On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems
Autor(a) principal: | |
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Data de Publicação: | 2013 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da FURG (RI FURG) |
Texto Completo: | http://repositorio.furg.br/handle/1/3730 |
Resumo: | We investigate level-set-type methods for solving ill-posed problems with discontinuous (piecewise constant) coefficients. The goal is to identify the level sets as well as the level values of an unknown parameter function on a model described by a nonlinear ill-posed operator equation. The PCLS approach is used here to parametrize the solution of a given operator equation in terms of a L2 level-set function, i.e. the level-set function itself is assumed to be a piecewise constant function. Two distinct methods are proposed for computing stable solutions of the resulting ill-posed problem: the first is based on Tikhonov regularization, while the second is based on the augmented Lagrangian approach with total variation penalization. Classical regularization results (Engl H W et al 1996 Mathematics and its Applications (Dordrecht: Kluwer)) are derived for the Tikhonov method. On the other hand, for the augmented Lagrangian method, we succeed in proving the existence of (generalized) Lagrangian multipliers in the sense of (Rockafellar R T and Wets R J-B 1998 Grundlehren der Mathematischen Wissenschaften (Berlin: Springer)). Numerical experiments are performed for a 2D inverse potential problem (Hettlich F and Rundell W 1996 Inverse Problems 12 251–66), demonstrating the capabilities of both methods for solving this ill-posed problem in a stable way (complicated inclusions are recovered without any a priori geometrical information on the unknown parameter). |
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On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problemsWe investigate level-set-type methods for solving ill-posed problems with discontinuous (piecewise constant) coefficients. The goal is to identify the level sets as well as the level values of an unknown parameter function on a model described by a nonlinear ill-posed operator equation. The PCLS approach is used here to parametrize the solution of a given operator equation in terms of a L2 level-set function, i.e. the level-set function itself is assumed to be a piecewise constant function. Two distinct methods are proposed for computing stable solutions of the resulting ill-posed problem: the first is based on Tikhonov regularization, while the second is based on the augmented Lagrangian approach with total variation penalization. Classical regularization results (Engl H W et al 1996 Mathematics and its Applications (Dordrecht: Kluwer)) are derived for the Tikhonov method. On the other hand, for the augmented Lagrangian method, we succeed in proving the existence of (generalized) Lagrangian multipliers in the sense of (Rockafellar R T and Wets R J-B 1998 Grundlehren der Mathematischen Wissenschaften (Berlin: Springer)). Numerical experiments are performed for a 2D inverse potential problem (Hettlich F and Rundell W 1996 Inverse Problems 12 251–66), demonstrating the capabilities of both methods for solving this ill-posed problem in a stable way (complicated inclusions are recovered without any a priori geometrical information on the unknown parameter).2013-08-22T21:31:12Z2013-08-22T21:31:12Z2013info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfCEZZARO, Adriano de; LEITÃO, Antônio; TAI, Xue-cheng. On piecewise constant level-set (PCLS) methods for the identification of discontinuous parameters in ill-posed problems. Inverse Problems, v. 29, p. 1- 23, 2013. Disponível em:<http://mtm.ufsc.br/~aleitao/public/reprints/pap2013-clt-IP.pdf>. Acesso em: 28 fev. 2013.http://repositorio.furg.br/handle/1/3730doi:10.1088/0266-5611/29/1/015003engCezaro, Adriano deLeitão, AntônioTai, Xue-Chenginfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da FURG (RI FURG)instname:Universidade Federal do Rio Grande (FURG)instacron:FURG2014-09-24T12:55:52Zoai:repositorio.furg.br:1/3730Repositório InstitucionalPUBhttps://repositorio.furg.br/oai/request || http://200.19.254.174/oai/requestopendoar:2014-09-24T12:55:52Repositório Institucional da FURG (RI FURG) - Universidade Federal do Rio Grande (FURG)false |
dc.title.none.fl_str_mv |
On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems |
title |
On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems |
spellingShingle |
On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems Cezaro, Adriano de |
title_short |
On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems |
title_full |
On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems |
title_fullStr |
On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems |
title_full_unstemmed |
On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems |
title_sort |
On piecewise constant level-set (pcls) methods for the identification of discontinuous parameters in ill-posed problems |
author |
Cezaro, Adriano de |
author_facet |
Cezaro, Adriano de Leitão, Antônio Tai, Xue-Cheng |
author_role |
author |
author2 |
Leitão, Antônio Tai, Xue-Cheng |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Cezaro, Adriano de Leitão, Antônio Tai, Xue-Cheng |
description |
We investigate level-set-type methods for solving ill-posed problems with discontinuous (piecewise constant) coefficients. The goal is to identify the level sets as well as the level values of an unknown parameter function on a model described by a nonlinear ill-posed operator equation. The PCLS approach is used here to parametrize the solution of a given operator equation in terms of a L2 level-set function, i.e. the level-set function itself is assumed to be a piecewise constant function. Two distinct methods are proposed for computing stable solutions of the resulting ill-posed problem: the first is based on Tikhonov regularization, while the second is based on the augmented Lagrangian approach with total variation penalization. Classical regularization results (Engl H W et al 1996 Mathematics and its Applications (Dordrecht: Kluwer)) are derived for the Tikhonov method. On the other hand, for the augmented Lagrangian method, we succeed in proving the existence of (generalized) Lagrangian multipliers in the sense of (Rockafellar R T and Wets R J-B 1998 Grundlehren der Mathematischen Wissenschaften (Berlin: Springer)). Numerical experiments are performed for a 2D inverse potential problem (Hettlich F and Rundell W 1996 Inverse Problems 12 251–66), demonstrating the capabilities of both methods for solving this ill-posed problem in a stable way (complicated inclusions are recovered without any a priori geometrical information on the unknown parameter). |
publishDate |
2013 |
dc.date.none.fl_str_mv |
2013-08-22T21:31:12Z 2013-08-22T21:31:12Z 2013 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
CEZZARO, Adriano de; LEITÃO, Antônio; TAI, Xue-cheng. On piecewise constant level-set (PCLS) methods for the identification of discontinuous parameters in ill-posed problems. Inverse Problems, v. 29, p. 1- 23, 2013. Disponível em:<http://mtm.ufsc.br/~aleitao/public/reprints/pap2013-clt-IP.pdf>. Acesso em: 28 fev. 2013. http://repositorio.furg.br/handle/1/3730 doi:10.1088/0266-5611/29/1/015003 |
identifier_str_mv |
CEZZARO, Adriano de; LEITÃO, Antônio; TAI, Xue-cheng. On piecewise constant level-set (PCLS) methods for the identification of discontinuous parameters in ill-posed problems. Inverse Problems, v. 29, p. 1- 23, 2013. Disponível em:<http://mtm.ufsc.br/~aleitao/public/reprints/pap2013-clt-IP.pdf>. Acesso em: 28 fev. 2013. doi:10.1088/0266-5611/29/1/015003 |
url |
http://repositorio.furg.br/handle/1/3730 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da FURG (RI FURG) instname:Universidade Federal do Rio Grande (FURG) instacron:FURG |
instname_str |
Universidade Federal do Rio Grande (FURG) |
instacron_str |
FURG |
institution |
FURG |
reponame_str |
Repositório Institucional da FURG (RI FURG) |
collection |
Repositório Institucional da FURG (RI FURG) |
repository.name.fl_str_mv |
Repositório Institucional da FURG (RI FURG) - Universidade Federal do Rio Grande (FURG) |
repository.mail.fl_str_mv |
|
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1807384406530195456 |