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Summary Reliability analysis with both aleatory and epistemic uncertainties is investigated in this paper. The aleatory uncertainties are described with random variables, and epistemic uncertainties are tackled with evidence theory. To estimate the bounds of failure probability, several methods have been proposed. However, the existing methods suffer the dimensionality challenge of epistemic variables. To get rid of this challenge, a so‐called random‐set based Monte Carlo simulation (RS‐MCS) method derived from the theory of random sets is offered. Nevertheless, RS‐MCS is also computational expensive. So an active learning Kriging (ALK) model that only rightly predicts the sign of performance function is introduced and closely integrated with RS‐MCS. The proposed method is termed as ALK‐RS‐MCS. ALK‐RS‐MCS accurately predicts the bounds of failure probability using as few function calls as possible. Moreover, in ALK‐RS‐MCS, an optimization method based on Karush–Kuhn–Tucker conditions is proposed to make the estimation of failure probability interval more efficient based on the Kriging model. The efficiency and accuracy of the proposed approach are demonstrated with four examples. Copyright © 2016 John Wiley & Sons, Ltd.

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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.