P-box fields: definition, analysis and practical applications

• Applicant: Professor Dr. Matthias Faes
• Subject Area: Engineering Design, Machine Elements, Product Development ; Mechanics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 530326817

Since structures are nowadays designed up to their performance limits, uncertainty quantification (UQ) is being increasingly applied in computer aided engineering to account for the fact that not all parameters of the computer models can be exactly quantified. Especially when these parameters vary within a single component, such accurate quantification is challenging due to typical limitations on the available data in terms of spatial and/or stochastic resolution. Classically, two competing modelling approaches exist: random fields, based on the well-established framework of probability theory, and interval fields, which start from an interval description of the uncertainty. Both approaches represent extreme situations. Indeed, a random field description provides a wealth of information on the spatial uncertainty, but also requires data with high spatial resolution and replicated experiments. Interval approaches on the other hand require only very limited data to provide an objective and accurate estimation of the uncertainty, however at the cost that only information on the bounds of the responses of the model is available. Further, often, relevant information needs to be discarded in the process to accommodate the interval paradigms. As such, there is a clear gap for uncertainty models to represent spatial uncertainty which can deal with realistic, limited, datasets, while still providing useful information to the analyst. These methods should furthermore be capable of aggregating all sources of available information. The objective of this project is to develop p-box fields to accommodate these needs, based on the framework of imprecise probabilities. The project builds on prior work of the applicant, where he introduced p-box fields based on an extension of translation theory towards imprecise probabilities. However promising results were obtained, many open questions are still to be answered. These questions concern the joint intervalrandom nature of the realisations of the p-box field, the bounds on the moments, resp., moments on the bounds of its realisations, as well as the definition and interpretation of the autocorrelation structure. It is furthermore unclear how p-box fields need to be extended towards the modelling of multivariate phenomena. Finally, there is still a clear lack of efficient numerical schemes to propagate p-box fields through expensive to calculate black box models. This project aims at answering these questions by deepening the initial work of the applicant on the theoretical foundation of p-box fields, as well as by proposing novel approaches to model the autocorrelation in a p-box field. Further, also advanced numerical schemes will be developed based on Bayesian integration that allow for a fast and accurate propagation of the uncertainty. These developments are foreseen in a close link with relevant application domains such as designing parts produced via Additive Manufacturing, and Geotechnical engineering.

• DFG Programme: Research Grants