Document Type

Journal Article

Department/Unit

Department of Mathematics

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 Year

2009

Journal Title

Advances in Computational Mathematics

Volume number

30

Issue number

4

Publisher

Springer

First Page (page number)

339

Last Page (page number)

354

Referreed

1

DOI

10.1007/s10444-008-9071-x

ISSN (print)

10197168

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