Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 21, 2015

On fractional Tikhonov regularization

  • Daniel Gerth EMAIL logo , Esther Klann , Ronny Ramlau and Lothar Reichel

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.

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

Received: 2014-7-17
Revised: 2014-12-17
Accepted: 2015-3-5
Published Online: 2015-5-21
Published in Print: 2015-12-1

© 2015 by De Gruyter

Downloaded on 3.2.2023 from https://www.degruyter.com/document/doi/10.1515/jiip-2014-0050/html
Scroll Up Arrow