Set inversion under functional uncertainties with joint meta-models

In this paper we propose an efficient sampling strategy to solve inversion problem under functional uncertainty. This approach aims to characterize region of a control space defined by exceedance above prescribed threshold. This study is motivated by an application on identifying the set of control parameters leading to meet the pollutant emission standards of a vehicle under driving profile uncertainties. In that context, the constrained response in the inversion problem is here formulated as the expectation over the functional random variable only known through a set of realizations and the unknown set is thus associated with the control variables. As often in industrial applications, this problem involves high-fidelity and time-consuming computational models. We thus proposed an approach that makes use of Gaussian Process meta-models built on the joint space of control and uncertain input variables. Specifically, we define a design criterion based on uncertainty in the excursion of the Gaussian Process and derive tractable expressions for the variance reduction in such a framework. Applications to analytical examples, followed by the automotive industrial test case show the accuracy and the efficiency brought by the proposed procedure.