Stabilizing radial basis functions techniques for a local boundary integral method

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 ke...

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Autor principal: Ponzellini Marinelli, Luciano
Formato: Artículo
Lenguaje:Inglés
Publicado: Unión Matemática Argentina 2023
Materias:
MATEMATICA
FUNCIONES MATEMÁTICAS
ANALISIS MATEMÁTICO
ECUACIONES DIFERENCIALES
Acceso en línea:https://repositorio.uca.edu.ar/handle/123456789/16414
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Repositorio Institucional de la Universidad Católica Argentina (UCA) de Universidad Católica Argentina
id I33-R139-123456789-16414
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spelling I33-R139-123456789-164142023-05-30T05:01:45Z Stabilizing radial basis functions techniques for a local boundary integral method Ponzellini Marinelli, Luciano MATEMATICA FUNCIONES MATEMÁTICAS ANALISIS MATEMÁTICO ECUACIONES DIFERENCIALES 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. 2023-05-29T22:34:20Z 2023-05-29T22:34:20Z 2023 Artículo 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 1669-9637 (online) 0041-6932 (impreso) https://repositorio.uca.edu.ar/handle/123456789/16414 10.33044/revuma.2901 eng Acceso abierto http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Unión Matemática Argentina Revista de la Unión Matemática Argentina. 2023. 64 (2)
institution Universidad Católica Argentina
institution_str I-33
repository_str R-139
collection Repositorio Institucional de la Universidad Católica Argentina (UCA)
language Inglés
topic MATEMATICA
FUNCIONES MATEMÁTICAS
ANALISIS MATEMÁTICO
ECUACIONES DIFERENCIALES
spellingShingle MATEMATICA
FUNCIONES MATEMÁTICAS
ANALISIS MATEMÁTICO
ECUACIONES DIFERENCIALES
Ponzellini Marinelli, Luciano
Stabilizing radial basis functions techniques for a local boundary integral method
topic_facet MATEMATICA
FUNCIONES MATEMÁTICAS
ANALISIS MATEMÁTICO
ECUACIONES DIFERENCIALES
description 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.
format Artículo
author Ponzellini Marinelli, Luciano
author_facet Ponzellini Marinelli, Luciano
author_sort Ponzellini Marinelli, Luciano
title Stabilizing radial basis functions techniques for a local boundary integral method
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
publisher Unión Matemática Argentina
publishDate 2023
url https://repositorio.uca.edu.ar/handle/123456789/16414
work_keys_str_mv AT ponzellinimarinelliluciano stabilizingradialbasisfunctionstechniquesforalocalboundaryintegralmethod
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