Uncertainty propagation of p-boxes using sparse polynomial chaos expansions

Abstract

In modern engineering, physical processes are modelled and analysed using advanced computer simulations, such as finite element models. Furthermore, concepts of reliability analysis and robust design are becoming popular, hence, making efficient quantification and propagation of uncertainties an important aspect. In this context, a typical workflow includes the characterization of the uncertainty in the input variables. In this paper, input variables are modelled by probability-boxes (p-boxes), accounting for both aleatory and epistemic uncertainty. The propagation of p-boxes leads to p-boxes of the output of the computational model. A two-level meta-modelling approach is proposed using non-intrusive sparse polynomial chaos expansions to surrogate the exact computational model and, hence, to facilitate the uncertainty quantification analysis. The capabilities of the proposed approach are illustrated through applications using a benchmark analytical function and two realistic engineering problem settings. They show that the proposed two-level approach allows for an accurate estimation of the statistics of the response quantity of interest using a small number of evaluations of the exact computational model. This is crucial in cases where the computational costs are dominated by the runs of high-fidelity computational models.

Structural reliability analysis for p-boxes using multi-level meta-models

“…The definition of parametric p-boxes indicates a hierarchical model where the distribution of X is defined conditionally on its distribution parameters. Hence, nested simulation algorithms can be applied in the context of ISRA, as discussed in Eldred and Swiler (2009);Schöbi and Sudret (2015b). In other words, the bounds of the failure probability can be found by estimating the conditional failure probabilities P f |θ = P (G (X θ ) ≤ 0), where X θ is a conditional distribution with F X θ (x) = F X (x|θ), and minimizing/maximizing it with respect to θ to get the bounds P f = min θ∈D Θ P f |θ and P f = max θ∈D Θ P f |θ .…”

Evaluation of failure probability under parameter epistemic uncertainty: application to aerospace system reliability assessment

“…From a numerical point of view, a first way of computing this integral relies on evaluating pointwise the inner integral for each realization θ of [14][15][16]: this leads to the nested reliability approach (presented in subsection 3.1). The second way consists in evaluating it by treating both basic variables and uncertain distribution parameters together and by integrating simultaneously on both domains (but still respecting the conditioning) as suggested in [17]: this is the augmented reliability approach (presented in subsection 3.2).…”

“…In practice, it consists in computing several P f (θ ) for a range of realizations θ of the vector of uncertain parameters . It has been widely used in literature, in various contexts, such as rare event probability estimation with Kriging-based approach in [14], probability-based tolerance analysis of products in [15] or uncertainty propagation using probability-boxes and polynomial chaos expansions in [16].…”

“…(7c) (see [16] for a similar definition). Thus, this augmented input space has a dimension of k + d (k uncertain distribution parameters j and d random basic variables X i ).…”

“…A classical approach, known as the "nested reliability approach" (NRA), is to consider sampling of uncertain distribution parameters and to perform a nested reliability analysis for each realization of these parameters (see, for example [14] for reliability assessment under epistemic uncertainty on distribution parameters given by intervals, [15] in the context of probability-based tolerance analysis of products or [16] for a coupling with Sparse Polynomial Chaos Expansions). Nonetheless, it implies to repeat several times a costly reliability analysis, which can be unaffordable for most cases in a complex industrial environment.…”