Uncertainty and robustness analysis for models with functional inputs and outputs.

Summary The objective of this thesis is to solve a problem of inversion under uncertainties of expensive functions to be evaluated within the framework of the parameterization of the control of a system of depollution of vehicles.The effect of these uncertainties is taken into account through the expectation of the quantity of interest. A difficulty lies in the fact that the uncertainty is partly due to a known functional input through a given sample. We propose two approaches based on an approximation of the costly code by Gaussian processes and a reduction of the dimension of the functional variable by a Karhunen-Loève method.The first approach consists in applying a SUR (Stepwise Uncertainty Reduction) inversion method on the expectation of the quantity of interest. At each evaluation point in the control space, the expectation is estimated by a gluttonous functional quantization method that provides a discrete representation of the functional variable and an efficient sequential estimation from the given sample of the functional variable.The second approach consists in applying the SUR method directly on the quantity of interest in the joint space of the control variables and the uncertain variables. A strategy of enrichment of the design of experiments dedicated to the inversion under functional uncertainties and exploiting the properties of Gaussian processes is proposed.These two approaches are compared on toy functions and are applied to an industrial case of after-treatment of exhaust gases of a vehicle. The problem is to determine the control settings of the system allowing the respect of the pollution control standards in the presence of uncertainties on the driving cycle.