Externally Funded Project

Inverse problems with sparsity constraints

FWF Project P19496
Runtime: 01.06.2007-31.03.2011

Project Team
Project Abstract

The combination of sparse signal reconstruction and ill-posed inverse problems is a growing area of research and will be investigated in this project. More specifically we are dealing with the development of new regularization techniques for nonlinear ill-posed inverse problems with sparsity constraints. The development of algorithms for the case of linear problems in Tikhonov functional form has been a main focus of research throughout the past years. A solution for the linear case was presented four years ago by Daubechies et al. [DDDM04]. In 2005 we presented such a concept for nonlinear problems and first results look very promising even though we are far from having a complete analysis of this case. Therefore the first goal of this project will be to merge together the topics of sparse signal reconstruction and nonlinear inverse problems and to create thereby a foundation for a number of applications, such as the analysis of the dynamics of cellular networks, medical imaging, and rotor dynamics. All these three applications result in nonlinear ill-posed problems whose solutions are assumed to be sparse. Therefore the methods to be developed in this project will contribute to removing the flaws of the algorithms currently in use. First simple results show that the natural approach will be to aim for iterative algorithms. With this in mind the canonical work program for this project is:

Keywords and AMS Classification