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Non-linear approximation and adaptivity: breaking complexity in numerical modelling and data representation

Objective

The main research objective of this network is the joint development, analysis, implementation and optimization 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 implicitly 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 infonnation 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 anisotropy singularities. We shall investigate these difficulties and come out with robust adaptive discretization tools that are in addition well fitted for specific problems: variational discretizations of PDE's arising in real life applications, progressive encoding in multimedia, noise reduction and inverse problems in medical imaging.

Funding Scheme

NET - Research network contracts

Coordinator

NATIONAL RESEARCH COUNCIL OF ITALY
Address
Via De Marini 6
16149 Genova
Italy

Participants (12)

AACHEN UNIVERSITY OF TECHNOLOGY
Germany
Address
Templergraben 55
52056 Aachen
CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE
France
Address
Rue F. Joliot Curie, 39
13453 Marseille
INSTITUT NATIONAL POLYTECHNIQUE DE GRENOBLE
France
Address
Domaine Universitaire
38041 Grenoble
POLITECNICO DI TORINO
Italy
Address
Corso Duca Degli Abruzzi 24
10129 Torino
RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITAET BONN
Germany
Address
Wegelerstrasse 6
53115 Bonn
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZUERICH
Switzerland
Address
101,Raemistrasse 101
8092 Zurich
TECHNISCHE UNIVERSITAET CHEMNITZ
Germany
Address
Reichenhainerstraße 41
09107 Chemnitz
THE BOARD OF REGENTS OF THE UNIVERSITY OF WISCONSCIN SYSTEMS
United States
Address
West Dayton Street, 1210
53706 Madison
UNIVERSITAT DE VALENCIA
Spain
Address
50,C/dr. Moliner 50
46100 Burjasot
UNIVERSITE PIERRE ET MARIE CURIE - PARIS VI
France
Address
Rue Du Chevaleret 175
75013 Paris
UNIVERSITY OF WALES - BANGOR
United Kingdom
Address
Dean Street
LL57 1UT Bangor (Gwynedd)
UTRECHT UNIVERSITY
Netherlands
Address
6,Budapestlaan 6
3508 TA Utrecht