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A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry

Abstract

Based on the Regularized Functional Matching Pursuit (RFMP) algorithm for linear inverse problems, we present an analogous iterative greedy algorithm for nonlinear inverse problems, called RFMP_NL. In comparison to established methods for nonlinear inverse problems, the algorithm is able to combine very diverse types of basis functions, for example, localized and global functions. This is important, in particular, in geoscientific applications, where global structures have to be distinguished from local anomalies. Furthermore, in contrast to other methods, the algorithm does not require the solution of large linear systems. We apply the RFMP_NL to the nonlinear inverse problem of gravimetry, where gravitational data are inverted for the shape of the surface or inner layer boundaries of planetary bodies. This inverse problem is described by a nonlinear integral operator, for which we additionally provide the Fréchet derivative. Finally, we present two synthetic numerical examples to show that it is beneficial to apply the presented method to inverse gravimetric problems.

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Acknowledgements

The authors gratefully acknowledge the financial support by the School of Science and Technology of the University of Siegen, Germany. Furthermore, we would like to thank the anonymous reviewer for his valuable comments, which helped to improve the paper.

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Correspondence to Max Kontak.

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Kontak, M., Michel, V. A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry. Int J Geomath 9, 167–198 (2018). https://doi.org/10.1007/s13137-018-0110-6

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  • DOI: https://doi.org/10.1007/s13137-018-0110-6

Keywords

  • Greedy algorithm
  • Inverse gravimetry
  • Nonlinear inverse problem
  • Regularization

Mathematics Subject Classification

  • 65J22
  • 65R32
  • 35R30
  • 45Q05