The multi-mode system failure probability function (SFPF) can quantify how the distribution parameters of the random input vector affect the system safety and decouple the system reliability-based design optimization model. However, for a problem with a time-consuming implicit performance function and rare failure domain, efficiently solving the SFPF remains significantly challenging. Therefore, in this study, two efficient algorithms are proposed, namely, the meta model-based importance sampling and cross entropy-based importance sampling. The contributions of this study are twofold. The first is constructing a single-loop optimal importance sampling density (SL-OISD) method to decouple the double-loop framework for analyzing the SFPF. The second is establishing two methods to efficiently approximate the SL-OISD and complete the SFPF estimation. The first method is based on the meta model of the system performance function, which is abbreviated as SL-Meta-IS. The second method is based on minimizing the cross entropy between the Gaussian mixture density model and SL-OISD, which is abbreviated as SL-CE-IS. To reduce the number of evaluating the system performance function when approximating the SL-OISD, sampling the SL-OISD, and identifying the state of the samples for completing the SFPF estimation, an adaptive Kriging model of the system performance function is introduced into SL-Meta-IS and SL-CE-IS. Owing to decoupling the double-loop framework into a single-loop framework, replacing the time-consuming system performance function with the economic Kriging model, and employing importance sampling variance reduction techniques to address issues related to the rare failure domain, the proposed SL-Meta-IS and SL-CE-IS methods greatly enhance the efficiency of SFPF estimations. The numerical and practical examples demonstrate that the two proposed methods are superior to the existing algorithms; moreover, the efficiency of SL-CE-IS is higher than that of SL-Meta-IS.

中文翻译：

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基于元模型和交叉熵的重要性采样算法，用于有效求解系统故障概率函数

多模系统失效概率函数（SFPF）可以量化随机输入向量的分布参数如何影响系统安全，解耦基于系统可靠性的设计优化模型。然而，对于具有耗时的隐式性能函数和罕见故障域的问题，有效求解 SFPF 仍然具有很大的挑战性。因此，本研究提出了两种有效的算法，即基于元模型的重要性采样和基于交叉熵的重要性采样。这项研究的贡献是双重的。第一个是构建单环最优重要性采样密度（SL-OISD）方法来解耦用于分析 SFPF 的双环框架。第二是建立两种方法来有效地逼近SL-OISD并完成SFPF估计。第一种方法是基于系统性能函数的元模型，简称SL-Meta-IS。第二种方法是基于最小化高斯混合密度模型和SL-OISD（简称SL-CE-IS）之间的交叉熵。为了减少逼近 SL-OISD、对 SL-OISD 进行采样以及识别样本状态以完成 SFPF 估计时评估系统性能函数的次数，在 SL-OISD 中引入了系统性能函数的自适应 Kriging 模型。 Meta-IS 和 SL-CE-IS。由于将双环框架解耦为单环框架，用经济克里金模型代替耗时的系统性能函数，并采用重要性采样方差减少技术来解决与罕见故障域相关的问题，提出的SL- Meta-IS和SL-CE-IS方法极大地提高了SFPF估计的效率。数值和实际例子表明，所提出的两种方法优于现有算法；而且SL-CE-IS的效率高于SL-Meta-IS。