On the Robust Estimation of Small Failure Probabilities for Strong Nonlinear Models

Abstract

Structural reliability methods are nowadays a cornerstone for the design of robustly performing structures, thanks to advancements in modeling and simulation tools. Monte Carlo-based simulation tools have been shown to provide the necessary accuracy and flexibility. While standard Monte Carlo estimation of the probability of failure is not hindered in its applicability by approximations or limiting assumptions, it becomes computationally unfeasible when small failure probability needs to be estimated, especially when the underlying numerical model evaluation is time consuming. In this case, variance reduction techniques are commonly employed, allowing for the estimation of small failure probabilities with a reduced number of samples and model calls. As a competing approach to variance reduction techniques, surrogate models can be used to substitute the computationally expensive model and performance function with an easy to evaluate numerical function calibrated through a supervised learning procedure. Both these tools provide accurate results for structural application. However, particular care should be taken into account when the reliability problems deal with high-dimensional or strongly nonlinear structural performances since the accuracy of the estimate is largely dependent on choices made during the surrogate modeling process. In this work, we compare the performance of the most recent state-of-the-art advance Monte Carlo techniques and surrogate models when applied to strongly nonlinear performance functions. This will provide the analysts with an insight to the issues that could arise in these challenging problems and help to decide with confidence on which tool to select in order to achieve accurate estimation of the failure probabilities within feasible times with their available computational capabilities.

Robust propagation of probability boxes by interval predictor models

“…to those obtained using Gaussian Process metamodels. This paper presents a major generalisation of the results in [11], by considering epistemic uncertainty in probability distribution functions, i.e. probability boxes.…”

“…Then the entire uncertainty in P f is due to the inaccuracy of the bounds as a consequence of Eqn. (11). When using a standard Monte Carlo simulation, without a metamodel, this uncertainty may be found by calculating P f − P f , where the bounds of the probability of failure are found by solving…”

“…This principle explains the scaling of the proposed method to high dimensional spaces, as typically functions in high dimensional spaces also require a higher d as the function requires more effort to approximate correctly. In [11] an iterative method is proposed to reduce d as much as possible to lessen the curse of dimensionality for the proposed method. Having a surrogate model is useful in engineering analysis as it can be repurposed for something else besides the original analysis.…”

“…The present article advances [24] by introducing new algorithms to yield tighter bounds on the failure probability, comprised of a different sampling strategy and improved techniques for creating IPMs, and providing more extensive numerical experiments. In [11], it is demonstrated that using IPM metamodels for the structural reliability problem with random variables, as opposed to probability boxes, produces bounds on the failure probability which are roughly equivalent This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ ©2019.…”

Analytic Probabilistic Safety Analysis under Severe Uncertainty

“…Although efficient simulation approaches for dealing with imprecision have recently been proposed (e.g. using Line Sampling (de Angelis et al 2015), Subset simulation (Patelli et al 2011) and tools to deal with probability boxes (Patelli et al 2014;Patelli et al 2017b;Faes et al 2019)), their use in the nuclear industry is not yet widespread. In Ref.…”