We study two types of systems:
- Fluid-structure interaction systems
- Complex waves systems
Fluid-structure interaction systems appear in many applications: medicine (motion of the blood in veins and arteries), biology (animal locomotion in a fluid, such as swimming fishes or flapping birds but also locomotion of microorganisms, such as amoebas), civil engineering (design of bridges or any structure exposed to the wind or the flow of a river), naval architecture (design of boats and submarines, seeking of new propulsion sys- tems for underwater vehicles by imitating the locomotion of aquatic animals). The FSIS can be studied by modeling their motions through Partial Differential Equations (PDE) and/or Ordinary Differential Equations (ODE), as classical in fluid mechanics or in solid mechanics. This leads to the study of difficult nonlinear free boundary problems which constitute a rich and active domain of research over the last decades.
Complex waves systems are involved in a large number of applications in several areas of science and engineering: medicine (breast cancer detection, kidney stones destruc- tion, osteoporosis diagnosis, etc.), telecommunications (in urban or submarine environments, optical fibers, etc.), aeronautics (targets detection, aircraft noise reduction, etc.) and, in the longer term, quantum supercomputers. For direct problems, most theoretical issues are now widely understood. However, substantial efforts remain to be undertaken concerning the simulation of wave propagation in complex media. Such situations include heterogenous media with strong local variations of the physical properties (high frequency scattering, mul- tiple scattering media) or quantum fluids (Bose-Einstein condensates). In the first case for instance, the numerical simulation of such direct problems is a hard task, as it generally re- quires to solve ill-conditioned possibly indefinite large size problems, following from space or space-time discretizations of linear or nonlinear evolution PDE set on unbounded domains. For inverse problems, many questions are open at both the theoretical (identifiability, stability and robustness, etc.) and practical (reconstruction methods, approximation and convergence analysis, numerical algorithms, etc.) levels.