Person in charge of the Unit : Oui
We have developed expertise on various aspects of applied mathematics and on methods of mathematical physics. More precisely our research topics include inverse problems and their applications (optics, microscopy, medical imaging, non destructive testing), optimization (convex and non convex problems, iterative algorithms, proximal operators), continuum mechanics and electrodynamics (wave propagation in elasticity and electromagnetism), and applications of Fourier and wavelet analysis (signal processing, macroeconomics, seismology, ...).
The main objective of this proposal is to explore and develop new methods and optimized algorithms for the analysis of high-dimensional data under the assumption of structured sparsity and low-rank conditions. Special attention will be paid to the impact of the numerical methods in statistical applications, especially in the domain of variable selection. We will investigate the benefits of structured sparse and low-rank models and their algorithms in relation to multiscale sparse decompositions, the regularization of inverse problems and in medical imaging. By linking these concepts to convex optimization problems, the planned research also contributes to turning these theoretical concepts into practical and efficient techniques for the solution of large scale inverse problems.
Development of analytical and numerical methods for solving inverse problems and applications to experimental data processing in various fields (optics, microscopy, non destructive testing of materials, medical imaging and tomography).
In optical or acoustical miscroscopy, near-field imaging techniques allow to considerably enhance the classical far-field Rayleigh resolution limit. The project aims at estimating and improving the resolving power of near-field microscopes. This requires the development of better imaging models and the use of adequate image processing techniques.
The project aims at the improvement of the filtering techniques used to discriminate between trends and cycles in economic time series and also at the development of time-frequency (local Fourier bases) and time-scale methods (wavelet bases) for non-stationary data.