Sparse Reconstructions for Inverse Problems

The combination of sparse signal recovery and inverse ill - posed problems is a new research field that is under consideration in this project. In particular, we shall be concerned with the development of new regularizing algorithms that solve nonlinear inverse ill - posed problems with sparsity constraints. The concept of such algorithms but for linear problems in its Tikhonov variational form with sparsity constraints was under consideration several years and finally solved two years ago by Daubechies et. al. [DDDM04]. Quite recently, in 2005 the applicants have opened the door to algorithms for nonlinear ill - posed inverse problems with mixed sparsity and smoothness constraints. Very first results are achieved, but there are still many open questions. Thus the first goal of this project is glue the theories of sparse signal recovery and nonlinear inverse ill - posed problems tightly together and to built a first fundament for an abundant variety of applications. Of interest in this project are three applications, coming from the analysis of the dynamics of cellular networks, medical imaging and rotor dynamics. These three applications provide nonlinear inverse problems in which the solution is assumed to have a sparse expansion in dictionaries of different building blocks. Thus the second goal is to provide algorithms to reasonably solve the mentioned inverse problems and to overcome therewith the significant deficiencies in current used approaches. In principle, the algorithms to be developed amount to be very likely iterative schemes (so far, we have only considered the easiest cases). This defines essentially the course of this project: development of the schemes, investigation of convergence properties, very likely an acceleration of the schemes, analysis of resulting convergence rates, establishment of regularization theory (parameter rules), and, finally, in collaboration with the project partners, an implementation of the algorithms such that the developed mathematical machineries can be really used.

The combination of sparse signal recovery and inverse ill - posed problems is a new research field that is under consideration in this project. In particular, we shall be concerned with the development of new regularizing algorithms that solve nonlinear inverse ill - posed problems with sparsity constraints. The concept of such algorithms but for linear problems in its Tikhonov variational form with sparsity constraints was under consideration several years and finally solved two years ago by Daubechies et. al. [DDDM04]. Quite recently, in 2005 the applicants have opened the door to algorithms for nonlinear ill - posed inverse problems with mixed sparsity and smoothness constraints. Very first results are achieved, but there are still many open questions. Thus the first goal of this project is glue the theories of sparse signal recovery and nonlinear inverse ill - posed problems tightly together and to built a first fundament for an abundant variety of applications. Of interest in this project are three applications, coming from the analysis of the dynamics of cellular networks, medical imaging and rotor dynamics. These three applications provide nonlinear inverse problems in which the solution is assumed to have a sparse expansion in dictionaries of different building blocks. Thus the second goal is to provide algorithms to reasonably solve the mentioned inverse problems and to overcome therewith the significant deficiencies in current used approaches. In principle, the algorithms to be developed amount to be very likely iterative schemes (so far, we have only considered the easiest cases). This defines essentially the course of this project: development of the schemes, investigation of convergence properties, very likely an acceleration of the schemes, analysis of resulting convergence rates, establishment of regularization theory (parameter rules), and, finally, in collaboration with the project partners, an implementation of the algorithms such that the developed mathematical machineries can be really used.