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Sparsity in Inverse Geophysical Problems

In: Handbook of Geomathematics

Author

Listed:
  • Markus Grasmair

    (Norwegian University of Science and Technology, Department of Mathematics)

  • Markus Haltmeier

    (University of Innsbruck, Institute of Mathematics)

  • Otmar Scherzer

    (University of Vienna, Computational Science Center)

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.

Suggested Citation

  • Markus Grasmair & Markus Haltmeier & Otmar Scherzer, 2015. "Sparsity in Inverse Geophysical Problems," Springer Books, in: Willi Freeden & M. Zuhair Nashed & Thomas Sonar (ed.), Handbook of Geomathematics, edition 2, pages 1659-1687, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-54551-1_25
    DOI: 10.1007/978-3-642-54551-1_25
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