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Modèle de marge, analyse de sensibilité avec des marges et quantification d'incertitudes dans des graphes de fonctions pour des systèmes industriels complexes

Adrien Touboul 1, 2 
2 MATHRISK - Mathematical Risk Handling
UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris
Abstract : This thesis focuses on two problems motivated by simulation in industrial complex systems design. The first part is dedicated to model and to identify costly design margins during the design process. A first challenge is to have a unified model for all the margin practices. To achieve this goal, we investigate the fundamental basis of the margin concept and provide a mathematical object to model it. With this model, we exhibit how margins are taken, independently of the field. Then, various margin practices from different engineering fields are modeled within this framework. Finally, some tools are developed, inspired from the sensitivity analysis domain, to identify which margins contribute the most to a cost or to a loss of performance. These works thus propose a non ambiguous approach to a quantitative analysis of margins and open a broad range of perspectives to model design problems that include margins.The second part studies an approach to uncertainty quantification in a multidisciplinary context. The design process is modeled by the composition of computer codes, that is represented by a directed acyclic graph with a function associated to each node. The inputs of each code are random variables, that can either come from the outputs of other disciplines (external variable) or be modeled by the discipline (internal variable). The method investigated is based on sample reweighting and allows for disciplinary autonomy. First, at each node and for each external variable, some synthetic samples that do not follow the true law are generated and the respective outputs are computed. Second, a method is chosen to weight to outputs with respect to the inputs, and these weights are propagated to the graph. The final result is a weighted sample whose law approximates the true joint law of the random variables. We begin our study by investigating the efficiency of a particular weighting method, based on a Wasserstein distance criterion. We derive an explicit version of the weights in terms of Nearest-Neighbor, we prove the consistency and derive some theoretical rates of convergence in terms of expected Wasserstein distance. Then, we concentrate on the propagation of the weights. We first define a family of weighting methods, that we call WLAMs (Weighted Linear Approximation Methods), for which we establish a local consistency criterion. Under the assumption of local consistency at each node, we prove the convergence towards the true joint law in the whole graph. Finally, we explicitly define a discrete Bayesian network that simplifies numerical computations in the propagation part and apply it to an industrial case
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Adrien Touboul. Modèle de marge, analyse de sensibilité avec des marges et quantification d'incertitudes dans des graphes de fonctions pour des systèmes industriels complexes. Variables complexes [math.CV]. École des Ponts ParisTech, 2021. Français. ⟨NNT : 2021ENPC0017⟩. ⟨tel-03435011⟩

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