Strategies combining active learning Kriging (ALK) model and Monte Carlo simulation (MCS) method can accurately estimate the failure probability of a performance function with a minimal number of training points. That is because training points are close to the limit state surface and the size of approximation region can be minimized. However, the estimation of a rare event with very low failure probability remains an issue, because purely building the ALK model is time-demanding. This paper is intended to address this issue by researching the fusion of ALK model with kernel-density-estimation (KDE)-based importance sampling (IS) method. Two stages are involved in the proposed strategy. First, ALK model built in an approximation region as small as possible is utilized to recognize the most probable failure region(s) (MPFRs) of the performance function. Consequentially, the priori information for IS are obtained with as few training points as possible. In the second stage, the KDE method is utilized to build an instrumental density function for IS and the ALK model is continually updated by treating the important samples as candidate samples. The proposed method is termed as ALK-KDE-IS. The efficiency and accuracy of ALK-KDE-IS are compared with relevant methods by four complicated numerical examples.

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