Inverse problems and imaging

The workshop "Inverse problems and imaging" took place in Paris at the Institut Henri Poincaré on February 20-22, 2013. It was organized under the auspices of the French Mathematical Society (SMF) as a session of the "États de la recherche". Its main objective was to present recent developments on inverse problems and imaging. Most of these developments result from interactions between several domains of mathematics: analysis and control of partial differential equations, stochastic analysis, statistics, multiscale analysis. The three mini-courses given by Liliana Borcea (University of Michigan, USA), Hyeonbae Kang (Inha University, Korea), and Gunther Uhlmann (University of Washington, USA) can be considered as perfect illustrations of these fruitful interactions. Liliana Borcea addresses sensor array imaging in random media, more exactly in randomly perturbed waveguides. She introduces different imaging methods and study their resolution and stability properties. In his lecture Hyeonbae Kang reviews recent progress on imaging by generalized polarization tensors (GPTs), enhancement of near-cloaking by GPT-vanishing structures, cloaking by anomalous localized resonance, and analysis of stress concentration. He shows how the Neumann-Poincaré operator naturally arises in all these problems. Gunther Uhlmann considers transformation-optics based cloaking in acoustic and electromagnetic scattering. He presents various regularized approximate cloaking schemes.