Inverse Problems, Sparse reconstructions, Convergence rates, Optimization

将稀疏信号恢复与逆不适定问题相结合是本项目正在考虑的一个新的研究领域。特别是，我们将关注新的正则化算法，解决非线性逆不适定问题的稀疏约束的发展。这种算法的概念，但线性问题的吉洪诺夫变分形式与稀疏约束正在考虑几年，终于解决了两年前在[DDDM04]。最近，在2005年，申请人已经打开了大门，算法的非线性不适定的反问题与混合稀疏性和光滑性约束。取得了初步成果，但仍有许多悬而未决的问题。因此，本项目的首要目标是将稀疏信号恢复和非线性逆不适定问题的理论紧密结合在一起，为各种各样的应用奠定基础。在这个项目中感兴趣的是三个应用程序，来自蜂窝网络，医学成像和转子动力学的动力学分析。这三个应用程序提供了非线性反问题，其中的解决方案被假定为具有稀疏扩展字典的不同的积木。因此，第二个目标是提供算法来合理地解决所提到的逆问题，并克服目前使用的方法中的重大缺陷。原则上，要开发的算法量是非常可能的迭代方案（到目前为止，我们只考虑了最简单的情况）。这基本上定义了这个项目的过程：方案的开发，收敛特性的调查，很可能是方案的加速，所得收敛率的分析，正则化理论（参数规则）的建立，最后，与项目合作伙伴合作，实现算法，以便开发的数学机器可以真正使用。

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 in [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 to 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.