Performance Based Design and Eurocode

Codes and Standard

Consensus-based rare event estimation

In this paper, we introduce a new algorithm for rare event estimation based on adaptive importance sampling. We consider a smoothed version of the optimal importance sampling density, which is approximated by an ensemble of interacting particles. The particle dynamics is governed by a McKean-Vlasov stochastic differential equation, which was introduced and analyzed in (Carrillo et al., Stud. Appl. Math. 148:1069-1140, 2022) for consensus-based sampling and optimization of posterior distributions arising in the context of Bayesian inverse problems. We develop automatic updates for the internal parameters of our algorithm. This includes a novel time step size controller for the exponential Euler method, which discretizes the particle dynamics. The behavior of all parameter updates depends on easy to interpret accuracy criteria specified by the user. We show in numerical experiments that our method is competitive to state-of-the-art adaptive importance sampling algorithms for rare event estimation, namely a sequential importance sampling method and the ensemble Kalman filter for rare event estimation

Variance-based reliability sensitivity with dependent inputs using failure samples

Reliability sensitivity analysis is concerned with measuring the influence of a system's uncertain input parameters on its probability of failure. Statistically dependent inputs present a challenge in both computing and interpreting these sensitivity indices; such dependencies require discerning between variable interactions produced by the probabilistic model describing the system inputs and the computational model describing the system itself. To accomplish such a separation of effects in the context of reliability sensitivity analysis we extend on an idea originally proposed by Mara and Tarantola (2012) for model outputs unrelated to rare events. We compute the independent (influence via computational model) and full (influence via both computational and probabilistic model) contributions of all inputs to the variance of the indicator function of the rare event. We compute this full set of variance-based sensitivity indices of the rare event indicator using a single set of failure samples. This is possible by considering $d$ different hierarchically structured isoprobabilistic transformations of this set of failure samples from the original $d$-dimensional space of dependent inputs to standard-normal space. The approach facilitates computing the full set of variance-based reliability sensitivity indices with a single set of failure samples obtained as the byproduct of a single run of a sample-based rare event estimation method. That is, no additional evaluations of the computational model are required. We demonstrate the approach on a test function and two engineering problems

Bayesian improved cross entropy method with categorical mixture models

We employ the Bayesian improved cross entropy (BiCE) method for rare event estimation in static networks and choose the categorical mixture as the parametric family to capture the dependence among network components. At each iteration of the BiCE method, the mixture parameters are updated through the weighted maximum a posteriori (MAP) estimate, which mitigates the overfitting issue of the standard improved cross entropy (iCE) method through a novel balanced prior, and we propose a generalized version of the expectation-maximization (EM) algorithm to approximate this weighted MAP estimate. The resulting importance sampling distribution is proved to be unbiased. For choosing a proper number of components $K$ in the mixture, we compute the Bayesian information criterion (BIC) of each candidate $K$ as a by-product of the generalized EM algorithm. The performance of the proposed method is investigated through a simple illustration, a benchmark study, and a practical application. In all these numerical examples, the BiCE method results in an efficient and accurate estimator that significantly outperforms the standard iCE method and the BiCE method with the independent categorical distribution

Life Quality Index for Assessing Risk Acceptance in Geotechnical Engineering

Hazard

Bayesian inference with Subset Simulation: Strategies and improvements

Bayesian Updating with Structural reliability methods (BUS) reinterprets the Bayesian updating problem as a structural reliability problem; i.e. a rare event estimation. The BUS approach can be considered an extension of rejection sampling, where a standard uniform random variable is added to the space of random variables. Each generated sample from this extended random variable space is accepted if the realization of the uniform random variable is smaller than the likelihood function scaled by a constant c. The constant c has to be selected such that 1∕c is not smaller than the maximum of the likelihood function, which, however, is typically unknown a-priori. A c chosen too small will have negative impact on the efficiency of the BUS approach when combined with sampling-based reliability methods. For the combination of BUS with Subset Simulation, we propose an approach, termed aBUS, for adaptive BUS, that does not require c as input. The proposed algorithm requires only minimal modifications of standard BUS with Subset Simulation. We discuss why aBUS produces samples that follow the posterior distribution –even if 1∕c is selected smaller than the maximum of the likelihood function. The performance of aBUS in terms of the computed evidence required for Bayesian model class selection and in terms of the produced posterior samples is assessed numerically for different example problems. The combination of BUS with Subset Simulation (and aBUS in particular) is well suited for problems with many uncertain parameters and for Bayesian updating of models where it is computationally demanding to evaluate the likelihood function