Résumé des résultats (Abstract)
(Anglais)
|
The main research objective of this network is the joint development, analysis, implementation and optimisation of a variety of mathematical concepts and computational tools that help 'breaking complexity' in a variety of scientific computing tasks. Such tasks include both explicit data compression, as encountered in signal and image processing, and numerical simulation based on a mathematical model such as a partial differential or integral equation. In this last case, the object of interest is implicitely given by the model, and its compression needs to be optimally intertwined with the solution process. In both situations, we aim at developing mathematical representations, tailored data structures and fast resolution/processing algorithms, which are capable of optimally capturing (in an information theoretic sense) the possible hidden simplicity of the underlying object to be stored, processed or computed.
On a theoretical level, we shall gravitate around the pivoting mathematical concept of 'nonlinear approximation' with the aim of fully understanding the process of adaptively representing classes of functions by N optimally chosen parameters. On a more practical level, we shall investigate practical realizations of such optimal representations, which can be implemented by fast algorithms. Classical instances include adaptive finite elements and more recently wavelets, which are still the source of theoretical and practical limitations when dealing with complicated domains and anisotropic singularities. We shall investigate these difficulties and come out with robust adaptive discretisation tools that are in addition well fitted for specific problems: variational discretisations of PDE's arising in real life applications, progressive encoding in multimedia, noise reduction and inverse problems in medical imaging.
|
