ERC Starting Grant QUANTHOM 2014--2019
Quantitative methods in stochastic homogenization



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ERC QUANTHOM




Description of the project

The purpose of this project is to develop a quantitative theory of stochastic homogenization. Primary applications concern the analysis and computations for random models in physics and materials science, such as numerical methods for diffusion in heterogeneous media and the mathematical derivation of nonlinear elasticity theory from polymer-chain networks models.

The mathematical fields of concern are:
  • linear and nonlinear partial differential equations
  • calculus of the variations
  • statistical physics
  • probability theory
  • random geometry
  • numerical analysis
  • scientific computing
The scientific project contains three main parts.

The first part is devoted to the development of a quantitative theory of stochastic homogenization. The far-reaching goal is to quantitatively relate the statistics of the solution of a PDE with random coefficients to the statistics of its coefficients for the largest class of PDEs and coefficients possible. The prototypical equation considered is a linear elliptic equation in divergence form.

The second part is devoted to the development and the analysis of numerical methods to solve PDEs with rapidly oscillating coefficients, which are consistent and optimal with respect to the quantitative stochastic homogenization theory.

The third part is devoted to the mathematical derivation of nonlinear elasticity theory from polymer-chain networks models. The aim is threefold:
  • fully justify hyperelastic models from first principles in the form of the statistic physics of polymer-chain networks
  • develop and analyze a numerical strategy to approximate the associated energy density
  • confront the results to physical and mechanical experiments



Team and collaborators

Coordinator of the project:
  • Antoine Gloria, Université Libre de Bruxelles and Inria Lille - Nord Europe
PhD Students:
  • Mitia Duerinckx

Post-doctoral positions:
there are 4 post-doc positions available (each of two years).
Post-doctoral subjects include (but are not limited to) the projects above. Questions such as effective behavior of random models (quenched CLT for Markov processes, effective laws etc.) can be considered depending on the background of the candidate. If interested send me an e-mail.

Collaborators in mathematics:
  • MEPHYSTO team
  • Scott Armstrong
  • Daniel Marahrens
  • Jean-Christophe Mourrat
  • Stefan Neukamm
  • Felix Otto
  • Mathew Penrose
  • Marina Vidrascu
Collaborators in physics and materials science:
  • François Boué
  • Patrick Le Tallec
  • François Lequeux


Activities

Seminar at ULB: nonlinear analysis and PDE

Guests:
  • Daniel Marahrens (3-7/03/14)
  • Felix Otto (13-16/04/14)
  • Andrea Braides (14-16/05/14)
  • Jean-Christophe Mourrat (2-10/06/14)
  • Stefan Neukamm (4-7/06/14)
  • Marco Cicalese (10-14/06/14)


Events

Here is a list of events organized or sponsored by the ERC project.

Workshop on Relaxation, homogenization and dimensional reduction in hyperelasticity
Organized by Gilles Francfort (Paris 13), Martin Kruzik (Prague), and Antoine Gloria
Université Paris-Nord, March 25-27, 2014