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.

Faes, M. G. R., & Valdebenito, M. A. (2021). Fully decoupled reliabilitybased optimization of linear structures subject to Gaussian dynamic loading considering discrete design variables. Mech. Syst. Sig. Process., 156, 107616.
Abstract: Reliabilitybased optimization (RBO) offers the possibility of finding an optimal design for a system according to a prescribed criterion while explicitly taking into account the effects of uncertainty. However, due to the necessity of solving simultaneously a reliability problem nested in an optimization procedure, the corresponding computational cost is usually high, impeding the applicability of the methods. This computational cost is even further enlarged when one or several design variables must belong to a discrete set, due to the requirement of resorting to integer programming optimization algorithms. To alleviate this issue, this contribution proposes a fully decoupled approach for a specific class of problems, namely minimization of the failure probability of a linear system subjected to an uncertain dynamic load of the Gaussian type, under the additional constraint that the design variables are integervalued. Specifically, by using the operator norm framework, as developed by the authors in previous work, this paper shows that by reducing the RBO problem with discrete design variables to the solution of a single deterministic optimization problem followed by a single reliability analysis, a large gain in numerical efficiency can be obtained without compromising the accuracy of the resulting optimal design. The application and capabilities of the proposed approach are illustrated by means of three examples.

Song, J. W., Wei, P. F., Valdebenito, M. A., Faes, M., & Beer, M. (2021). Datadriven and active learning of variancebased sensitivity indices with Bayesian probabilistic integration. Mech. Syst. Sig. Process., 163, 108106.
Abstract: Variancebased sensitivity indices play an important role in scientific computation and data mining, thus the significance of developing numerical methods for efficient and reliable estimation of these sensitivity indices based on (expensive) computer simulators and/or data cannot be emphasized too much. In this article, the estimation of these sensitivity indices is treated as a statistical inference problem. Two principle lemmas are first proposed as rules of thumb for making the inference. After that, the posterior features for all the (partial) variance terms involved in the main and total effect indices are analytically derived (not in closed form) based on Bayesian Probabilistic Integration (BPI). This forms a datadriven method for estimating the sensitivity indices as well as the involved discretization errors. Further, to improve the efficiency of the developed method for expensive simulators, an acquisition function, named Posterior Variance Contribution (PVC), is utilized for realizing optimal designs of experiments, based on which an adaptive BPI method is established. The application of this framework is illustrated for the calculation of the main and total effect indices, but the proposed two principle lemmas also apply to the calculation of interaction effect indices. The performance of the development is demonstrated by an illustrative numerical example and three engineering benchmarks with finite element models.

Yuan, X. K., Liu, S. L., Faes, M., Valdebenito, M. A., & Beer, M. (2021). An efficient importance sampling approach for reliability analysis of timevariant structures subject to timedependent stochastic load. Mech. Syst. Sig. Process., 159, 107699.
Abstract: Structural performance is affected by deterioration processes and external loads. Both effects may change over time, posing a challenge for conducting reliability analysis. In such context, this contribution aims at assessing the reliability of structures where some of its parameters are modeled as random variables, possibly including deterioration processes, and which are subjected to stochastic load processes. The approach is developed within the framework of importance sampling and it is based on the concept of composite limit states, where the timedependent reliability problem is transformed into a series system with multiple performance functions. Then, an efficient twostep importance sampling density function is proposed, which splits timeinvariant parameters (random variables) from the timevariant ones (stochastic processes). This importance sampling scheme is geared towards a particular class of problems, where the performance of the structural system exhibits a linear dependency with respect to the stochastic load for fixed time. This allows calculating the reliability associated with the series system most efficiently. Practical examples illustrate the performance of the proposed approach.

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.
