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Regularization techniques in Banach spaces applied to inverse Problems

Working Group:Former WG Inverse Problems
Leadership: Prof. Dr. Armin Lechleiter
Processor: Dr.-Ing. Florian Bürgel
Dr. Marcel Rennoch
Funding: DFG - Deutsche Forschungsgemeinschaft
Project partner:
Time period: 16.07.2014 - 16.10.2017
Bild des Projekts Regularisierungsverfahren in Banachräumen angewandt auf inverse Streuprobleme

Considering inverse scattering problems in certain areas of application, e.g. material control, one is tempted to assume that the support of the contrast of an object consists only of a few non-zero entries, i.e. it is sparse. Respecting this assumption of sparsity during the reconstruction, one derives regularization techniques, which penalty terms are part of Banach spaces instead of more popular Hilbert spaces.

Although the concept of sparsity has become well-known in image and signal processing during the last decade, it is still less established in scattering theory. One task of this project is to develop regularization techniques for problems in inverse scattering theory and therewith to derive fast and efficient algorithms. As a consequence we hope to make our results more common used in existing cooperations with partners from different academic fields.


Publications

  1. F. Bürgel, K. Kazimierski, A. Lechleiter.
    IPscatt—A MATLAB Toolbox for the Inverse Medium Problem in Scattering.
    ACM Transactions on Mathematical Software, 45(4), 45:1–45:20, 2019.

    DOI: 10.1145/3328525

  2. F. Bürgel, K. Kazimierski, A. Lechleiter.
    A sparsity regularization and total variation based computational framework for the inverse medium problem in scattering.
    Journal of Computational Physics, 339, 1–30, 2017.

    DOI: 10.1016/j.jcp.2017.03.011

  3. A. Lechleiter, M. Rennoch.
    Non-linear Tikhonov Regularization in Banach Spaces for Inverse Scattering from Anisotropic Penetrable Media.
    Inverse Problems and Imaging, 11(1):151-176, 2017.

    DOI: 10.3934/ipi.2017008

  4. M. Rennoch.
    Regularization Methods in Banach Spaces Applied to Inverse Medium Scattering Problems.
    Dissertationsschrift, Universität Bremen, 2017.

    online at: Elektronische Bibliothek der Universität Bremen