Inverse potential problems, where a source is to be determined from measurements of the field at a distance from its support,arise in many scientific areas. Applications we have in mind here comprise the analysis of Electro-Encephalo-Graphic (EEG) data, where the location of foci of electric activity in the brain are to be found, as well as inverse magnetization problems in Paleomagnetis where the magnetization distribution is to be computed and yields important information on the past history of rocks. They share a common framework, namely the source is the divergence of a measure. However, the geometry of the measurements is different, as well as the fact that the field is harmonic in
Paleomagnetism but only piecewise harmonic in EEG (where several layers separate the source from the measurement place). The purpose of the thesis is twofold: to study regulatization techniques, based on constraining the total variation of the measure whose divergence is the source on the one hand, and to study the inverse propagation process of the field across the layers for the case of a piecewise harmonic field on the other hand. The geometry of the layers could typically be spherical or modeled on other smooth surfaces.
This study will offer an opportunity to balance theoretical work in harmonic analysis with numerical work in optimization. In particular, experiments on real data will be conducted.
Advisors : Laurent Baratchart (firstname.lastname@example.org), Juliette Leblond (email@example.com).