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Sparsity in Inverse Geophysical Problems

  • Markus Grasmair
  • Markus Haltmeier
  • Otmar Scherzer
Living reference work entry

Abstract

Many geophysical imaging problems are ill-posed in the sense that the solution does not depend continuously on the measured data. Therefore, their solutions cannot be computed directly but instead require the application of regularization. Standard regularization methods find approximate solutions with small L 2 norm. In contrast, sparsity regularization yields approximate solutions that have only a small number of nonvanishing coefficients with respect to a prescribed set of basis elements. Recent results demonstrate that these sparse solutions often much better represent real objects than solutions with small L 2 norm. In this survey, recent mathematical results for sparsity regularization are reviewed. As an application of the theoretical results, synthetic focusing in Ground Penetrating Radar is considered, which is a paradigm of inverse geophysical problem.

Keywords

Ground Penetrate Radar Tikhonov Regularization Augmented Lagrangian Method Constrain Minimization Problem Ricker Wavelet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work has been supported by the Austrian Science Fund (FWF) within the national research networks Industrial Geometry, project 9203-N12, Variational Imaging on Manifolds, project 11704, and Photoacoustic Imaging in Biology and Medicine, project S10505-N20. The authors thank Sylvia Leimgruber (alpS – Center for Natural Hazard Management in Innsbruck) and Harald Grossauer (University Innsbruck) for providing real life data sets.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Markus Grasmair
    • 1
  • Markus Haltmeier
    • 2
  • Otmar Scherzer
    • 3
  1. 1.Department of Mathematics, Norwegian University of Science and TechnologyTrondheimNorway
  2. 2.Institute of Mathematics, University of InnsbruckInnsbruckAustria
  3. 3.Computational Science Center, University of ViennaViennaAustria

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