Research

Dynamics of novel PDE models

Dynamical models consisting of evolutionary PDEs, mainly of hyperbolic type, appear classically in the applications studied by the team. Yet, the classical purely macroscopic approach is not able to account for some particular phenomena related to specific interactions occurring at lower (possibly micro) level. These phenomena can be of greater importance when dealing with particular applications, where the “first order” approximation given by the purely macroscopic approach reveals to be inadequate. Nevertheless, macroscopic models offer well known advantages, namely a sound analytical framework, fast numerical schemes, the presence of a low number of parameters to be calibrated, and efficient optimization procedures. Therefore, we are convinced of the interest of keeping this point of view as dominant, while completing the models with information on the dynamics at the small scale / microscopic level. This can be achieved through several techniques, such as:

  • Multi-scale modeling
  • Non-local flows
  • Measure-valued solutions
  • Mean-field games

Uncertainty in PDE simulations and control

Uncertainty quantification and analysis is an issue of utmost importance for engineering applications. Uncertainty appears at several stages of the modeling/transfer process: data assimilation, model calibration, model design, optimization, etc. For the applications targeted in this project, we aim at addressing the following issues:

  • Uncertainty in parameters and initial-boundary data
  • Robust design and control

Optimization and control algorithms for systems governed by PDEs

The focus here is on the methodological development and analysis of optimization algorithms and paradigms for PDE systems in general, keeping in mind our privileged domains of application in the way the problems are mathematically formulated. We consider in particular:

  • Hierarchical methods
  • Multi-objective descent algorithms
  • Quasi-Newton methods with high order derivatives
  • Games for ill-posed problems