Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift.
Autor(a) principal: | |
---|---|
Data de Publicação: | 2012 |
Outros Autores: | , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFOP |
Texto Completo: | http://www.repositorio.ufop.br/handle/123456789/1968 |
Resumo: | In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains X _ RN. This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function u 2 L2ðXÞ. |
id |
UFOP_cde2aed41bd2ac0a5477a340852ef162 |
---|---|
oai_identifier_str |
oai:localhost:123456789/1968 |
network_acronym_str |
UFOP |
network_name_str |
Repositório Institucional da UFOP |
repository_id_str |
3233 |
spelling |
Biezuner, Rodney JosuéErcole, GreyGiacchini, Breno LoureiroMartins, Eder Marinho2012-12-05T14:14:21Z2012-12-05T14:14:21Z2012BIEZUNER, R. J. et al. Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. Applied Mathematics and Computation, v. 219, n. 1, p. 360-375, 2012. Disponível em: <https://www.sciencedirect.com/science/article/pii/S0096300312006248>. Acesso em: 03 dez. 201200963003http://www.repositorio.ufop.br/handle/123456789/1968In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains X _ RN. This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function u 2 L2ðXÞ.LaplacianEigenvaluesEigenfunctionsFourier seriesInverse interation with shiftEigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift.info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleO periódico Applied Mathematics and Computation concede permissão para depósito do artigo no Repositório Institucional da UFOP. Número da licença: 3305300249129.info:eu-repo/semantics/openAccessengreponame:Repositório Institucional da UFOPinstname:Universidade Federal de Ouro Preto (UFOP)instacron:UFOPLICENSElicense.txtlicense.txttext/plain; charset=utf-81748http://www.repositorio.ufop.br/bitstream/123456789/1968/2/license.txt8a4605be74aa9ea9d79846c1fba20a33MD52ORIGINALARTIGO_EigenvaluesEigenfunctionsLaplacian.pdfARTIGO_EigenvaluesEigenfunctionsLaplacian.pdfapplication/pdf456109http://www.repositorio.ufop.br/bitstream/123456789/1968/1/ARTIGO_EigenvaluesEigenfunctionsLaplacian.pdf801ec714d6ae0ac02f0ba94262fd2e5cMD51123456789/19682019-03-15 12:56:58.034oai:localhost: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Repositório InstitucionalPUBhttp://www.repositorio.ufop.br/oai/requestrepositorio@ufop.edu.bropendoar:32332019-03-15T16:56:58Repositório Institucional da UFOP - Universidade Federal de Ouro Preto (UFOP)false |
dc.title.pt_BR.fl_str_mv |
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. |
title |
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. |
spellingShingle |
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. Biezuner, Rodney Josué Laplacian Eigenvalues Eigenfunctions Fourier series Inverse interation with shift |
title_short |
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. |
title_full |
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. |
title_fullStr |
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. |
title_full_unstemmed |
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. |
title_sort |
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. |
author |
Biezuner, Rodney Josué |
author_facet |
Biezuner, Rodney Josué Ercole, Grey Giacchini, Breno Loureiro Martins, Eder Marinho |
author_role |
author |
author2 |
Ercole, Grey Giacchini, Breno Loureiro Martins, Eder Marinho |
author2_role |
author author author |
dc.contributor.author.fl_str_mv |
Biezuner, Rodney Josué Ercole, Grey Giacchini, Breno Loureiro Martins, Eder Marinho |
dc.subject.por.fl_str_mv |
Laplacian Eigenvalues Eigenfunctions Fourier series Inverse interation with shift |
topic |
Laplacian Eigenvalues Eigenfunctions Fourier series Inverse interation with shift |
description |
In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains X _ RN. This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function u 2 L2ðXÞ. |
publishDate |
2012 |
dc.date.accessioned.fl_str_mv |
2012-12-05T14:14:21Z |
dc.date.available.fl_str_mv |
2012-12-05T14:14:21Z |
dc.date.issued.fl_str_mv |
2012 |
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.citation.fl_str_mv |
BIEZUNER, R. J. et al. Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. Applied Mathematics and Computation, v. 219, n. 1, p. 360-375, 2012. Disponível em: <https://www.sciencedirect.com/science/article/pii/S0096300312006248>. Acesso em: 03 dez. 2012 |
dc.identifier.uri.fl_str_mv |
http://www.repositorio.ufop.br/handle/123456789/1968 |
dc.identifier.issn.none.fl_str_mv |
00963003 |
identifier_str_mv |
BIEZUNER, R. J. et al. Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift. Applied Mathematics and Computation, v. 219, n. 1, p. 360-375, 2012. Disponível em: <https://www.sciencedirect.com/science/article/pii/S0096300312006248>. Acesso em: 03 dez. 2012 00963003 |
url |
http://www.repositorio.ufop.br/handle/123456789/1968 |
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.source.none.fl_str_mv |
reponame:Repositório Institucional da UFOP instname:Universidade Federal de Ouro Preto (UFOP) instacron:UFOP |
instname_str |
Universidade Federal de Ouro Preto (UFOP) |
instacron_str |
UFOP |
institution |
UFOP |
reponame_str |
Repositório Institucional da UFOP |
collection |
Repositório Institucional da UFOP |
bitstream.url.fl_str_mv |
http://www.repositorio.ufop.br/bitstream/123456789/1968/2/license.txt http://www.repositorio.ufop.br/bitstream/123456789/1968/1/ARTIGO_EigenvaluesEigenfunctionsLaplacian.pdf |
bitstream.checksum.fl_str_mv |
8a4605be74aa9ea9d79846c1fba20a33 801ec714d6ae0ac02f0ba94262fd2e5c |
bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 |
repository.name.fl_str_mv |
Repositório Institucional da UFOP - Universidade Federal de Ouro Preto (UFOP) |
repository.mail.fl_str_mv |
repositorio@ufop.edu.br |
_version_ |
1801685789727260672 |