Breaking the double loop: Operator norm theory as a tool to compute with imprecise probabilities

In engineering analysis, numerical models are being increasingly used for the approximation of the real-life behavior of components and structures. In this context, a designer is often faced with uncertain and inherently variable model quantities, which are respectively represented by epistemic and aleatory uncertainties. To ensure interpretability, and hence, correct engineering decisions, these sources of uncertainty must remain strictly separated during the analysis. In case an analyst is faced with combinations of epistemic and aleatory uncertainty, which can take the form of imprecise probabilities (e.g., stochastic quantities with imprecisely defined hyper-parameters) or hybrid uncertainties (combinations of stochastic quantities, intervals and/or imprecise probabilities), the computation of the bounds on the reliability involves solving a set of nested optimization problems (a.k.a., “the double loop”), where the calculation of the reliability of the structure has to be performed for each realisation of the epistemic uncertainty. 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, the paper shows that the computational efficiency of propagating these uncertainties can be reduced to solving two optimization problems and two calculations of the structural reliability. A case study involving a finite element model of a clamped plate is included to illustrate the application, efficiency and effectivity of the developed technique.