Publications and Preprints
I am mainly interested in multiscale problems with applications to continuum mechanics, both from the analytic and computational points of view. In particular, I focus on homogenization problems for PDEs, multiple integrals and discrete operators in various frameworks (periodic, stochastic, etc.).
I am looking more specifically at three issues:
A very longterm goal is to derive the theory of nonlinear elastodynamics (or nonlinear viscoelasticity) from the dynamics of stochastic networks of polymer chains for rubber elasticity. Although this is currently out of reach (think of general hyperbolic systems of conservation laws), some progress can be made for several simplified models or in restricted frameworks, such as:
The stochastic homogenization of linear elliptic equations is a theory which justifies that one can replace highly oscillating random coefficients by homogeneous deterministic ones. Standard homogenization results ensure that the solutions to the original (stochastic) and homogenized (deterministic) problems are close to each other. A quantitative theory aims at making quantitative these qualitative convergence results. Some definite progress was recently made on random discrete elliptic equations (see publication page). The longterm objective is to obtain optimal estimates and characterize the statistics of the solutions for linear and nonlinear, discrete and continuum elliptic equations.
From the computational point of view, the objective is twofold:
Roughly speaking, I try to fill the mathematical blanks in the following sketch, for the models above.
For details, see my publication page.
Students and post-docs
Stella Krell (Post-doc, Oct. 2010 --- Oct. 2011, financement ANDRA)
Maya de Buhan (Post-doc, Dec. 2010 --- Dec. 2011, financement INRIA)
Zakaria Habibi (Post-doc, Nov. 2011 --- Sep. 2012, financement ANDRA)
Mitia Duerinckx (PhD student, 2014 -- 2017, financement FNRS)
Thomas GallouŽt (Post-doc, Sep. 2014 -- Aug. 2015, ERC)
Christopher Shirley (Post-doc, Sept. 2015 -- Aug. 2018, ERC)
Antoine Benoit (Post-doc, Oct. 2015 -- Aug. 2017, ERC)
Matthias Ruf (Post-doc, Feb. 2017 -- Aug. 2019, ERC)
Laure Giovangigli (Post-doc, Sep. 2017 -- , ERC)