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
- Institución
- Pontificia Universidad Católica Argentina
- OAI Identificador
- oai:ucacris:123456789/16414
Ver los metadatos del registro completo
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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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1836638368651280384 |
score |
13.13397 |