Engineering analysis with probability boxes: A review on computational methods

Abstract

The consideration of imprecise probability in engineering analysis to account for missing, vague or incomplete data in the description of model uncertainties is a fast-growing field of research. Probability-boxes (p-boxes) are of particular interest in an engineering context, since they offer a mathematically straightforward description of imprecise probabilities, as well as allow for an intuitive visualisation. In essence, p-boxes are defined via lower and upper bounds on the cumulative distribution function of a random variable whose exact probability distribution is unknown. However, the propagation of p-boxes on model inputs towards bounds on probabilistic measures describing the uncertainty on the model responses is numerically still very demanding, and hence is subject of intensive research. In order to provide an overview on the available methods, this paper gives a state-of-the art review for the modelling and propagation of p-boxes with a special focus on structural reliability analysis.

Structural Safety of the Steel Hall under Dynamic Excitation Using the Relative Probabilistic Entropy Concept

“…It may also be valuable to apply the relative probabilistic entropy with the method based on the exact determination of probability distributions for the structural response [ 47 ], where the moments equations for the reliability index may not be necessary at all. Common application of the proposed approach with the so-called p-box methodology [ 48 ] may also be promising, because numerical values of the upper and lower bounds and the PDF formulas introduced in the p-box approach can increase the precision of interval simulation of both direct and relative probabilistic entropies.…”

Random set solutions to elliptic and hyperbolic partial differential equations

“…Following [20], problem (11) can be solved in the following way. First, the equations 1 − α tan α = 0, respectively α + 1 tan α = 0 (12) have two sequences of positive solutions denoted by α k = α k ( ) and α * k = α * k ( ), for even and odd k, respectively. The resulting eigenvalues and eigenfunctions are…”

“…Here is a short list of methods currently being developed for the reduction of computational cost in imprecise probability models, in particular, for imprecise random fields: Approximations by inner and outer bounds [2,3,57,64]; propagation methods for p-boxes [11,12,48,62,63]; methods employing polynomial chaos expansion [31,43,52]; probability plots, an enhancement of the first order reliability method (FORM) proposed by [26]. A very efficient method appears to be advanced line sampling [8,56] that intertwines the two required loops and can reduce M and N simultaneously.…”

“…[13,45]. Papers laying special emphasis on comparing stochastic and interval methods are [12,18,27,37,53,54]. Further references on modelling with random sets can be found at the end of Section 2.3.…”

A Decoupling Strategy for Reliability Analysis of Multidisciplinary System with Aleatory and Epistemic Uncertainties

“…In RBMDO, the probabilistic model on basis of a large amount of statistical data is the most common method to quantify aleatory uncertainty and it has achieved great success [1,2]. With the development of artificial intelligence technology, some machine learning and advanced statistical framework, such as probability boxes [3], are introduced into the field of reliability evaluation. Xiang et al [4] proposed a deep reinforcement learning-based sampling method for reliability analysis, which uses a deep neural network as agent to select test points automatically and construct the surrogate model for reliability assessment.…”