Stabilizing radial basis functions techniques for a local boundary integral method

Autores
Ponzellini Marinelli, Luciano
Año de publicación
2023
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Fil: Ponzellini Marinelli, Luciano. Universidad Nacional de Rosario; Argentina
Abstract: Radial basis functions (RBFs) have been gaining popularity recently in the development of methods for solving partial differential equations (PDEs) numerically. These functions have become an extremely effective tool for interpolation on scattered node sets in several dimensions. One key issue with infinitely smooth RBFs is the choice of a suitable value for the shape parameter ε, which controls the flatness of the function. It is observed that best accuracy is often achieved when ε tends to zero. However, the system of discrete equations from interpolation matrices becomes ill-conditioned. A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit, such as the RBFQR method and the RBF-GA method. We present these techniques in the context of boundary integral methods to improve the solution of PDEs with RBFs. These stable calculations open up new opportunities for applications and developments of local integral methods based on local RBF approximations. Numerical results for a small shape parameter that stabilizes the error are presented. Accuracy and comparisons are also shown for elliptic PDEs.
Fuente
Revista de la Unión Matemática Argentina. 2023. 64 (2)
Materia
MATEMATICA
FUNCIONES MATEMÁTICAS
ANALISIS MATEMÁTICO
ECUACIONES DIFERENCIALES
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/4.0/
Repositorio
Repositorio Institucional (UCA)
Institución
Pontificia Universidad Católica Argentina
OAI Identificador
oai:ucacris:123456789/16414

id RIUCA_7c8f66a6335cf9c30dd4a9c0ac6a5299
oai_identifier_str oai:ucacris:123456789/16414
network_acronym_str RIUCA
repository_id_str 2585
network_name_str Repositorio Institucional (UCA)
spelling Stabilizing radial basis functions techniques for a local boundary integral methodPonzellini Marinelli, LucianoMATEMATICAFUNCIONES MATEMÁTICASANALISIS MATEMÁTICOECUACIONES DIFERENCIALESFil: Ponzellini Marinelli, Luciano. Universidad Nacional de Rosario; ArgentinaAbstract: Radial basis functions (RBFs) have been gaining popularity recently in the development of methods for solving partial differential equations (PDEs) numerically. These functions have become an extremely effective tool for interpolation on scattered node sets in several dimensions. One key issue with infinitely smooth RBFs is the choice of a suitable value for the shape parameter ε, which controls the flatness of the function. It is observed that best accuracy is often achieved when ε tends to zero. However, the system of discrete equations from interpolation matrices becomes ill-conditioned. A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit, such as the RBFQR method and the RBF-GA method. We present these techniques in the context of boundary integral methods to improve the solution of PDEs with RBFs. These stable calculations open up new opportunities for applications and developments of local integral methods based on local RBF approximations. Numerical results for a small shape parameter that stabilizes the error are presented. Accuracy and comparisons are also shown for elliptic PDEs.Unión Matemática Argentina2023info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfhttps://repositorio.uca.edu.ar/handle/123456789/164141669-9637 (online)0041-6932 (impreso)10.33044/revuma.2901Ponzellini Marinelli, L. Stabilizing radial basis functions techniques for a local boundary integral method [en línea]. Revista de la Unión Matemática Argentina. 2023. 64 (2). doi: 10.33044/revuma.2901. Disponible en: https://repositorio.uca.edu.ar/handle/123456789/16414Revista de la Unión Matemática Argentina. 2023. 64 (2)reponame:Repositorio Institucional (UCA)instname:Pontificia Universidad Católica Argentinaenginfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/4.0/2025-07-03T10:59:17Zoai:ucacris:123456789/16414instacron:UCAInstitucionalhttps://repositorio.uca.edu.ar/Universidad privadaNo correspondehttps://repositorio.uca.edu.ar/oaiclaudia_fernandez@uca.edu.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:25852025-07-03 10:59:17.963Repositorio Institucional (UCA) - Pontificia Universidad Católica Argentinafalse
