On convergent numerical algorithms for unsymmetric collocation
- PDF / 439,920 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 74 Downloads / 222 Views
On convergent numerical algorithms for unsymmetric collocation Cheng-Feng Lee · Leevan Ling · Robert Schaback
Received: 5 March 2007 / Accepted: 15 January 2008 / Published online: 7 May 2008 © Springer Science + Business Media, LLC 2008
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. Keywords Radial basis function · Kansa’s method · Convergence · Error bounds · Linear optimization · Effective condition number · High precision computation Mathematics Subject Classifications (2000) 65N12 · 65N15 · 65N35
Communicated by Juan Manuel Peña. Preprint submitted to Advances in Computational Mathematics. © All rights reserved to the authors. Generated by LATEX on April 3, 2008. C.-F. Lee Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan L. Ling (B) Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong e-mail: [email protected] R. Schaback Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Lotzestraße 16-18, 37083 Göttingen, Germany
340
C.-F. Lee et al.
1 Introduction Many successful applications of recently developed mesh-free methods can be found in different Mathematics, Physics and Engineering journals; for example, see [1–10]. In this paper, we are interested in the radial basis function (RBF) method for solving partial differential equations (PDE) in strong form. We consider PDE in the general form of Lu = f, L : U → F ,
(1)
where L is a linear operator between two normed linear spaces U and F . The PDE (1) can be solved by a sufficiently smooth solution u∗ ∈ U that generates the data f := Lu∗ ∈ F . Equivalently, we write (1) as an uncountably infinite set of simultaneous scalar equations λ[u] = fλ ∈ R, for all λ ∈ .
(2)
In general, a single set of functionals contains several types of differential or boundary operators. Discretization consists of replacing the infinite set by some finite unstructured subsets m := {λ1 , . . . , λm }. The space spanned by these functionals can be called the test space, and is the infinite test set. In order to obtain a numerical approximation u, we assume that the
Data Loading...
