École Nationale Supérieure de
Techniques Avancées

Abstract

This Master thesis deals with rare event probability estimation with non parametric importance sampling. The rare event corresponds here to the failure of a complex system in an aerospace context. When the random failure of the system induces huge consequences, then system reliability becomes crucial. The use of importance sampling to estimate the failure probability can improve the convergence rate of the probability estimator, compared to the usual Monte Carlo method. Importance sampling consists in considering an auxiliary sampling density which generate more failing samples than Monte Carlo method. The non parametric aspect corresponds then in the non parametric learning of this auxiliary density. The non parametric method is handy when estimating a failure probability knowing only a few elements. Kernel density estimation is a non parametric density estimation method which has been proven. However this approach becomes less efficient when the dimension of the density to be learnt goes up. The solution exposed in this Master thesis is then to estimate the auxiliary density of the importance sampling by decomposing it like a product of marginal densities and a copula. The use of copulas allows to define in a better way the dependence between the input variables between each other, especially when the number of inputs is large. The marginals are each evaluated with a kernel density estimator of dimension 1. The results obtained in this manuscript show that this copula-marginal decomposition of the auxiliary density improves the precision of the failure probability estimation by importance sampling. However some of the limits of importance sampling prevent this technique to be extended to a complete adaptive algorithm.