Operator norm theory as an efficient tool to propagate hybrid uncertainties and calculate imprecise probabilities

Abstract

This paper presents a highly efficient and effective approach to bound the responses and probability of failure of linear systems where the model parameters are subjected to combinations of epistemic and aleatory uncertainty. These combinations can take the form of imprecise probabilities or hybrid uncertainties. Typically, such computations involve solving a nested double loop problem, where the propagation of the aleatory uncertainty has to be performed for each realisation of the epistemic uncertainty. Apart from near-trivial cases, such computation is intractable without resorting to surrogate modeling schemes. In this paper, a method is presented to break this double loop by virtue of the operator norm theorem. Indeed, in case linear models are considered and under the restriction that the model definition cannot be subject to aleatory uncertainty, the paper shows that the computational efficiency, quantified by the required number of model evaluations, of propagating these parametric uncertainties can be improved by several orders of magnitude.Two case studies involving a finite element model of a clamped plate and a six-story building are included to illustrate the application of the developed technique, as well as its computational merit in comparison to existing double-loop approaches.

Augmented reliability analysis for estimating imprecise first excursion probabilities in stochastic linear dynamics

“…In this sense, it should be noted that the scheme proposed in Section 3 can consider epistemic uncertainty on the loading as well. Studies on the effect of imprecise stochastic loads on the bounds on the first excursion probability of linear structures can be found in previous work of the authors [21,22,23]. These studies acknowledge the importance of including imprecision in the definition of stochastic loads modelled by advanced auto-correlation methods.…”

“…Furthermore, by virtue of Directional Importance Sampling, it is possible to derive an explicit approximation of the probability of failure as a function of the interval-valued hyper-parameters of the uncertain input quantities. The paper is complementary to earlier work by the authors, presented in [21,22,23], in the sense that it presents a conceptually totally different framework for the propagation of imprecise probabilities towards the bounds on first excursion probabilities. Indeed, whereas the work in [21,22,23] focuses on the operator norm framework to decouple the double loop, the methods presented in this paper are based on solving the problem in an augmented space.…”

“…The paper is complementary to earlier work by the authors, presented in [21,22,23], in the sense that it presents a conceptually totally different framework for the propagation of imprecise probabilities towards the bounds on first excursion probabilities. Indeed, whereas the work in [21,22,23] focuses on the operator norm framework to decouple the double loop, the methods presented in this paper are based on solving the problem in an augmented space.…”

“…Decoupling approaches, such as presented in recent work by some of the authors (see [21,22,23]) are proven to be extremely efficient, but their application is limited to linear models. Finally, the idea of using augmented space methods was introduced by [24] in the context of probabilistic failure analysis and further developed for sensitivity analysis and reliability-based optimization in, e.g.…”

Decision-Making with Risk under Interval Uncertainty Based on Area Metrics

“…Current methods to deal with uncertainty include probability theory [19,20], fuzzy theory [21][22][23], and Dempster-Shafer (D-S) evidence theory [24,25]. D-S evidence theory has a strong ability to deal with epistemic uncertainty.…”