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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.

Dang, C., Wei, P. F., Faes, M. G. R., Valdebenito, M. A., & Beer, M. (2022). Interval uncertainty propagation by a parallel Bayesian global optimization method. Appl. Math. Model., 108, 220–235.
Abstract: This paper is concerned with approximating the scalar response of a complex computational model subjected to multiple input interval variables. Such task is formulated as finding both the global minimum and maximum of a computationally expensive blackbox function over a prescribed hyperrectangle. On this basis, a novel nonintrusive method, called `tripleengine parallel Bayesian global optimization', is proposed. The method begins by assuming a Gaussian process prior (which can also be interpreted as a surrogate model) over the response function. The main contribution lies in developing a novel infill sampling criterion, i.e., tripleengine pseudo expected improvement strategy, to identify multiple promising points for minimization and/or maximization based on the past observations at each iteration. By doing so, these identified points can be evaluated on the real response function in parallel. Besides, another potential benefit is that both the lower and upper bounds of the model response can be obtained with a single run of the developed method. Four numerical examples with varying complexity are investigated to demonstrate the proposed method against some existing techniques, and results indicate that significant computational savings can be achieved by making full use of prior knowledge and parallel computing.

Dang, C., Wei, P. F., Faes, M. G. R., Valdebenito, M. A., & Beer, M. (2022). Parallel adaptive Bayesian quadrature for rare event estimation. Reliab. Eng. Syst. Saf., 225, 108621.
Abstract: Various numerical methods have been extensively studied and used for reliability analysis over the past several decades. However, how to understand the effect of numerical uncertainty (i.e., numerical error due to the discretization of the performance function) on the failure probability is still a challenging issue. The active learning probabilistic integration (ALPI) method offers a principled approach to quantify, propagate and reduce the numerical uncertainty via computation within a Bayesian framework, which has not been fully investigated in context of probabilistic reliability analysis. In this study, a novel method termed `Parallel Adaptive Bayesian Quadrature' (PABQ) is proposed on the theoretical basis of ALPI, and is aimed at broadening its scope of application. First, the Monte Carlo method used in ALPI is replaced with an importance ball sampling technique so as to reduce the sample size that is needed for rare failure event estimation. Second, a multipoint selection criterion is proposed to enable parallel distributed processing. Four numerical examples are studied to demonstrate the effectiveness and efficiency of the proposed method. It is shown that PABQ can effectively assess small failure probabilities (e.g., as low as 10(7)) with a minimum number of iterations by taking advantage of parallel computing.

Ding, C., Dang, C., Valdebenito, M. A., Faes, M. G. R., Broggi, M., & Beer, M. (2023). Firstpassage probability estimation of highdimensional nonlinear stochastic dynamic systems by a fractional momentsbased mixture distribution approach. Mech. Syst. Sig. Process., 185, 109775.
Abstract: Firstpassage probability estimation of highdimensional nonlinear stochastic dynamic systems is a significant task to be solved in many science and engineering fields, but remains still an open challenge. The present paper develops a novel approach, termed 'fractional momentsbased mixture distribution', to address such challenge. This approach is implemented by capturing the extreme value distribution (EVD) of the system response with the concepts of fractional moment and mixture distribution. In our context, the fractional moment itself is by definition a highdimensional integral with a complicated integrand. To efficiently compute the fractional moments, a parallel adaptive sampling scheme that allows for sample size extension is developed using the refined Latinized stratified sampling (RLSS). In this manner, both variance reduction and parallel computing are possible for evaluating the fractional moments. From the knowledge of loworder fractional moments, the EVD of interest is then expected to be reconstructed. Based on introducing an extended inverse Gaussian distribution and a log extended skewnormal distribution, one flexible mixture distribution model is proposed, where its fractional moments are derived in analytic form. By fitting a set of fractional moments, the EVD can be recovered via the proposed mixture model. Accordingly, the firstpassage probabilities under different thresholds can be obtained from the recovered EVD straightforwardly. The performance of the proposed method is verified by three examples consisting of two test examples and one engineering problem.

Zhao, W. H., Yang, L. C., Dang, C., Rocchetta, R., Valdebenito, M., & Moens, D. (2022). Enriching stochastic model updating metrics: An efficient Bayesian approach using BrayCurtis distance and an adaptive binning algorithm. Mech. Syst. Sig. Process., 171, 108889.
Abstract: In practical engineering, experimental data is not fully in line with the true system response due to various uncertain factors, e.g., parameter imprecision, model uncertainty, and measurement errors. In the presence of mixed sources of aleatory and epistemic uncertainty, stochastic model updating is a powerful tool for model validation and parameter calibration. This paper investigates the use of BrayCurtis (BC) distance in stochastic model updating and proposes a Bayesian approach addressing a scenario where the dataset contains multiple outliers. In the proposed method, a BC distancebased uncertainty quantification metric is employed, that rewards models for which the discrepancy between observations and simulated samples is small while penalizing those which exhibit large differences. To improve the computational efficiency, an adaptive binning algorithm is developed and embedded into the Bayesian approximate computation framework. The merit of this algorithm is that the number of bins is automatically selected according to the difference between the experimental data and the simulated data. The effectiveness and efficiency of the proposed method is verified via two numerical cases and an engineering case from the NASA 2020 UQ challenge. Both static and dynamic cases with explicit and implicit propagation models are considered.
