Sorbonne Université

Master de Sciences & Technologies

M2 Mathématiques & Applications (Sorbonne Université)

Waves and imaging: Concepts, Theory and Applications

H. Haddar École Polytechnique

Thèmes abordés :

This course is an introduction to inverse scattering problems where waves are used to obtain information on an unknown media. This information can be the geometry of defects or anomalies as in non destructive testings and medical imaging or the shape of a target in radar and sonar applications. One may also be interested in complementing the geometry with quantitative or qualitative estimates on the physical parameters of the probed domain. We shall discuss various mathematical aspects of this inverse problem and different solution methodologies. The course outline is as follows:

  1. Introduction of some basic notions and mathematical tools for the study of scattering problems at fixed frequency (Radiation Condition, Fredholm theory, The Rellich lemma and unique continuation principle).
  2. Introduction of different settings for the inverse problem according to targeted applications. Discussion of some linearization approaches: Born approximation, Time reversal and Synthetic Aperture Radar principles, to solve the geometrical inverse problem. This is the occasion to also introduce the notion of ill-posed problems and give some rudiments on regularization theory.
  3. Discussion of some aspects of the full nonlinear inverse problem:
    • Study of uniqueness issues in the framework of geometrical optic solutions.
    • Presentation and analysis of some nonlinear inversion algorithms.
    • Presentation and analysis of so-called sampling methods to solve the geometrical inverse problem.

Référence :

Inverse Scattering Theory and Transmission Eigenvalues, F. Cakoni, D. Colton, and H. Haddar, SIAM publications, 88, 2016, CBMS Series.