Discrete gradient flows for shape optimization and applications

Autores
Dogan, G.; Morin, Pedro; Nochetto, R.H.; Verani, M.
Año de publicación
2007
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation, its full discretization and the resulting linear algebraic systems. Our approach hinges on the following essential features: shape differential calculus, a semi-implicit time discretization and a finite element method for space discretization. We use shape differential calculus to express variations of bulk and surface energies with respect to domain changes. The semi-implicit time discretization allows us to track the domain boundary without an explicit parametrization, and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity. We propose a Schur complement approach to solve the resulting linear systems efficiently. We discuss applications of this framework to image segmentation, optimal shape design for PDE, and surface diffusion, along with the choice of suitable scalar products in each case. We illustrate the method with several numerical experiments, some developing pinch-off and topological changes in finite time.
Fil: Dogan, G.. University of Maryland; Estados Unidos
Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Nochetto, R.H.. University of Maryland; Estados Unidos
Fil: Verani, M.. Politecnico di Milano; Italia
Materia
FINITE ELEMENTS
GRADIENT FLOW
IMAGE SEGMENTATION
SCALAR PRODUCT
SEMI-IMPLICIT DISCRETIZATION
SHAPE OPTIMIZATION
SURFACE DIFFUSION
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:11336/84269
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/84269
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Discrete gradient flows for shape optimization and applicationsDogan, G.Morin, PedroNochetto, R.H.Verani, M.FINITE ELEMENTSGRADIENT FLOWIMAGE SEGMENTATIONSCALAR PRODUCTSEMI-IMPLICIT DISCRETIZATIONSHAPE OPTIMIZATIONSURFACE DIFFUSIONhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation, its full discretization and the resulting linear algebraic systems. Our approach hinges on the following essential features: shape differential calculus, a semi-implicit time discretization and a finite element method for space discretization. We use shape differential calculus to express variations of bulk and surface energies with respect to domain changes. The semi-implicit time discretization allows us to track the domain boundary without an explicit parametrization, and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity. We propose a Schur complement approach to solve the resulting linear systems efficiently. We discuss applications of this framework to image segmentation, optimal shape design for PDE, and surface diffusion, along with the choice of suitable scalar products in each case. We illustrate the method with several numerical experiments, some developing pinch-off and topological changes in finite time.Fil: Dogan, G.. University of Maryland; Estados UnidosFil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Nochetto, R.H.. University of Maryland; Estados UnidosFil: Verani, M.. Politecnico di Milano; ItaliaElsevier Science Sa2007-08info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/84269Dogan, G.; Morin, Pedro; Nochetto, R.H.; Verani, M.; Discrete gradient flows for shape optimization and applications; Elsevier Science Sa; Computer Methods in Applied Mechanics and Engineering; 196; 37-40 SPEC. ISS.; 8-2007; 3898-39140045-7825CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.cma.2006.10.046info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:38:56Zoai:ri.conicet.gov.ar:11336/84269instacron: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 09:38:57.153CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Discrete gradient flows for shape optimization and applications
title Discrete gradient flows for shape optimization and applications
spellingShingle Discrete gradient flows for shape optimization and applications
Dogan, G.
FINITE ELEMENTS
GRADIENT FLOW
IMAGE SEGMENTATION
SCALAR PRODUCT
SEMI-IMPLICIT DISCRETIZATION
SHAPE OPTIMIZATION
SURFACE DIFFUSION
title_short Discrete gradient flows for shape optimization and applications
title_full Discrete gradient flows for shape optimization and applications
title_fullStr Discrete gradient flows for shape optimization and applications
title_full_unstemmed Discrete gradient flows for shape optimization and applications
title_sort Discrete gradient flows for shape optimization and applications
dc.creator.none.fl_str_mv Dogan, G.
Morin, Pedro
Nochetto, R.H.
Verani, M.
author Dogan, G.
author_facet Dogan, G.
Morin, Pedro
Nochetto, R.H.
Verani, M.
author_role author
author2 Morin, Pedro
Nochetto, R.H.
Verani, M.
author2_role author
author
author
dc.subject.none.fl_str_mv FINITE ELEMENTS
GRADIENT FLOW
IMAGE SEGMENTATION
SCALAR PRODUCT
SEMI-IMPLICIT DISCRETIZATION
SHAPE OPTIMIZATION
SURFACE DIFFUSION
topic FINITE ELEMENTS
GRADIENT FLOW
IMAGE SEGMENTATION
SCALAR PRODUCT
SEMI-IMPLICIT DISCRETIZATION
SHAPE OPTIMIZATION
SURFACE DIFFUSION
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 present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation, its full discretization and the resulting linear algebraic systems. Our approach hinges on the following essential features: shape differential calculus, a semi-implicit time discretization and a finite element method for space discretization. We use shape differential calculus to express variations of bulk and surface energies with respect to domain changes. The semi-implicit time discretization allows us to track the domain boundary without an explicit parametrization, and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity. We propose a Schur complement approach to solve the resulting linear systems efficiently. We discuss applications of this framework to image segmentation, optimal shape design for PDE, and surface diffusion, along with the choice of suitable scalar products in each case. We illustrate the method with several numerical experiments, some developing pinch-off and topological changes in finite time.
Fil: Dogan, G.. University of Maryland; Estados Unidos
Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Nochetto, R.H.. University of Maryland; Estados Unidos
Fil: Verani, M.. Politecnico di Milano; Italia
description We present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation, its full discretization and the resulting linear algebraic systems. Our approach hinges on the following essential features: shape differential calculus, a semi-implicit time discretization and a finite element method for space discretization. We use shape differential calculus to express variations of bulk and surface energies with respect to domain changes. The semi-implicit time discretization allows us to track the domain boundary without an explicit parametrization, and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity. We propose a Schur complement approach to solve the resulting linear systems efficiently. We discuss applications of this framework to image segmentation, optimal shape design for PDE, and surface diffusion, along with the choice of suitable scalar products in each case. We illustrate the method with several numerical experiments, some developing pinch-off and topological changes in finite time.
publishDate 2007
dc.date.none.fl_str_mv 2007-08
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/11336/84269
Dogan, G.; Morin, Pedro; Nochetto, R.H.; Verani, M.; Discrete gradient flows for shape optimization and applications; Elsevier Science Sa; Computer Methods in Applied Mechanics and Engineering; 196; 37-40 SPEC. ISS.; 8-2007; 3898-3914
0045-7825
CONICET Digital
CONICET
url http://hdl.handle.net/11336/84269
identifier_str_mv Dogan, G.; Morin, Pedro; Nochetto, R.H.; Verani, M.; Discrete gradient flows for shape optimization and applications; Elsevier Science Sa; Computer Methods in Applied Mechanics and Engineering; 196; 37-40 SPEC. ISS.; 8-2007; 3898-3914
0045-7825
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1016/j.cma.2006.10.046
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 Elsevier Science Sa
publisher.none.fl_str_mv Elsevier Science Sa
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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