# A Semi-Parallel Active Learning Method Based on Kriging for Structural Reliability Analysis
## 1 Introduction  
In engineering practice, reliability analysis is crucial in determining the likelihood that a system would fail due to random variables [1; 2; 3]. A performance function \(G(x)\) characterizes the response of the system, when \(G(x)>0\) the system is safe, otherwise, it is unsafe. The Monte Carlo Simulation (MCS) is still the most used reliability analysis technique [4; 5; 6]. It is thought to be the most precise and reliable method for system reliability analysis and comprises a random population \(n_{MC}\) that is assessed based on the performance function. The failure probability \(P_{f}\) is then calculated as the ratio of \(n_{G\leq 0}\) to the sampling \(n_{MC}\) using Equation (1). \[P_{f}=\frac{n_{G\leq 0}}{n_{MC}} \tag{1}\]  
In order to obtain a higher accuracy of analysis, the larger population \(n_{MC}\) is better, i.e., \(n_{MC}=1\times 10^{6}\). However, it is computationally expensive and time-consuming to evaluate the performance function for the entire population, especially when performance function is implicit, is a complex system [7; 8; 9], or is the finite element analysis (FEA) [10; 11]. It is highly necessary to study efficient reliability analysis approaches. Aside from numerical simulation methods such as MCS and its advanced tactics, such as importance sample(IS) [12], subset simulation (SS) [13], and line sampling [14], the most commonly utilized approaches are analytical methods and surrogate model methods. The operational efficacy of analytical techniques, such as the first-order reliability method (FORM) and second-order reliability method (SORM), are considerable [11; 15]. The Taylor approximation of the performance function at the most likely point is the basis of these techniques. However, for extremely nonlinear situations or when trying to solve the implicit function problem, the inaccuracy based on the methods could be unacceptable [16]. Due to these difficulties, the simple-to-evaluate surrogate model methods are an essential component of reliability analysis [11]. The surrogate model methods aim at approximating the performance function using the strategic design of experiments (DoE). The method results in a sufficiently accurate prediction of the performance function with a lower computing demand [17; 18; 19; 20]. Over the last few decades, many surrogate model methods have been developed, including response surface methods (RSM) [21; 22; 23], polynomial chaos expansion (PCE) [24; 25; 26], support vector machine (SVM) [27; 28; 29; 30], Kriging method [31; 32; 33; 34], artificial neural networks (ANN) [35; 36], and others. [PERSON] et al. [29] proposed a regression-model-adaptive algorithm based on the Bayesian support vector machine. The learning function of this algorithm uses penalty function to select sample points efficiently. Based on the generalized polynomial chaos expansion, [PERSON] et al. [37] performed stochastic structural modal analysis, which includes uncertain parameters. Taking advantage of the global prediction and local interpolation capability of PCE and Kriging, [PERSON] et al. [26] proposed a hybrid method, which used the U learning function to drive the active learning function. Since there are multiple failure modes, [PERSON] and [PERSON] [38] developed several Kriging metamodels and emphasized the AK-SYS sampling technique. However, in actual engineering, many studies combine IS and SS with Kriging surrogate model methods to solve the small failure probability problem [39]. [PERSON] et al. [40] combine a stratified IS method with the adaptive Kriging to estimate reliability. Of the above methods, the Kriging method shows significant advantages [41; 42]. First, Kriging is a reliable interpolation technique. The performance function's precise value is predicted at a point in DoE [43; 44; 45; 46]. Moreover, it estimates the local variance of the predictions in addition to the expected values without increasing the computational cost [11; 41].