AN OVERVIEW ON UNCERTAINTY QUANTIFICATION AND PROBABILISTIC LEARNING ON MANIFOLDS IN MULTISCALE MECHANICS OF MATERIALS

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.