On convergent numerical algorithms for unsymmetric collocation

Cheng Feng Lee, Leevan LING*, Robert Schaback

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

32 Citations (Scopus)

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 radial basis functions (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.

Original languageEnglish
Pages (from-to)339-354
Number of pages16
JournalAdvances in Computational Mathematics
Volume30
Issue number4
DOIs
Publication statusPublished - May 2009

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Convergence
  • Effective condition number
  • Error bounds
  • High precision computation
  • Kansa's method
  • Linear optimization
  • Radial basis function

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