Journal of Scientific Computing

, Volume 78, Issue 3, pp 1526–1549 | Cite as

An \(\ell ^2-\ell ^q\) Regularization Method for Large Discrete Ill-Posed Problems

  • Alessandro BucciniEmail author
  • Lothar Reichel


Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. Regularization aims to reduce this sensitivity. Typically, regularization methods replace the original problem by a minimization problem with a fidelity term and a regularization term. Recently, the use of a p-norm to measure the fidelity term, and a q-norm to measure the regularization term, has received considerable attention. The relative importance of these terms is determined by a regularization parameter. This paper discussed how the latter parameter can be determined with the aid of the discrepancy principle. We primarily focus on the situation when \(p=2\) and \(0<q\le 2\), where we note that when \(0<q<1\), the minimization problem may be non-convex.


\(\ell ^2-\ell ^q\) minimization Ill-posed problem Iterative method 

Mathematics Subject Classification

65F10 65R32 90C26 



The authors would like to thank the referees for their insightful comments that improved the presentation. The first author is a member of the INdAM Research group GNCS and his work is partially founded by the group. The second author is supported in part by NSF Grants DMS-1720259 and DMS-1729509.


  1. 1.
    Brill, M., Schock, E.: Iterative solution of ill-posed problems—a survey, in Model Optimization in Exploration Geophysics, ed. A. Vogel, Vieweg, Braunschweig, pp. 17–37 (1987)Google Scholar
  2. 2.
    Buccini, A.: Regularizing preconditioners by non-stationary iterated Tikhonov with general penalty term. Appl. Numer. Math. 116, 64–81 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cai, J.-F., Chan, R.H., Shen, L., Shen, Z.: Simultaneously inpainting in image and transformed domains. Numer. Math. 112, 509–533 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cai, J.-F., Chan, R.H., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comput. Harmonic Anal. 24, 131–149 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cai, J.-F., Chan, R.H., Shen, Z.: Linearized Bregman iterations for frame-based image deblurring. SIAM J. Imaging Sci. 2, 226–252 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cai, J.-F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8, 337–369 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chan, R.H., Liang, H.X.: Half-quadratic algorithm for \(\ell _p\)-\(\ell _q\) problems with applications to TV-\(\ell _1\) image restoration and compressive sensing. In: Proceedings of Efficient Algorithms for Global Optimization Methods in Computer Vision, Lecture Notes in Comput. Sci. # 8293, pp. 78–103. Springer, Berlin (2014)Google Scholar
  8. 8.
    Daniel, J.W., Gragg, W.B., Kaufman, L., Stewart, G.W.: Reorthogonalization and stable algorithms for updating the Gram–Schmidt QR factorization. Math. Comput. 30, 772–795 (1976)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Donatelli, M., Hanke, M.: Fast nonstationary preconditioned iterative methods for ill-posed problems with application to image deblurring. Inverse Probl. 29, 095008 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Donatelli, M., Huckle, T., Mazza, M., Sesana, D.: Image deblurring by sparsity constraint on the Fourier coefficients. Numer. Algorithms 72, 341–361 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Estatico, C., Gratton, S., Lenti, F., Titley-Peloquin, D.: A conjugate gradient like method for \(p\)-norm minimization in functional spaces. Numer. Math. 137, 895–922 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gazzola, S., Nagy, J.G.: Generalized Arnoldi–Tikhonov method for sparse reconstruction. SIAM J. Sci. Comput. 36, B225–B247 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gazzola, S., Novati, P., Russo, M.R.: On Krylov projection methods and Tikhonov regularization. Electron. Trans. Numer. Anal. 44, 83–123 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hanke, M., Hansen, P.C.: Regularization methods for large-scale problems. Surv. Math. Ind. 3, 253–315 (1993)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hanke, M., Groetsch, C.W.: Nonstationary iterated Tikhonov regularization. J. Optim. Theory Appl. 98, 37–53 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hansen, P.C.: Rank Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)CrossRefGoogle Scholar
  18. 18.
    Hansen, P.C., Nagy, J.G., O’Leary, D.P.: Deblurring Images: Matrices, Spectra, and Filtering. SIAM, Philadelphia (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Hiriart-Urruty, J.-P., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, New York (2004)zbMATHGoogle Scholar
  20. 20.
    Huang, G., Lanza, A., Morigi, S., Reichel, L., Sgallari, F.: Majorization-minimization generalized Krylov subspace methods for \(\ell _p-\ell _q\) optimization applied to image restoration. BIT Numer. Math. 57, 351–378 (2017)CrossRefzbMATHGoogle Scholar
  21. 21.
    Huang, G., Reichel, L., Yin, F.: Projected nonstationary iterated Tikhonov regularization. BIT Numer. Math. 56, 467–487 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Huang, J., Donatelli, M., Chan, R.H.: Nonstationary iterated thresholding algorithms for image deblurring. Inverse Probl. Imaging 7, 717–736 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lampe, J., Reichel, L., Voss, H.: Large-scale Tikhonov regularization via reduction by orthogonal projection. Linear Algebra Appl. 436, 2845–2865 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lanza, A., Morigi, S., Reichel, L., Sgallari, F.: A generalized Krylov subspace method for \(\ell _p-\ell _q\) minimization. SIAM J. Sci. Comput. 37, S30–S50 (2015)CrossRefzbMATHGoogle Scholar
  25. 25.
    Rodríguez, P., Wohlberg, B.: Efficient minimization method for a generalized total variation functional. IEEE Trans. Image Process. 18, 322–332 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Serra-Capizzano, S.: A note on antireflective boundary conditions and fast deblurring models. SIAM J. Sci. Comput. 25, 1307–1325 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wolke, R., Schwetlick, H.: Iteratively reweighted least squares: algorithms, convergence analysis, and numerical comparisons. SIAM J. Sci. Stat. Comput. 9, 907–921 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA

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