An overview of the author’s works, many of which were carried out in collaboration,
is presented. The first part concerns the quantification of uncertainties for
complex engineering science systems for which analyses are now carried out
using large numerical simulation models. More recently, machine learning
methods have appeared in this field to address certain problems of nonconvex
optimization under uncertainties and inverse identification, which are not affordable
with standard computer resources. Thus the second part is relative to the
presentation of a method of probabilistic learning on manifolds recently proposed
for the case of small data and which makes it possible to build statistical
surrogate models useful to perform probabilistic inferences. The illustrations are
mainly focused on the multiscale analyses of microstructures made up of
heterogeneous continuous materials, which cannot be described in terms of
constituents and which are modeled with stochastic apparent quantities at
mesoscale.
