Dang, C., Valdebenito, M. A., Faes, M. G. R., Wei, P. F., & Beer, M. (2022). Structural reliability analysis: A Bayesian perspective. Struct. Saf., 99, 102259.
Abstract: Numerical methods play a dominant role in structural reliability analysis, and the goal has long been to produce a failure probability estimate with a desired level of accuracy using a minimum number of performance function evaluations. In the present study, we attempt to offer a Bayesian perspective on the failure probability integral estimation, as opposed to the classical frequentist perspective. For this purpose, a principled Bayesian Failure Probability Inference (BFPI) framework is first developed, which allows to quantify, propagate and reduce numerical uncertainty behind the failure probability due to discretization error. Especially, the posterior variance of the failure probability is derived in a semianalytical form, and the Gaussianity of the posterior failure probability distribution is investigated numerically. Then, a Parallel AdaptiveBayesian Failure Probability Learning (PABFPL) method is proposed within the Bayesian framework. In the PABFPL method, a varianceamplified importance sampling technique is presented to evaluate the posterior mean and variance of the failure probability, and an adaptive parallel active learning strategy is proposed to identify multiple updating points at each iteration. Thus, a novel advantage of PABFPL is that both prior knowledge and parallel computing can be used to make inference about the failure probability. Four numerical examples are investigated, indicating the potential benefits by advocating a Bayesian approach to failure probability estimation.

Fina, M., Lauff, C., Faes, M. G. R., Valdebenito, M. A., Wagner, W., & Freitag, S. (2023). Bounding imprecise failure probabilities in structural mechanics based on maximum standard deviation. Struct. Saf., 101, 102293.
Abstract: This paper proposes a framework to calculate the bounds on failure probability of linear structural systems whose performance is affected by both random variables and interval variables. This kind of problems is known to be very challenging, as it demands coping with aleatoric and epistemic uncertainty explicitly. Inspired by the framework of the operator norm theorem, it is proposed to consider the maximum standard deviation of the structural response as a proxy for detecting the crisp values of the interval parameters, which yield the bounds of the failure probability. The scope of application of the proposed approach comprises linear structural systems, whose properties may be affected by both aleatoric and epistemic uncertainty and that are subjected to (possibly imprecise) Gaussian loading. Numerical examples indicate that the application of such proxy leads to substantial numerical advantages when compared to a traditional doubleloop approach for coping with imprecise failure probabilities. In fact, the proposed framework allows to decouple the propagation of aleatoric and epistemic uncertainty.

Yuan, X. K., Liu, S. L., Valdebenito, M. A., Gu, J., & Beer, M. (2021). Efficient procedure for failure probability function estimation in augmented space. Struct. Saf., 92, 102104.
Abstract: An efficient procedure is proposed to estimate the failure probability function (FPF) with respect to design variables, which correspond to distribution parameters of basic structural random variables. The proposed procedure is based on the concept of an augmented reliability problem, which assumes the design variables as uncertain by assigning a prior distribution, transforming the FPF into an expression that includes the posterior distribution of those design variables. The novel contribution of this work consists of expressing this target posterior distribution as an integral, allowing it to be estimated by means of sampling, and no distribution fitting is needed, leading to an efficient estimation of FPF. The proposed procedure is implemented within three different simulation strategies: Monte Carlo simulation, importance sampling and subset simulation; for each of these cases, expressions for the coefficient of variation of the FPF estimate are derived. Numerical examples illustrate performance of the proposed approaches.