dc.title.none.fl_str_mv Stabilizing radial basis functions techniques for a local boundary integral method
title Stabilizing radial basis functions techniques for a local boundary integral method
spellingShingle Stabilizing radial basis functions techniques for a local boundary integral method
Ponzellini Marinelli, Luciano
MATEMATICA
FUNCIONES MATEMÁTICAS
ANALISIS MATEMÁTICO
ECUACIONES DIFERENCIALES
title_short Stabilizing radial basis functions techniques for a local boundary integral method
title_full Stabilizing radial basis functions techniques for a local boundary integral method
title_fullStr Stabilizing radial basis functions techniques for a local boundary integral method
title_full_unstemmed Stabilizing radial basis functions techniques for a local boundary integral method
title_sort Stabilizing radial basis functions techniques for a local boundary integral method
dc.creator.none.fl_str_mv Ponzellini Marinelli, Luciano
author Ponzellini Marinelli, Luciano
author_facet Ponzellini Marinelli, Luciano
author_role author
dc.subject.none.fl_str_mv MATEMATICA
FUNCIONES MATEMÁTICAS
ANALISIS MATEMÁTICO
ECUACIONES DIFERENCIALES
topic MATEMATICA
FUNCIONES MATEMÁTICAS
ANALISIS MATEMÁTICO
ECUACIONES DIFERENCIALES
dc.description.none.fl_txt_mv Fil: Ponzellini Marinelli, Luciano. Universidad Nacional de Rosario; Argentina
Abstract: Radial basis functions (RBFs) have been gaining popularity recently in the development of methods for solving partial differential equations (PDEs) numerically. These functions have become an extremely effective tool for interpolation on scattered node sets in several dimensions. One key issue with infinitely smooth RBFs is the choice of a suitable value for the shape parameter ε, which controls the flatness of the function. It is observed that best accuracy is often achieved when ε tends to zero. However, the system of discrete equations from interpolation matrices becomes ill-conditioned. A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit, such as the RBFQR method and the RBF-GA method. We present these techniques in the context of boundary integral methods to improve the solution of PDEs with RBFs. These stable calculations open up new opportunities for applications and developments of local integral methods based on local RBF approximations. Numerical results for a small shape parameter that stabilizes the error are presented. Accuracy and comparisons are also shown for elliptic PDEs.
description Fil: Ponzellini Marinelli, Luciano. Universidad Nacional de Rosario; Argentina
publishDate 2023
dc.date.none.fl_str_mv 2023
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 https://repositorio.uca.edu.ar/handle/123456789/16414
1669-9637 (online)
0041-6932 (impreso)
10.33044/revuma.2901
Ponzellini Marinelli, L. Stabilizing radial basis functions techniques for a local boundary integral method [en línea]. Revista de la Unión Matemática Argentina. 2023. 64 (2). doi: 10.33044/revuma.2901. Disponible en: https://repositorio.uca.edu.ar/handle/123456789/16414
url https://repositorio.uca.edu.ar/handle/123456789/16414
identifier_str_mv 1669-9637 (online)
0041-6932 (impreso)
10.33044/revuma.2901
Ponzellini Marinelli, L. Stabilizing radial basis functions techniques for a local boundary integral method [en línea]. Revista de la Unión Matemática Argentina. 2023. 64 (2). doi: 10.33044/revuma.2901. Disponible en: https://repositorio.uca.edu.ar/handle/123456789/16414
dc.language.none.fl_str_mv eng
language eng
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/4.0/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/4.0/
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Unión Matemática Argentina
publisher.none.fl_str_mv Unión Matemática Argentina
dc.source.none.fl_str_mv Revista de la Unión Matemática Argentina. 2023. 64 (2)
reponame:Repositorio Institucional (UCA)
instname:Pontificia Universidad Católica Argentina
reponame_str Repositorio Institucional (UCA)
collection Repositorio Institucional (UCA)
instname_str Pontificia Universidad Católica Argentina
repository.name.fl_str_mv Repositorio Institucional (UCA) - Pontificia Universidad Católica Argentina
repository.mail.fl_str_mv claudia_fernandez@uca.edu.ar
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score 13.13397