A practical stratified importance sampling strategy for extremely rare event estimation in high dimensional systems

Estimating extremely rare events in complex systems represents a significant and persistent challenge in reliability analysis. This difficulty is further compounded in high-dimensional settings, where the target probability mass becomes exceedingly difficult to capture, introducing new obstacles to accurate estimation. To address these challenges, this paper proposes a practical and adaptive stratified importance sampling (IS) strategy for reliability analysis involving high-dimensional inputs and extremely rare events. The proposed method makes several key contributions. First, it theoretically demonstrates that within existing stratified IS frameworks, the accuracy of estimating intermediate failure events does not impact the final rare-event probability estimate. Leveraging this insight, we design an adaptive stratified strategy that prioritizes rapid progression toward the failure domain over precise intermediate event estimation, thereby avoiding the computational inefficiency inherent in exhaustively exploring intermediate failure regions. At the core of our approach is a novel sampling density construction. We develop a mixture IS density based on a bi-domain K-means clustering technique, integrated with variance-enlarged sampling. This design enables the method to not only converge efficiently toward the failure region but also accurately capture diverse and challenging failure modes, including discontinuous patterns and those with sharp gradient transitions. To mitigate the significant estimation errors that can arise from minor deviations in failure information in high-dimensional problems, an optimization mechanism is innovatively introduced in the final sampling layer. This allows for the accurate identification of the most probable failure information in high-dimensional space, leading to precise estimation of extremely rare events in high-dimensional systems. The effectiveness of the proposed method is validated through a series of benchmark problems in reliability analysis, encompassing high dimensionality, small failure probabilities, and complex failure domains. Results consistently show that our method outperforms existing approaches in both capturing varied failure modes and delivering efficient, high-precision estimates, particularly for high-dimensional systems.