Document Type

Journal Article

Department/Unit

Department of Mathematics

Language

English

Abstract

In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa’s method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa’s method are examined and verified in arbitrary–precision computations. Numerical examples confirm with the theories that the modified Kansa’s method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian RBFs. Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa’s method. In double–precision, even for very large RBF shape parameters, we show that the modified Kansa’s method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process.

Publication Date

2009

Source Publication Title

Advances in Computational Mathematics

Volume

30

Issue

4

Start Page

339

End Page

354

Publisher

Springer

Peer Reviewed

1

DOI

10.1007/s10444-008-9071-x

Link to Publisher's Edition

http://dx.doi.org/10.1007/s10444-008-9071-x

ISSN (print)

10197168

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