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An overview on uncertainty quantification and probabilistic learning on manifolds in multiscale mechanics of materials

Christian Soize

Vol. 11 (2023), No. 1, 87–174
DOI: 10.2140/memocs.2023.11.87
Abstract

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.

Keywords
uncertainty quantification, probabilistic learning, stochastic homogenization, heterogeneous material, multiscale mechanics
Mathematical Subject Classification
Primary: 60G60, 60J20, 62M40, 74Q05
Milestones
Received: 7 February 2023
Accepted: 13 May 2023
Published: 23 October 2023

Communicated by Francesco dell'Isola
Authors
Christian Soize
Université Gustave Eiffel
MSME UMR 8208 CNRS
77454 Marne-la-Vallée
France