A novel method which combines the active learning Kriging (ALK) model with important sampling is proposed in this paper. The main aim of the proposed method is to solve problems with very small failure probability and multiple failure regions. A surrogate limit state surface (LSS) which strikes a balance between the Kriging mean and variance is proposed. In each iteration, important samples of the surrogate LSS are generated, optimal training points are chosen, the Kriging model is updated and the surrogate LSS is refined. After several iterations, the surrogate LSS will converge to the true LSS. To obtain all the local and global most probable points (MPPs) on the surrogate LSS in each iteration, a recently proposed evolutionary algorithm from the field of multimodal optimization is introduced. In this way, none of the potential failure regions is missed and the unbiasedness of the proposed method is guaranteed. The contribution factor of each MPP is defined and a weighted multimodal instrumental sampling density is formulated. In this way, more attention is paid to the more important failure regions and training points are further saved. The performance of the proposed method is verified by six case studies.

中文翻译：

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结合克里格模型和基于多峰优化的重要性采样的主动学习方法，用于估计小故障概率

提出了一种将主动学习克里格模型与重要采样相结合的新方法。提出的方法的主要目的是解决具有非常小的故障概率和多个故障区域的问题。提出了一种替代极限状态表面（LSS），该状态表面在Kriging均值和方差之间取得平衡。在每次迭代中，将生成替代LSS的重要样本，选择最佳训练点，更新Kriging模型，并优化替代LSS。经过几次迭代后，替代LSS将收敛到真正的LSS。为了在每次迭代中获得代理LSS上的所有局部和全局最可能点（MPP），从多峰优化领域介绍了最近提出的进化算法。通过这种方式，不会丢失任何潜在的故障区域，并且可以保证所提方法的无偏性。定义每个MPP的贡献因子，并制定加权的多峰仪器采样密度。这样，可以将更多的注意力放在更重要的故障区域上，从而进一步节省了培训点。六个案例研究验证了该方法的性能。