A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions

Autores
Agnelli, Juan Pablo; de Cezaro, Adriano; Leitao Antonio
Año de publicación
2018
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We propose and analyse a regularization method for parameter identification problems modeled by ill-posed nonlinear operator equations, where the parameter to be identified is a piecewise constant function taking known values. Following (De Cezaro et al 2013 Inverse Problems 29 015003), a piecewise constant level set approach is used to represent the unknown parameter, and a corresponding Tikhonov functional is defined on an appropriated space of level set functions. Additionally, a suitable constraint is enforced, resulting that minimizers of our Tikhonov functional belong to the set of piecewise constant level set functions. In other words, the original parameter identification problem is rewritten in the form of a constrained optimization problem, which is solved using an augmented Lagrangian method. We prove the existence of zero duality gaps and the existence of generalized Lagrangian multipliers. Moreover, we extend the analysis in De Cezaro et al's work (2013 Inverse Problems 29 015003), proving convergence and stability of the proposed parameter identification method. A primal-dual algorithm is proposed to compute approximate solutions of the original inverse problem, and its convergence is proved. Numerical examples are presented: this algorithm is applied to a 2D diffuse optical tomography problem. The numerical results are compared with the ones in Agnelli et al (2017 ESAIM: COCV 23 663-83) demonstrating the effectiveness of this primal-dual algorithm.
Fil: Agnelli, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Fil: de Cezaro, Adriano. Universidade Federal Rio Grande Do Sul; Brasil
Fil: Leitao Antonio. Universidade Federal de Santa Catarina; Brasil
Materia
AUGMENTED LAGRANGIAN METHOD
ILL-POSED PROBLEMS
LEVEL-SET APPROACH
REGULARIZATION
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/89275
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/89275
network_acronym_str CONICETDig
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network_name_str CONICET Digital (CONICET)
spelling A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutionsAgnelli, Juan Pablode Cezaro, AdrianoLeitao AntonioAUGMENTED LAGRANGIAN METHODILL-POSED PROBLEMSLEVEL-SET APPROACHREGULARIZATIONhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We propose and analyse a regularization method for parameter identification problems modeled by ill-posed nonlinear operator equations, where the parameter to be identified is a piecewise constant function taking known values. Following (De Cezaro et al 2013 Inverse Problems 29 015003), a piecewise constant level set approach is used to represent the unknown parameter, and a corresponding Tikhonov functional is defined on an appropriated space of level set functions. Additionally, a suitable constraint is enforced, resulting that minimizers of our Tikhonov functional belong to the set of piecewise constant level set functions. In other words, the original parameter identification problem is rewritten in the form of a constrained optimization problem, which is solved using an augmented Lagrangian method. We prove the existence of zero duality gaps and the existence of generalized Lagrangian multipliers. Moreover, we extend the analysis in De Cezaro et al's work (2013 Inverse Problems 29 015003), proving convergence and stability of the proposed parameter identification method. A primal-dual algorithm is proposed to compute approximate solutions of the original inverse problem, and its convergence is proved. Numerical examples are presented: this algorithm is applied to a 2D diffuse optical tomography problem. The numerical results are compared with the ones in Agnelli et al (2017 ESAIM: COCV 23 663-83) demonstrating the effectiveness of this primal-dual algorithm.Fil: Agnelli, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: de Cezaro, Adriano. Universidade Federal Rio Grande Do Sul; BrasilFil: Leitao Antonio. Universidade Federal de Santa Catarina; BrasilIOP Publishing2018-10-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/89275Agnelli, Juan Pablo; de Cezaro, Adriano; Leitao Antonio; A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions; IOP Publishing; Inverse Problems; 34; 12; 1-10-20180266-5611CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://iopscience.iop.org/article/10.1088/1361-6420/aae04dinfo:eu-repo/semantics/altIdentifier/doi/10.1088/1361-6420/aae04dinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:24:21Zoai:ri.conicet.gov.ar:11336/89275instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:24:21.985CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions
title A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions
spellingShingle A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions
Agnelli, Juan Pablo
AUGMENTED LAGRANGIAN METHOD
ILL-POSED PROBLEMS
LEVEL-SET APPROACH
