Set inversion under functional uncertainties with Gaussian Process Regression defined in the joint space of control and uncertain

Abstract : In this paper we propose an efficient sampling strategy to solve an inversion problem subjected to functional uncertainties. More precisely, we aim at characterizing a control variable region defined by exceedance above a prescribed threshold of specific Quantities of Interest (QoT). This study is motivated by an automotive industrial application consisting in the identification of the set of values of control variables of a gas after-treatment device, in line with pollutant emission standards of a vehicle under driving profile uncertainties. In that context, driving profile uncertainties are modelled by a functional random variable and the constrained response in the inversion problem is formulated as the expectation over this functional random variable only known through a set of realizations. As often in industrial applications, this problem involves time-consuming computational models. We thus propose an approach that uses Gaussian Process meta-models built on the joint space of control and uncertain input variables. Specifically, we define a learning criterion based on uncertainty in the excursion of the Gaussian Process and derive tractable expressions for 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 procedure we propose.