On convergent numerical algorithms for unsymmetric collocation

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. In double–precision, even for very large RBF shape parameters, we show that the modified Kansa’s method can be solved stably by subspace selection using the greedy algorithm. A benchmark algorithm are used to verify the optimality of the selection process. Preprint submitted to Advances in Computational Mathematics. c © All rights reserved to the authors. Generated by LTEX on October 15, 2007. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong. ([email protected]) Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Lotzestraße 16-18, D-37083 Göttingen, Germany. Running title. Convergent Unsymmetric Collocation