REGULARIZATION
title_short A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions
title_full A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions
title_fullStr A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions
title_full_unstemmed A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions
title_sort A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions
dc.creator.none.fl_str_mv Agnelli, Juan Pablo
de Cezaro, Adriano
Leitao Antonio
author Agnelli, Juan Pablo
author_facet Agnelli, Juan Pablo
de Cezaro, Adriano
Leitao Antonio
author_role author
author2 de Cezaro, Adriano
Leitao Antonio
author2_role author
author
dc.subject.none.fl_str_mv AUGMENTED LAGRANGIAN METHOD
ILL-POSED PROBLEMS
LEVEL-SET APPROACH
REGULARIZATION
topic AUGMENTED LAGRANGIAN METHOD
ILL-POSED PROBLEMS
LEVEL-SET APPROACH
REGULARIZATION
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We propose and analyse a regularization method for parameter identification problems modeled by ill-posed nonlinear operator equations, where the parameter to be identified is a piecewise constant function taking known values. Following (De Cezaro et al 2013 Inverse Problems 29 015003), a piecewise constant level set approach is used to represent the unknown parameter, and a corresponding Tikhonov functional is defined on an appropriated space of level set functions. Additionally, a suitable constraint is enforced, resulting that minimizers of our Tikhonov functional belong to the set of piecewise constant level set functions. In other words, the original parameter identification problem is rewritten in the form of a constrained optimization problem, which is solved using an augmented Lagrangian method. We prove the existence of zero duality gaps and the existence of generalized Lagrangian multipliers. Moreover, we extend the analysis in De Cezaro et al's work (2013 Inverse Problems 29 015003), proving convergence and stability of the proposed parameter identification method. A primal-dual algorithm is proposed to compute approximate solutions of the original inverse problem, and its convergence is proved. Numerical examples are presented: this algorithm is applied to a 2D diffuse optical tomography problem. The numerical results are compared with the ones in Agnelli et al (2017 ESAIM: COCV 23 663-83) demonstrating the effectiveness of this primal-dual algorithm.
Fil: Agnelli, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Fil: de Cezaro, Adriano. Universidade Federal Rio Grande Do Sul; Brasil
Fil: Leitao Antonio. Universidade Federal de Santa Catarina; Brasil
description We propose and analyse a regularization method for parameter identification problems modeled by ill-posed nonlinear operator equations, where the parameter to be identified is a piecewise constant function taking known values. Following (De Cezaro et al 2013 Inverse Problems 29 015003), a piecewise constant level set approach is used to represent the unknown parameter, and a corresponding Tikhonov functional is defined on an appropriated space of level set functions. Additionally, a suitable constraint is enforced, resulting that minimizers of our Tikhonov functional belong to the set of piecewise constant level set functions. In other words, the original parameter identification problem is rewritten in the form of a constrained optimization problem, which is solved using an augmented Lagrangian method. We prove the existence of zero duality gaps and the existence of generalized Lagrangian multipliers. Moreover, we extend the analysis in De Cezaro et al's work (2013 Inverse Problems 29 015003), proving convergence and stability of the proposed parameter identification method. A primal-dual algorithm is proposed to compute approximate solutions of the original inverse problem, and its convergence is proved. Numerical examples are presented: this algorithm is applied to a 2D diffuse optical tomography problem. The numerical results are compared with the ones in Agnelli et al (2017 ESAIM: COCV 23 663-83) demonstrating the effectiveness of this primal-dual algorithm.
publishDate 2018
dc.date.none.fl_str_mv 2018-10-01
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/89275
Agnelli, Juan Pablo; de Cezaro, Adriano; Leitao Antonio; A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions; IOP Publishing; Inverse Problems; 34; 12; 1-10-2018
0266-5611
CONICET Digital
CONICET
url http://hdl.handle.net/11336/89275
identifier_str_mv Agnelli, Juan Pablo; de Cezaro, Adriano; Leitao Antonio; A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions; IOP Publishing; Inverse Problems; 34; 12; 1-10-2018
0266-5611
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://iopscience.iop.org/article/10.1088/1361-6420/aae04d
info:eu-repo/semantics/altIdentifier/doi/10.1088/1361-6420/aae04d
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv IOP Publishing
publisher.none.fl_str_mv IOP Publishing
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.070432

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