Méthodes probabilistes pour l'estimation de probabilités de défaillance

Abstract : To evaluate the profitability of a production before the launch of the manufacturing process, most industrial companies use numerical simulation. This allows to test virtually several configurations of the parameters of a given product and to decide on its performance (defined by the specifications). In order to measure the impact of industrial process fluctuations on product performance, we are particularly interested in estimating the probability of failure of the product. Since each simulation requires the execution of a complex and expensive calculation code, it is not possible to perform a sufficient number of tests to estimate this probability using, for example, a Monte-Carlo method. Under the constraint of a limited number of code calls, we propose two very different estimation methods. The first is based on the principles of Bayesian estimation. Our observations are the results of numerical simulation. The probability of failure is seen as a random variable, the construction of which is based on that of a random process to model the expensive calculation code. To correctly define this model, the Kriging method is used. Conditionally to the observations, the posterior distribution of the random variable, which models the probability of failure, is inaccessible. To learn about this distribution, we construct approximations of the following characteristics: expectation, variance, quantiles... We use the theory of stochastic orders to compare random variables and, more specifically, the convex order. The construction of an optimal experimental design is ensured by the implementation of a sequential experimental planning procedure, based on the principle of the SUR ("Stepwise Uncertainty Reduction") strategies. The second method is an iterative procedure, particularly adapted to the case where the probability of failure is very small, i.e. the redoubt event is rare. The expensive calculation code is represented by a function that is assumed to be Lipschitz continuous. At each iteration, this hypothesis is used to construct approximations, by default and by excess, of the probability of failure. We show that these approximations converge towards the true value with the number of iterations. In practice, they are estimated using the Monte-Carlo method known as splitting method. The proposed methods are relatively simple to implement and the results they provide can be easily interpreted. We test them on various examples, as well as on a real case from STMicroelectronics.
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Lucie Bernard. Méthodes probabilistes pour l'estimation de probabilités de défaillance. Probabilités [math.PR]. Université de Tours, 2019. Français. ⟨tel-02279258⟩

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