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In this article, the convergence of the often used iterative softthresholding algorithm for the solution of linear operator equations in infinite dimensional Hilbert spaces is analyzed in detail. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis. The analysis bases on new techniques like the… (More)

- Kanglin Chen, Dirk A. Lorenz
- Journal of Mathematical Imaging and Vision
- 2011

The problem of finding an interpolating image between two given images in an image sequence is considered. The problem is formulated as an optimal control problem governed by a transport equation, i.e. we aim at finding a flow field which transports the first image as close as possible to the second image. This approach bears similarities with the Horn and… (More)

- Kristian Bredies, Dirk A. Lorenz, Stefan Reiterer
- J. Optimization Theory and Applications
- 2015

Numerical algorithms for a special class of non-smooth and non-convex minimization problems in infinite dimensional Hilbert spaces are considered. The functionals under consideration are the sum of a smooth and non-smooth functional, both possibly non-convex. We propose a generalization of the gradient projection method and analyze its convergence… (More)

- Dirk A. Lorenz, Thomas Pock
- Journal of Mathematical Imaging and Vision
- 2014

In this paper, we propose an inertial forward-backward splitting algorithm to compute a zero of the sum of two monotone operators, with one of the two operators being co-coercive. The algorithm is inspired by the accelerated gradient method of Nesterov, but can be applied to a much larger class of problems including convex-concave saddle point problems and… (More)

- Kristian Bredies, Dirk A. Lorenz
- SIAM J. Scientific Computing
- 2008

A new iterative algorithm for the solution of minimization problems in infinitedimensional Hilbert spaces which involve sparsity constraints in form of `p-penalties is proposed. In contrast to the well-known algorithm considered by Daubechies, Defrise and De Mol, it uses hard instead of soft shrinkage. It is shown that the hard shrinkage algorithm is a… (More)

- Kristian Bredies, Dirk A. Lorenz, Peter Maass
- Comp. Opt. and Appl.
- 2009

- L Denis, D A Lorenz, D Trede
- 2009

The orthogonal matching pursuit (OMP) is a greedy 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… (More)

- Dirk A. Lorenz, Marc E. Pfetsch, Andreas M. Tillmann
- ACM Trans. Math. Softw.
- 2015

The problem of finding a minimum ℓ<sub>1</sub>-norm solution to an underdetermined linear system is an important problem in compressed sensing, where it is also known as <i>basis pursuit</i>. We propose a heuristic optimality check as a general tool for ℓ<sub>1</sub>-minimization, which often allows for early termination by “guessing” a… (More)

- Loïc Denis, Dirk Lorenz, Eric Thiébaut, Corinne Fournier, Dennis Trede
- Optics letters
- 2009

Inline digital holograms are classically reconstructed using linear operators to model diffraction. It has long been recognized that such reconstruction operators do not invert the hologram formation operator. Classical linear reconstructions yield images with artifacts such as distortions near the field-of-view boundaries or twin images. When objects… (More)

- Jan Lorenz, Dirk A. Lorenz
- IEEE Trans. Automat. Contr.
- 2010

A new theorem on conditions for convergence to consensus of a multiagent time-dependent timediscrete dynamical system is presented. The theorem is build up on the notion of averaging maps. We compare this theorem to Moreau’s Theorem and his proposed set-valued Lyapunov theory (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005). We give examples… (More)