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In this article, the convergence of the often used iterative soft-thresholding 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)

The problem of the generation of an intermediate 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. This approach bears similarities with the Horn & Schunck method for optical flow calculation but in fact the model is quite different. The images are… (More)

A new iterative algorithm for the solution of minimization problems in infinite-dimensional 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)

- R Griesse, D A Lorenz
- 2007

Minimization problems in ℓ 2 for Tikhonov functionals with sparsity constraints are considered. Sparsity of the solution is ensured by a weighted ℓ 1 penalty term. The necessary and sufficient condition for optimality is shown to be slantly differentiable (Newton differentiable), hence a semismooth Newton method is applicable. Local superlinear convergence… (More)

The starting point for this paper is the well known equivalence between convolution filtering with a rescaled Gaussian and the solution of the heat equation. In the first chapters we analyze the equivalence between multi-scale convolution filtering, linear smoothing methods based on continuous wavelet transforms and the solutions of linear diffusion… (More)

- Dirk A Lorenz
- 2008

This paper addresses the regularization by sparsity constraints by means of weighted ℓ p penalties for 0 ≤ p ≤ 2. For 1 ≤ p ≤ 2 special attention is payed to convergence rates in norm and to source conditions. As main results it is proven that one gets a convergence rate of √ δ in the 2-norm for 1 < p ≤ 2 and in the 1-norm for p = 1 as soon as the unknown… (More)

A new theorem on conditions for convergence to consensus of a multiagent time-dependent time-discrete 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)

- 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)