Minisymposia Abstracts

Speaker: Stefan Kindermann
RICAM, JJohannes Kepler University of Linz
Title: Convergence of heuristic parameter choice rules
Abstract: A heuristic parameter choice rule determines a regularization parameter without knowledge of the noise level. By a well-known result of Bakushinskii this cannot yield convergence of the worst-case error for any regularization scheme for ill-posed problems.
However, recently a convergence analysis was established, where instead of the worst case (all possible data error are considered) the situation for restricted data noise was considered. We comment on recent results of general minimization-based heuristic parameter choice rules for linear ill-posed problems in Hilbert spaces. It turns out that under not too strong restrictions on the data noise, convergence for general regularization schemes can be shown. Moreover, under additional assumption suboptimal and optimal convergence rates can be proven.
In this talk we outline the main ingredients of the convergence analysis and describe the situations in which heuristic parameter choice rules are successful or might fail. Finally we discuss open problems and recent results in this field.

Speaker: Tomoya Takeuchi
North Carolina State University
Title: A regularization parameter for nonsmooth regularization
Abstract: We develop a novel criterion for choosing regularization parameters in nonsmooth Tikhonov functionals. The proposed criterion is solely based on the value function and applicable to a broad range of nonsmooth models. An efficient numerical algorithm for computing the minimizer is developed, and its convergence properties are also discussed. Numerical results for several common nonsmooth models are presented, including deblurring natural images subjected to impulsive noise. The numerical results indicate the proposed criterion can yield accurate results, which are comparable with that by the discrepancy principle and the optimal choice, and the algorithm merits a fast and steady convergence.

Speaker: Ewout van den Berg
Stanford University
Title: A root-finding approach for sparse recovery
Abstract: The use of L1 regularization in optimization to obtain sparse solutions has become common practice over the past few years. Its success has motivated the introduction of a variety of other types of regularization, each suitable for a particular type of sparsity. However, despite all theoretical advances, there are still only a very few specialized codes that can successfully deal with the large-scale problem instances arising in practice.
In this talk we present an algorithm that can efficiently solve large-scale instances of a variety of sparse recovery problems, including L1, sign-constrained L1, and joint-sparse recovery. The algorithm exploits properties of the Pareto curve, which describes the optimal trade-off between objective value and misfit. We explore possible generalizations, and discuss how the algorithm applies to the more recent problems of matrix completion and robust principal component analysis. Finally, we compare the performance of our algorithm to existing solvers.
This is joint work with Michael Friedlander, University of British Columbia, Vancouver.
Speaker: Uno Hämarik
Institute of Applied Mathematics, Tartu University
Title: On parameter choice in Tikhonov regularization in case of different information about noise level

Speaker: Antoine Laurain
Humboldt University of Berlin
Title: Multiphase image segmentation based on shape and topological sensitivity
Abstract: Topological sensitivity analysis is performed for a piecewise constant Mumford-Shah type functional dealing with linear ill-posed problems. In the identity case, the functional corresponds to the classical Mumford-Shah functional: topological and shape derivatives are then combined in order to derive an algorithm for image segmentation with fully automatized initialization. We also present result for the case of the Radon transform. We may extend this functional to investigate the segmentation of modulated images due to, e.g., coil sensitivities.

Speaker: Rosemery Renaut
Arizona State University
Title: Parameter choice for TV regulation via Tikhonov methods
Abstract: TBA

Speaker: Christian Clason
Institute of Mathematics and Scientific Computing and University of Graz
Title: Parameter choice for nonsmooth problems
Abstract: TBA

Speaker: Shuai Lu
Fudan University
Title: Numerical differentiation by a Tikhonov regulation method based on the discrete cosine transform
Abstract: In this paper, we discuss a classical ill-posed problem on numerical differentiation by a Tikhonov regularization method based on the discrete cosine transform (DCT). After implementing an eigenvalue decomposition of the second order difference matrix in the penalty we obtain an explicit minimizer of the Tikhonov functional in a linear combination of DCT-2 components. The choice of the regularization parameter thus can be properly chosen after calling a single variable bounded nonlinear function minimization. Moreover, we propose two approaches to weaken the Gibbs phenomena appearing near the boundary of the reconstructed derivatives. Several numerical examples are provided to show the computational efficiency of our proposed algorithm. Finally the extension to the multidimensional numerical differentiation leads the approach to a wider scope.

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