Uncertainty quantification by geometric characterization of sensitivity spaces

We propose a systematic procedure for both aleatory and epistemic uncertainty quantification of numerical simulations through geometric characteristics of global sensitivity spaces. Two mathematical concepts are used to characterize the geometry of these spaces and to identify possible impacts of variability in data or changes in the models or solution procedures: the dimension of the maximal free generator subspace in vector spaces and the principal angles between subspaces. We show how these characters can be used as indications on the aleatory and epistemic uncertainties. In the case of large dimensional parameter spaces, these characterizations are established for quantile-based extreme scenarios and a multi-point moment-based sensitivity direction permits to propose a directional uncertainty quantification concept for directional extreme scenarios (DES). The approach is non-intrusive and exploits in parallel the elements of existing mono-point gradient-based design platforms. The ingredients of the paper are illustrated on a model problem with the Burgers equation with control and on a constrained aerodynamic performance analysis problem.