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On the convergence theorem for the regularized functional matching pursuit (RFMP) algorithm

Abstract

The RFMP is an iterative regularization method for a class of linear inverse problems. It has proved to be applicable to problems which occur, for example, in the geosciences. In the early publications (Fischer in Sparse Regularization of a Joint Inversion of Gravitational Data and NormalMode Anomalies, 2011; Fischer and Michel in Inverse Probl 28:065012, 2012), it was shown that the iteration converges in the unregularized case to an exact solution. In Michel (in: Freeden, Nashed, Sonar (eds) Handbook of geomathematics, 2nd edn, Springer, Berlin, pp 2121–2147, 2015) and Michel and Telschow (Int J Geomath 5:195–224, 2014), it was later shown (for two different scenarios) that the iteration also converges in the regularized case, where the limit of the iteration is the solution of the Tikhonov-regularized normal equation. However, the condition of these convergence proofs cannot be satisfied and, therefore, has to be weakened, as it was pointed out for the convergence theorem of the related iterated regularized orthogonal functional matching pursuit algorithm in Michel and Telschow (SIAM J Numer Anal 54:262–287, 2016). Moreover, the convergence proof in Michel (2015) contained a minor error. For these reasons, we reformulate here the convergence theorem for the RFMP and its proof. We also use this opportunity to extend the algorithm for an arbitrary infinite-dimensional separable Hilbert space setting. In addition, we particularly elaborate the cases of non-injective and non-surjective operators.

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References

  • Engl, H.W., Hanke-Bourgeois, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  • Fischer, D.: Sparse Regularization of a Joint Inversion of Gravitational Data and Normal Mode Anomalies. Ph.D. thesis, Geomathematics Group, Department of Mathematics, University of Siegen, Verlag Dr. Hut, Munich (2011)

  • Fischer, D., Michel, V.: Sparse regularization of inverse gravimetry—case study: spatial and temporal mass variations in South America. Inverse Probl. 28, 065012 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Groetsch, C.W.: Inverse Problems in the Mathematical Sciences. Springer, Wiesbaden (1993)

    Book  MATH  Google Scholar 

  • Hofmann, B.: Regularization for Applied Inverse and Ill-Posed Problems. Teubner, Leipzig (1986)

    Book  MATH  Google Scholar 

  • Louis, A.K.: Inverse und schlecht gestellte Probleme. Teubner, Stuttgart (1989)

    Book  MATH  Google Scholar 

  • Lu, S., Pereverzev, S.V.: Regularization Theory for Ill-Posed Problems-Selected Topics. de Gruyter, Berlin, Boston (2013)

    Book  MATH  Google Scholar 

  • Michel, V.: RFMP—An iterative best basis algorithm for inverse problems in the geosciences. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn, pp. 2121–2147. Springer, Berlin, Heidelberg (2015)

    Chapter  Google Scholar 

  • Michel, V., Telschow, R.: A non-linear approximation method on the sphere. Int. J. Geomath. 5, 195–224 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  • Michel, V., Telschow, R.: The regularized orthogonal functional matching pursuit for ill-posed inverse problems. SIAM J. Numer. Anal. 54, 262–287 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  • Rieder, A.: Keine Probleme mit inversen Problemen. Vieweg, Wiesbaden (2003)

    Book  MATH  Google Scholar 

  • Telschow, R.: An Orthogonal Matching Pursuit for the Regularization of Spherical Inverse Problems. Ph.D. thesis, Geomathematics Group, Department of Mathematics, University of Siegen, Verlag Dr. Hut, Munich (2014)

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Acknowledgements

We gratefully acknowledge the support by the German Research Foundation (DFG), Project MI 655/10-1.

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Correspondence to Volker Michel.

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Michel, V., Orzlowski, S. On the convergence theorem for the regularized functional matching pursuit (RFMP) algorithm. Int J Geomath 8, 183–190 (2017). https://doi.org/10.1007/s13137-017-0095-6

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  • DOI: https://doi.org/10.1007/s13137-017-0095-6

Keywords

  • Convergence
  • Greedy algorithm
  • Inverse problem
  • Matching pursuit
  • RFMP
  • Tikhonov-Phillips regularization

Mathematics Subject Classification

  • 65J10
  • 65J20
  • 65J22
  • 86A22