TY - JOUR
T1 - On approximate cardinal preconditioning methods for solving PDEs with radial basis functions
AU - Brown, Damian
AU - LING, Leevan
AU - Kansa, Edward
AU - Levesley, Jermy
N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.
PY - 2005/4
Y1 - 2005/4
N2 - The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. This preconditioning technique is sufficiently general that it can be easily applied to many differential operators. In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson's, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance.
AB - The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. This preconditioning technique is sufficiently general that it can be easily applied to many differential operators. In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson's, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance.
KW - Cardinal basis function
KW - Partial differential equation
KW - Preconditioner
KW - Radial basis function
UR - http://www.scopus.com/inward/record.url?scp=18144408028&partnerID=8YFLogxK
U2 - 10.1016/j.enganabound.2004.05.006
DO - 10.1016/j.enganabound.2004.05.006
M3 - Journal article
AN - SCOPUS:18144408028
SN - 0955-7997
VL - 29
SP - 343
EP - 353
JO - Engineering Analysis with Boundary Elements
JF - Engineering Analysis with Boundary Elements
IS - 4
ER -