## Section: New Results

### Reliability and Computer Experiments

Participants : Yves Auffray, Pierre Barbillon, Gilles Celeux, Pierre Connault, Pascal Massart.

In the framework of a convention with EDF, Gilles Celeux worked in collaboration with Yannick Lefebvre and Étienne de Rocquigny (EDF) on the resolution of not linear inverse problems for the quantification of uncertainties in a physical model. More precisely, noisy observed data (Y) were dependent, through a known but complex and expensive function H from non-observed data X . The aim is to estimate parameters of the probability distribution of the non observed data (X) and the variance of the noise. The problem has a missing data structure and can be solved with an EM-type algorithm coupled to an iterative linearisation of the function H ([10] ). However, the linearisations of the function H can be poor and this scheme misleading. Pierre Barbillon, Gilles Celeux and Agnès Grimaud (Université de Marseille) proposed a non-linearised method coupling the use of the Stochastic EM algorithm with a MCMC method and a Kriging approximation of the H function. The algorithm and its practical figures are described in [63] and is compared with iterated linear approximation on the basis of numerical experiments on simulated and real data sets. Situations where this non linear approach is to be preferred to linearisation are highlighted.

In aircraft equipment, fatigue is one of the first cause of ruptures. Morevover fatigue ruptures appear brutally and can be catastrophics: important material damage, human death. The fatigue rupture is a complex random process: it is influenced by numerous and various factors of the production process and environment. The large number of factors, and the strong variability of some of them, yield any expertise very difficult. select develops a collaboration with SAFRAN via the Phd of Pierre Connault, supervised by Pascal Massart and Patrick Pamphile (Université Paris-Sud). Variable selection methods (CART, LASSO) have been used to perform an efficient statistical control of aircraft equipment production processes. LASSO is a regularisation method for linear regression. It minimizes the sum of squared errors, with a penalty on the sum of the absolute values of the coefficients. The objectif of Pierre Connault is to calibrate automatically that penalty. Moreover, a probabilistic model of fatigue has been proposed to extrapolate results on tets tubes to assess the reliability of the entire equipment.

Yves Auffray and Pierre Barbillon have proposed a more natural and general definition of a conditionally positive definite kernel in [61] . From this definition, a full generalization of Aronszajna's theorem has been shown. It states for a conditionally positive definite kernel, the existence of a unique reproducing kernel semi-Hilbert space. Furthermore, they provided the interpolation operator on this space and showed that it is a generalization of the standard ones.

In the computer experiments field, the goal is to approximate an expensive black box function from
a limited number of evaluations. The choice of these evaluations i.e. the choice of a design of
(computer) experiments is a major issue.
Yves Auffray and Pierre Barbillon have justified with Jean-Michel Marin (Université de Montpellier)
to take a design satisfying to the maximin criterion by using results from the approximation
theory literature. In the case where the black box function is to be approximated on a hypercubic domain,
the standard strategy consists of taking a maximin design within a class of Latin hypercube Designs.
It can be done thanks to a well-known algorithm of Morris and Mitchell (1992). However, the Latin hypercube
sampling is pointless in a non hypercubic domain. In [62] , they proposed
a simulated annealing algorithm, implemented in *C* , which aims at obtaining a maximin design
in any bounded connected domain. They have proved the convergence of their algorithm.