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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
Article
  • 75 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

65F10 65R32 90C26 

Notes

Acknowledgements

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.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA

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