Skip to main content

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


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14


  1. Groetsch, C. W.: Integral equations of the first kind, inverse problems and regularization: a crash course. J. Phys.: Conf. Ser. 73, 012001 (2007)

    Google Scholar 

  2. Groetsch, C. W.: Elements of Applicable Functional Analysis Monographs and Textbooks in Pure and Applied Mathematics, vol. 55. Dekker, New York and Basel (1980)

    Google Scholar 

  3. Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven (1923)

    MATH  Google Scholar 

  4. Wing, G. M.: A Primer on Integral Equations of the First Kind: the Problem of Deconvolution and Unfolding. SIAM, Philadelphia (1991)

  5. Atkinson, K. E.: The Numerical Solution of Integral Equations of the Second Kind, vol. 552. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  6. McNeill, J. D.: Electromagnetic Terrain Conductivity Measurement at Low Induction Numbers. Technical Report TN-6, Geonics Limited, Mississauga (1980)

  7. Díaz de Alba, P., Fermo, L., van der Mee, C., Rodriguez, G.: Recovering the electrical conductivity of the soil via a linear integral model. J. Comput. Appl. Math. 352, 132–145 (2019)

    MathSciNet  Article  Google Scholar 

  8. Hendrickx, J.M.H., Borchers, B., Corwin, D.L., Lesch, S.M., Hilgendorf, A.C., Schlue, J.: Inversion of soil conductivity profiles from electromagnetic induction measurements. Soil Sci. Soc. Am. J. 66(3), 673–685 (2002). Package NONLINEM38 available at

    Google Scholar 

  9. Wang, R., Xu, Y.: Functional reproducing kernel Hilbert spaces for nonpoint-evaluation functional data. Appl. Comput. Harmon. Anal. 46 (3), 569–623 (2019)

    MathSciNet  Article  Google Scholar 

  10. Wang, R., Xu, Y.: Regularization in a functional reproducing kernel Hilbert space. J. Complex. 101567 (2021)

  11. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68(3), 337–404 (1950)

    MathSciNet  Article  Google Scholar 

  12. Cui, M., Lin, Y.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science Publishers, New York (2008)

    MATH  Google Scholar 

  13. Wahba, G.: An introduction to reproducing kernel Hilbert spaces and why they are so useful. In: Proceedings of the 13th IFAC Symposium on System Identification (SYSID 2003) (2003)

  14. Berlinet, A., Thomas-Agnan, C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics. Springer, New York (2004)

    Book  Google Scholar 

  15. Cucker, F., Smale, S.: On the mathematical foundations of learning. Bull. Am. Math. Soc. 39(1), 1–49 (2002)

    MathSciNet  Article  Google Scholar 

  16. Rodriguez, G., Seatzu, S.: Numerical solution of the finite moment problem in a reproducing kernel Hilbert space. J. Comput. Appl. Math. 33(3), 233–244 (1990)

    MathSciNet  Article  Google Scholar 

  17. Castro, L. P., Itou, H., Saitoh, S.: Numerical solutions of linear singular integral equations by means of Tikhonov regularization and reproducing kernels. Houston J. Math 38(4), 1261–1276 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Rodriguez, G., Seatzu, S.: On the numerical inversion of the Laplace transform in reproducing kernel Hilbert spaces. IMA J. Numer. Anal. 13, 463–475 (1993)

    MathSciNet  Article  Google Scholar 

  19. Sawano, Y., Fujiwara, H., Saitoh, S.: Real inversion formulas of the Laplace transform on weighted function spaces. Complex Anal. Oper. Theory 2 (3), 511–521 (2008)

    MathSciNet  Article  Google Scholar 

  20. Evgeniou, T., Pontil, M., Poggio, T.: Regularization networks and support vector machines. Adv. Comput. Math. 13(1), 1–50 (2000)

    MathSciNet  Article  Google Scholar 

  21. Díaz de Alba, P., Fermo, L., Pes, F., Rodriguez, G.: Minimal-norm RKHS solution of an integral model in geo-electromagnetism. In: 21st International Conference on Computational Science and Its Applications (ICCSA), Cagliari, Italy, pp. 21–28, September 13–16, 2021. (2021)

  22. Pes, F., Rodriguez, G.: The minimal-norm Gauss-Newton method and some of its regularized variants. Electron. Trans. Numer. Anal. 53, 459–480 (2020)

