^{2}(x_{i})\), they are the same point for high probability. So it will only add one new point to the current training points. Oppositely, a point with a low correlation \(r(x_{i})\) to the current training points is most likely different from the point based on the U function. Then, both of the two points will be added to the current training points parallelly. Step 6. The stopping criterion of the learning process. If the best next points are identified with the learning functions, the values of the corresponding learning functions are compared to the stopping criterion in Equation (18). \[min(U_{min},\ U_{Amin})\geq\gamma \tag{18}\]  
Step 7. Update DoE with new points. If the stopping criterion in Step 6 is not satisfied, the learning carries on and the new points are evaluated on the performance function. The learning process turns back to Step 3 to refine the new surrogate model with the updated DoE composed of \(n_{\text{TP}}+1\) or \(n_{\text{TP}}+2\) points. Step 8. Compute \(COV\big{(}\hat{P}_{\text{f}}\big{)}\). If the stopping criterion in Step 6 is achieved, then the learning is finished and the Kriging surrogate model is precise enough. As a result, it remains to be seen if \(n_{\text{MC}}\) is an appropriate size for the MCS. If the size is too small, the probability of failure is trustless. Here, it is acceptable to have a coefficient of variation below 2%.