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Regularized minimal-norm solution of an overdetermined system of first kind integral equations

Abstract

Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method to compute the minimal-norm solution in the presence of boundary constraints. The algorithm stems from the Riesz representation theorem and operates in a reproducing kernel Hilbert space. Since the resulting linear system is strongly ill-conditioned, we construct a regularization method depending on a discrete parameter. It is based on the expansion of the minimal-norm solution in terms of the singular functions of the integral operator defining the problem. Two estimation techniques are tested for the automatic determination of the regularization parameter, namely, the discrepancy principle and the L-curve method. Numerical results concerning two artificial test problems demonstrate the excellent performance of the proposed method. Finally, a particular model typical of geophysical applications, which reproduces the readings of a frequency domain electromagnetic induction device, is investigated. The results show that the new method is extremely effective when the sought solution is smooth, but produces significant information even for non-smooth solutions.

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Acknowledgments

The authors would like to thank an anonymous referee for his insightful comments that lead to improvements of the presentation.

Funding

Luisa Fermo, Federica Pes, and Giuseppe Rodriguez are partially supported by Regione Autonoma della Sardegna research project “Algorithms and Models for Imaging Science [AMIS]” (RASSR57257, intervento finanziato con risorse FSC 2014-2020 - Patto per lo Sviluppo della Regione Sardegna). Luisa Fermo is partially supported by INdAM-GNCS 2020 project “Approssimazione multivariata ed equazioni funzionali per la modellistica numerica”. Patricia Díaz de Alba, Federica Pes, and Giuseppe Rodriguez are partially supported by INdAM-GNCS 2020 project “Tecniche numeriche per l’analisi delle reti complesse e lo studio dei problemi inversi”. Patricia Díaz de Alba gratefully acknowledges Fondo Sociale Europeo REACT EU - Programma Operativo Nazionale Ricerca e Innovazione 2014-2020 and Ministero dell’Universit‘a e della Ricerca for the financial support. Federica Pes gratefully acknowledges CRS4 (Centro di Ricerca, Sviluppo e Studi Superiori in Sardegna) for the financial support of her Ph.D. scholarship.

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Correspondence to Federica Pes.

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The author Giuseppe Rodriguez is a member of the editorial board of Numerical Algorithms. The authors declare no other conflict of interest.

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Dedicated to Claude Brezinski on the occasion of his 80th birthday.

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All authors have equally contributed to the development of this manuscript.

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Data sharing not applicable to this article. Only synthetic datasets were generated and analyzed during the current study. They can be generated by the computational code.

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The computational code is only prototypal, but it is available from the authors upon request.

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Patricia Díaz de Alba, Luisa Fermo and Giuseppe Rodriguez contributed equally to this work.

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de Alba, P., Fermo, L., Pes, F. et al. Regularized minimal-norm solution of an overdetermined system of first kind integral equations. Numer Algor (2022). https://doi.org/10.1007/s11075-022-01282-2

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  • DOI: https://doi.org/10.1007/s11075-022-01282-2

Keywords

  • Fredholm integral equations
  • Riesz representation theorem
  • Reproducing kernel Hilbert space
  • Linear inverse problems
  • Regularization
  • FDEM induction

Mathematics Subject Classification (2010)

  • 65R30
  • 65R32
  • 45Q05
  • 86A22