    MathSciNet  Article  Google Scholar 

  23. Pes, F., Rodriguez, G.: A doubly relaxed minimal-norm Gauss–Newton method for underdetermined nonlinear least-squares problems. Appl. Numer. Math. 171, 233–248 (2022)

    MathSciNet  Article  Google Scholar 

  24. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Texts in Applied Mathematics, vol. 12. Springer, New York (1991)

    Google Scholar 

  25. Hille, E.: Introduction to general theory of reproducing kernels. Rocky Mt. J. Math. 2(3), 321–368 (1972)

    MathSciNet  Article  Google Scholar 

  26. Yosida, K.: Functional Analysis. Classics in Mathematics. Springer, Berlin (1995)

    Book  Google Scholar 

  27. Hansen, P. C.: Regularization Tools: Version 4.0 for Matlab 7.3. Numer. Algorithms 46, 189–194 (2007)

    MathSciNet  Article  Google Scholar 

  28. Stewart, G. W.: Matrix Algorithms: Volume 1: Basic Decompositions. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  29. Goodman, T. N. T., Micchelli, C. A., Rodriguez, G., Seatzu, S.: On the Cholesky factorization of the Gram matrix of locally supported functions. BIT 35(2), 233–257 (1995)

    MathSciNet  Article  Google Scholar 

  30. Goodman, T. N. T., Micchelli, C. A., Rodriguez, G., Seatzu, S.: On the limiting profile arising from orthonormalizing shifts of exponentially decaying functions. IMA J. Numer. Anal. 18(3), 331–354 (1998)

    MathSciNet  Article  Google Scholar 

  31. Goodman, T. N. T., Micchelli, C. A., Rodriguez, G., Seatzu, S.: On the Cholesky factorisation of the Gram matrix of multivariate functions. SIAM J. Matrix Anal. Appl. 22(2), 501–526 (2000)

    MathSciNet  Article  Google Scholar 

  32. Goodman, T. N. T., Micchelli, C. A., Rodriguez, G., Seatzu, S.: Spectral factorization of Laurent polynomials. Adv. Comput. Math. 7, 429–454 (1997)

    MathSciNet  Article  Google Scholar 

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

    Book  Google Scholar 

  34. Kress, R.: Linear Integral Equation. Springer, Berlin (1999)

    Book  Google Scholar 

  35. Hansen, P. C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  36. Alqahtani, A., Gazzola, S., Reichel, L., Rodriguez, G.: On the block Lanczos and block Golub-Kahan reduction methods applied to discrete ill-posed problems. Numer. Linear Algebra Appl. 2021, 2376 (2021)

    MathSciNet  MATH  Google Scholar 

  37. Gazzola, S., Onunwor, E., Reichel, L., Rodriguez, G.: On the Lanczos and Golub–Kahan reduction methods applied to discrete ill-posed problems. Numer. Linear Algebra Appl. 23, 187–204 (2016)

    MathSciNet  Article  Google Scholar 

  38. Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)

  39. Reichel, L., Rodriguez, G.: Old and new parameter choice rules for discrete ill-posed problems. Numer. Algorithms 63(1), 65–87 (2013)

    MathSciNet  Article  Google Scholar 

  40. Morozov, V. A.: The choice of parameter when solving functional equations by regularization. Dokl. Akad. Nauk. SSSR 175, 1225–1228 (1962)

    Google Scholar 

  41. Hansen, P. C.: Analysis of the discrete ill-posed problems by means of the L-curve. SIAM Rev. 34, 561–580 (1992)

    MathSciNet  Article  Google Scholar 

  42. Hansen, P. C., O’Leary, D. P.: The use of the L-curve in the regularization of discrete ill–posed problems. SIAM J. Sci. Comput. 14, 1487–1503 (1993)

    MathSciNet  Article  Google Scholar 

  43. Hansen, P. C., Jensen, T. K., Rodriguez, G.: An adaptive pruning algorithm for the discrete L-curve criterion. J. Comput. Appl. Math. 198(2), 483–492 (2007)

    MathSciNet  Article  Google Scholar 

  44. Baart, M. L.: The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned linear least-squares problems. IMA J. Numer. Anal. 2, 241–247 (1982)

    MathSciNet  Article  Google Scholar 

Download references


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


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Federica Pes.

Ethics declarations

Conflict of interest

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

Additional information

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

Author contribution

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

Availability of data and materials

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.

Code availability

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

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


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

Mathematics Subject Classification (2010)

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