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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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Volume 23, Issue 6

Issues

Mixed spatially varying L2-BV regularization of inverse ill-posed problems

Gisela L. Mazzieri
  • Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Centro Científico Tecnológico CONICET-Santa Fe, Colectora Ruta Nacional 168 km. 472, Paraje “El Pozo”, C.P. 3000 Santa Fe, Argentina; and Departamento de Matemática, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, 3000 Santa Fe, Argentina
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/ Ruben D. Spies
  • Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Centro Científico Tecnológico CONICET-Santa Fe, Colectora Ruta Nacional 168 km. 472, Paraje “El Pozo”, C.P. 3000 Santa Fe, Argentina; and Departamento de Matemática, Facultad de Ingeniería Química, Universidad Nacional del Litoral, 3000 Santa Fe, Argentina
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/ Karina G. Temperini
  • Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Centro Científico Tecnológico CONICET-Santa Fe, Colectora Ruta Nacional 168 km. 472, Paraje “El Pozo”, C.P. 3000 Santa Fe, Argentina; and Departamento de Matemática, Facultad de Humanidades y Ciencias, Universidad Nacional del Litoral, 3000 Santa Fe, Argentina
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Published Online: 2014-12-11 | DOI: https://doi.org/10.1515/jiip-2014-0034

Abstract

Several generalizations of the traditional Tikhonov–Phillips regularization method have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers which allow the capturing of diverse properties of the exact solution (e.g. edges, discontinuities, borders, etc.). However, in some problems in which it is known that the regularity of the exact solution is heterogeneous and/or anisotropic, it is reasonable to think that a much better option could be the simultaneous use of two or more penalizers of different nature. Such is the case, for instance, in some image restoration problems in which preservation of edges, borders or discontinuities is an important matter. In this work we present some results on the simultaneous use of penalizers of L2 and of bounded variation (BV) type. For particular cases, existence and uniqueness results are proved. Open problems are discussed and results to signal restoration problems are presented.

Keywords: Inverse problem; ill-posed; regularization

MSC: 65J20; 47A52

About the article

Received: 2014-04-30

Revised: 2014-10-01

Accepted: 2014-10-05

Published Online: 2014-12-11

Published in Print: 2015-12-01


Funding Source: Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET

Award identifier / Grant number: PIP 2010-2012 Nro. 0219

Funding Source: Agencia Nacional de Promoción Científica y Tecnológica, ANPCyT

Award identifier / Grant number: PICT 2008-1301

Funding Source: Universidad Nacional del Litoral

Award identifier / Grant number: CAI+D 2009-PI-62-315

Funding Source: Universidad Nacional del Litoral

Award identifier / Grant number: CAI+D PJov 2011 Nro. 50020110100055

Funding Source: Universidad Nacional del Litoral

Award identifier / Grant number: CAI+D PI 2011 Nro. 50120110100294

Funding Source: Air Force Office of Scientific Research, AFOSR

Award identifier / Grant number: FA9550-14-1-0130


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 23, Issue 6, Pages 571–585, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2014-0034.

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[2]
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[3]
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Journal of Mathematical Analysis and Applications, 2017, Volume 450, Number 1, Page 427

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