(untitled) (REC 2026 - 11th International Workshop on Reliable Engineering Computing: Reliability Computations in a Data and Model-Driven World, 2026)
Forward uncertainty propagation of hybrid uncertainties from inputs to the outputs, and estimating the resultant bounds of failure probability under insufficient information is a critical issue for the safety and reliability assessment. The two curses—namely the double-loop structure and the extremely small failure probability—pose challenges to the numerical efficiency and global convergence of existing methods when solving this problem. The recently developed Collaborative and Adaptive Bayesian Optimization (CABO) has been recognized as an appealing scheme for breaking the double-loop curse. Under this methodology framework, this work presents a novel algorithm, named combining Bayesian active learning and stratified sampling (BAL-SS), for efficiently estimating the bounds of small failure probability. Specifically, a new learning function, with the aim of adaptive balance of exploration and exploitation, is developed for implementing Bayesian optimization during the marginal space of epistemic uncertainty, and then develops a Bayesian quadrature rule that combines active learning and stratified sampling to estimate the target small failure probabilities. The proposed BAL-SS algorithm is devised by collaborative implementation of the above two key procedures, in the marginal supports of epistemic and aleatory uncertainties, to sequentially approach the global optima and estimate the corresponding failure probabilities with desired accuracy. Effectiveness of the proposed method is ultimately demonstrated with numerical benchmark study and practical engineering applications.