Preconditioning for radial basis functions with domain decomposition methods

Authors:
Leevan Ling;E. J. Kansa
Affiliations:
-;-
Venue:
Mathematical and Computer Modelling: An International Journal
Year:
2004

Citing 2
Cited 12

Fast Solution of the Radial Basis Function Interpolation Equations: Domain Decomposition Methods

SIAM Journal on Scientific Computing
Overlapping domain decomposition method by radial basis functions

Applied Numerical Mathematics

Boundary knot method based on geodesic distance for anisotropic problems

Journal of Computational Physics
A note on the meshless method using radial basis functions

Computers & Mathematics with Applications
Radial function collocation solution of partial differential equations in irregular domains

International Journal of Computing Science and Mathematics
Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform

Journal of Computational and Applied Mathematics
The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems

Journal of Computational Physics
The role of the multiquadric shape parameters in solving elliptic partial differential equations

Computers & Mathematics with Applications
The uselessness of the Fast Gauss Transform for summing Gaussian radial basis function series

Journal of Computational Physics
Application of the RBF meshless method to the solution of the radiative transport equation

Journal of Computational Physics
An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions

Journal of Complexity
Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite interval

Computers & Mathematics with Applications
Stable Computations with Gaussian Radial Basis Functions

SIAM Journal on Scientific Computing
A robust method of thin plate spline and its application to DEM construction

Computers & Geosciences

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Abstract

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(N^2) 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.