Journal of Engineering Mathematics

, Volume 93, Issue 1, pp 113–129 | Cite as

Fractional regularization matrices for linear discrete ill-posed problems

  • Michiel E. Hochstenbach
  • Silvia Noschese
  • Lothar Reichel


The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices \(A^\mathrm{T}\!A\) (for Tikhonov regularization) and A (for Lavrentiev regularization) as regularization matrices, where A is the matrix that defines the linear discrete ill-posed problem. Both small- and large-scale problems are considered.


Fractional Lavrentiev regularization Fractional power regularization matrix  Fractional Tikhonov regularization Ill-posed problem 



M. E. Hochstenbach was supported by an NWO Vidi Grant. L. Reichel was supported in part by NSF Grant DMS-1115385.


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Michiel E. Hochstenbach
    • 1
  • Silvia Noschese
    • 2
  • Lothar Reichel
    • 3
  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Dipartimento di Matematica “Guido Castelnuovo”SAPIENZA Università di RomaRomeItaly
  3. 3.Department of Mathematical SciencesKent State UniversityKentUSA

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