Dennis Trede

Dirk A. Lorenz, Stefan Schiffler, Dennis Trede

The Tikhonov regularization of linear ill-posed problems with an $\ell^1$ penalty is considered. We recall results for linear convergence rates and results on exact recovery of the support. Moreover, we derive conditions for exact support recovery which are especially applicable in the case of ill-posed problems, where other conditions, e.g. based on the so-called coherence or the restricted isometry property are usually not applicable. The obtained results also show that the regularized solutions do not only converge in the $\ell^1$-norm but also in the vector space $\ell^0$ (when considered as the strict inductive limit of the spaces $\R^n$ as $n$ tends to infinity). Additionally, the relations between different conditions for exact support recovery and linear convergence rates are investigated. With an imaging example from digital holography the applicability of the obtained results is illustrated, i.e. that one may check a priori if the experimental setup guarantees exact recovery with Tikhonov regularization with sparsity constraints.

Loic Denis, Dirk A. Lorenz, Dennis Trede

The orthogonal matching pursuit (OMP) is an algorithm to solve sparse approximation problems. Sufficient conditions for exact recovery are known with and without noise. In this paper we investigate the applicability of the OMP for the solution of ill-posed inverse problems in general and in particular for two deconvolution examples from mass spectrometry and digital holography respectively. In sparse approximation problems one often has to deal with the problem of redundancy of a dictionary, i.e. the atoms are not linearly independent. However, one expects them to be approximatively orthogonal and this is quantified by the so-called incoherence. This idea cannot be transfered to ill-posed inverse problems since here the atoms are typically far from orthogonal: The ill-posedness of the operator causes that the correlation of two distinct atoms probably gets huge, i.e. that two atoms can look much alike. Therefore one needs conditions which take the structure of the problem into account and work without the concept of coherence. In this paper we develop results for exact recovery of the support of noisy signals. In the two examples in mass spectrometry and digital holography we show that our results lead to practically relevant estimates such that one may check a priori if the experimental setup guarantees exact deconvolution with OMP. Especially in the example from digital holography our analysis may be regarded as a first step to calculate the resolution power of droplet holography.