Highly efficient Bayesian updating using metamodels: An adaptive Kriging-based approach

Bayesian updating offers a powerful tool for probabilistic calibration and uncertainty quantification of models as new observations become available. By reformulating Bayesian updating into a structural reliability problem via introducing an auxiliary random variable, the state-of-the-art Bayesian updating with structural reliability method (BUS) has showcased large potential to achieve higher accuracy and efficiency compared with conventional approaches based on Markov Chain Monte Carlo simulations. However, BUS faces a number of limitations. The transformed reliability problem often involves a very rare event especially when the number of observations increases. This along with the fact that conventional reliability analysis techniques are not efficient, and often not capable of accurately estimating the probability of rare events, unavoidably lead to a very large number of evaluations of the likelihood function and simultaneously insufficient accuracy of the derived posterior distributions. To overcome these limitations, we propose Simple Rejection Sampling with Multiple Auxiliary Random Variables (SRS-MARV), where the limit state function in BUS is decomposed into a system reliability problem with multiple limit state functions. The main advantage of this approach is that the acceptance rate of each decomposed limit state function is significantly improved, which facilitates effective integration of adaptive Kriging-based reliability analysis into SRS-MARV. Moreover, a new stopping criterion is proposed for efficient, adaptive training of the Kriging model. The proposed method called BUAK is shown to be highly computationally efficient and accurate based on results of comprehensive investigations for three diverse benchmark problems. Compared to the state-of-the-art methods, BUAK substantially reduces the computational demand by one to three orders of magnitude, therefore, facilitating the application of Bayesian updating to computationally very intensive models.