An error-based stopping criterion for spherical decomposition-based adaptive Kriging model and rare event estimation

Surrogate model-based reliability analysis methods, especially the adaptive Kriging model, have received increasing attention due to their high accuracy and efficiency for reliability analysis. In the context of the practical engineering and a design target of high reliability, the adaptive Kriging is combined with more efficient sampling techniques such as the spherical decomposition-based Monte Carlo simulation (SDMCS). In this paper, a new stopping criterion is tailored for the adaptive Kriging model with SDMCS (AKSDMCS). The relative error of the failure probability obtained by AKSDMCS is first derived. Then, the effect of samples with high probabilities of wrong classification on the relative error of failure probability is quantified. In each sub-region of SDMCS, it can be found that the number of points in wrong classification follows a binomial distribution. The expected upper bound of the relative error of the failure probability is subsequently formulated. Finally, a new stopping criterion designed for AKSDMCS is developed. Three applications are used to validate the performance of the proposed stopping criterion and the results showcase that the adaptive Kriging with the proposed stopping criterion can stop the active training process at an appropriate stage and significantly mitigate the computational effort of rare event estimation.