TY - JOUR
T1 - Preconditioning for radial basis functions with domain decomposition methods
AU - LING, Leevan
AU - Kansa, E. J.
N1 - Funding Information:
The MQ radial basis function (MQ-RBF) for the interpolation problem has been shown by various authors to possess some very powerful properties. Madych and Nelson \[4,5\] proved that *The research of this author was supported by a Natural Science and Engineering Research Council of Canada postgraduate scholarship.
PY - 2004/12
Y1 - 2004/12
N2 - In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N2) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c∼1/√N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.
AB - In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N2) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c∼1/√N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.
KW - Approximate cardinal basis function
KW - Domain decomposition
KW - Partial differential equation
KW - Preconditioner
KW - Radial basis function
UR - http://www.scopus.com/inward/record.url?scp=18144408406&partnerID=8YFLogxK
U2 - 10.1016/j.mcm.2005.01.002
DO - 10.1016/j.mcm.2005.01.002
M3 - Journal article
AN - SCOPUS:18144408406
SN - 0895-7177
VL - 40
SP - 1413
EP - 1427
JO - Mathematical and Computer Modelling
JF - Mathematical and Computer Modelling
IS - 13
ER -