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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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1569-3945
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Volume 23, Issue 6

Issues

On fractional Tikhonov regularization

Daniel Gerth
  • Johannes Kepler University Linz, Doctoral Program “Computational Mathematics”, Altenbergerstraße 69, A-4040 Linz, Austria
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/ Esther Klann / Ronny Ramlau
  • Radon Institute for Computational and Applied Mathematics (RICAM) and Johannes Kepler University Linz, Industrial Mathematics Institute and Doctoral Program “Computational Mathematics”, Altenbergerstraße 69, A-4040 Linz, Austria
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/ Lothar Reichel
Published Online: 2015-05-21 | DOI: https://doi.org/10.1515/jiip-2014-0050

Abstract

It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth, i.e., the approximate solution may lack many details that the desired exact solution might possess. Two different approaches, both referred to as fractional Tikhonov methods have been introduced to remedy this shortcoming. This paper investigates the convergence properties of these methods by reviewing results published previously by various authors. We show that both methods are order optimal when the regularization parameter is chosen according to the discrepancy principle. The theory developed suggests situations in which the fractional methods yield approximate solutions of higher quality than Tikhonov regularization in standard form. Computed examples that illustrate the behavior of the methods are presented.

Keywords: Ill-posed problem; regularization method; fractional Tikhonov; filter function; discrepancy principle

MSC: 65F22; 65R30; 65R32

About the article

Received: 2014-07-17

Revised: 2014-12-17

Accepted: 2015-03-05

Published Online: 2015-05-21

Published in Print: 2015-12-01


Funding Source: Austrian Science Fund (FWF)

Award identifier / Grant number: W1214-N15

Funding Source: Austrian Science Fund (FWF)

Award identifier / Grant number: T529-N18

Funding Source: NSF

Award identifier / Grant number: DMS-1115385


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 23, Issue 6, Pages 611–625, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2014-0050.

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