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

•New method to propagate hybrid uncertainties is introduced.•The method fully decouples the double loop during uncertainty propagation.•Method applicable to linear models with epistemic uncertainty.•Two case studies show a gain in computational efficiencyof orders of magnitude .•The obtained results are exact for linear models. 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.