Rigorous verification, validation, uncertainty quantification and certification through concentration-of-measure inequalities

Abstract

We apply concentration-of-measure inequalities to the quantification of uncertainties in the performance of engineering systems. Specifically, we envision uncertainty quantification in the context of certification, i.e., as a tool for deciding whether a system is likely to perform safely and reliably within design specifications. We show that concentration-of-measure inequalities rigorously bound probabilities of failure and thus supply conservative certification criteria. In addition, they supply unambiguous quantitative definitions of terms such as margins, epistemic and aleatoric uncertainties, verification and validation measures, confidence factors, and others, as well as providing clear procedures for computing these quantities by means of concerted simulation and experimental campaigns. We also investigate numerically the tightness of concentration-of-measure inequalities with the aid of an imploding ring example. Our numerical tests establish the robustness and viability of concentration-of-measure inequalities as a basis for certification in that particular example of application.