# A Semi-Parallel Active Learning Method Based on Kriging for Structural Reliability Analysis
## 1 Introduction
Thanks to the variance, the sequential construction of the surrogate model is carried out using the Kriging-based active learning reliability method. After the initial surrogate model is built with a small number of initial training points, more training points are added until the surrogate model accurately represents the performance function. In the process of developing a surrogate model, a learning function is essential for choosing the optimal training points. The criterion for choosing the following optimum training points is based on the learning function. Various Kriging-based surrogate model reliability techniques have been developed by various learning functions. Based on the cumulative distribution function (CDF) and probability density function (PDF), the expected improvement (EI) learning function is constructed [47; 48], then the efficient global optimization (EGO) is proposed. On the basis of EGO, the efficient global reliability analysis (EGRA) uses the expected feasibility function to determine training points [18; 49]. The active learning reliability method combines Kriging with Monte Carlo Simulation (AK-MCS) and constructs a learning function that can predict the probability of the correct sign of the performance function [4; 50]. In addition, other leaning functions are also available, [PERSON] [51] proposed an H learning function based on the information entropy. [PERSON] [52] proposed an LIF learning function, and the studies on learning functions can be found in References. The core of all the methods is the strategies of surrogate model construction and training points selection. EGO focuses on the expected improvement of model error, but the stopping criterion of learning process has no uniform standard, which may be difficult to converge for some problems. EGRA and LIF are similar to EGO. The theoretical accuracy of AK-MCS is 95.45%, and the stopping criterion is relatively stable. However, its learning process mainly focuses on the accuracy of the classification of test points, and lacks the accuracy of the overall model. To further improve the efficiency of reliability methods, a semi-parallel active learning method based on Kriging is proposed in this paper. It takes into account the relationship between training points and sample points comprehensively but seldom increases the computational amount of the learning process and can consider more sample points to participate in the learning. Furthermore, the new training points are added based on the new learning process semi-parallel to update the surrogate model. Because of these innovations, it involves fewer iterations and is more efficient than traditional parallel learning or sequential learning. The remainder of this paper is organized as follows: in Section 2, the Kriging method and current Kriging-based reliability techniques are discussed. The new method and the implemental procedure are described in Section 3. Three examples followed in Section 4 and an engineering application in Section 5 verifies the effectiveness and practicability of the proposed method. Conclusions are made in Section 6.