Active Learning Kriging Model Combining With Kernel-Density-Estimation-Based Importance Sampling Method for the Estimation of Low Failure Probability

Abstract

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.

Active learning method combining Kriging model and multimodal‐optimization‐based importance sampling for the estimation of small failure probability

“…in which [U, U] is the prescribed searching domain (like [−6,6] n ). 40 In practice, G(u) may be a multimodal function, and henceĜ Sur (u) is multimodal. Moreover, in the initial stage,Ĝ Sur (u) can be very inaccurate.…”

“…Although a great deal of work has been done, some fatal issues remain unsolved during fusing the ALK model and IS methods. Firstly, Markov Chain Monte Carlo (MCMC) sampling is widely utilized to obtain important samples 37‐40 . Because the acceptance ratio of new candidates shrinks along with the increase of dimension, the computational time of the learning process is remarkably increased during dealing with problems with a little more random variables 38 .…”

“…Firstly, Markov Chain Monte Carlo (MCMC) sampling is widely utilized to obtain important samples. [37][38][39][40] Because the acceptance ratio of new candidates shrinks along with the increase of dimension, the computational time of the learning process is remarkably increased during dealing with problems with a little more random variables. 38 Moreover, the samples generated by MCMC are frequently trapped in one failure region, 43 and the accuracy of the existing methods cannot be guaranteed.…”

“…This method was called ALK‐CRA in 39 . The fusion of ALK model with the kernel‐density‐estimation‐based IS method (KDE‐IS) was researched and the method named ALK‐KDE‐IS was proposed in Reference 40. The fusion of ALK model and KDE‐IS for hybrid reliability analysis was researched in Reference 41.…”

Advanced time-dependent reliability analysis based on adaptive sampling region with Kriging model

“…For the time-independent structure, the reliability index β is defined as the shortest distance from the origin to the failure region in the standard normal space. According to the experience, 27–29 a hypersphere centered in the origin with a radius 1 . 2 − 1 . 5 β can cover all the most probable failure regions, 30 and this hypersphere is the minimal candidate region 31 for identifying the most probable failure regions.…”