Collection(s) : Cours spécialisés
Paru le 01/02/2018 | Relié VII-230 pages
Professionnels
Lectures on Elliptic Methods for Hybrid Inverse Problems
In recent years, several new imaging modalities have been developed in order to be able to detect physical parameters simultaneously at a high spatial resolution and with a high sensitivity to contrast. These new approaches typically rely on the interaction of two physical imaging methods, and the corresponding mathematical models are the so-called hybrid, or coupled-physics, inverse problems. The combination of two physical modalities poses new mathematical challenges : the analysis of this new class of inverse problems requires the use of various mathematical tools, often of independent interest. This book intends to provide a first comprehensive course on some of these tools (mainly related to elliptic partial differential equations) and on their applications to hybrid inverse problems.
For certain topics, such as the observability of the wave equation, the generalisation of the Radó-Kneser-Choquet Theorem to the conductivity equation, complex geometrical optics solutions and the Runge approximation property, we review well-known results. The material is presented with a clear focus on the intended applications to inverse problems. On other topics, including the regularity theory and the study of small-volume perturbations for Maxwell's equations, scattering estimates for the Helmholtz equation and the study of non-zero constraints for solutions of certain PDE, we discuss several new results. We then show how all these tools can be applied to the analysis of the parameter reconstruction for some hybrid inverse problems : Acousto-Electric tomography, Current Density Impedance Imaging, Dynamic Elastography, Thermoacoustic and Photoacoustic Tomography.
Giovanni S. Alberti obtained his D. Phil. at Oxford University and, after two postdocs in Paris at the École Normale Supérieure and in Zürich at ETH, is now Assistant Professor at the University of Genoa. His research focuses on partial differential equations and applied harmonic analysis. In particular, he has recently worked on Maxwell's equations and on the mathematical theory of multi-dimensional wavelets and shearlets. He is also interested in the interactions of these areas with imaging, as in inverse problems in PDE and compressed sensing.
Yves Capdeboscq is an Associate Professor at the University of Oxford. He is interested in problems arising from multiple scales interactions in partial differential equations, particularly in homogenization theory and parameter identification via non invasive measurements. Previously he was at Université de Versailles-Saint-Quentin-en-Yvelines, before that at INSA in Rennes, earlier at Rutgers University and he prepared his thesis at the Commissariat à l'Énergie Atomique in Saclay.
