On the ill-conditioned nature of C∞ RBF strong collocation. (May 2017)

• Record Type: Journal Article
• Authors: Kansa, E.J.; Holoborodko, P.
• Is Part Of: Engineering analysis with boundary elements. Volume 78(2017)
• Page Start: 26
• Page End: 30
• Publication Date: 2017-05
• Subjects: Meshless radial basis functions -- Multiquadrics -- Gaussians -- Partial differential equations -- Integralequations -- Multi-precision tools; Boundary element methods -- Periodicals; Engineering mathematics -- Periodicals; Équations intégrales de frontière, Méthodes des -- Périodiques; Mathématiques de l'ingénieur -- Périodiques; Boundary element methods; Engineering mathematics; Periodicals; 620.00151
• DOI: 10.1016/j.enganabound.2017.02.006
• Languages: English
• ISSNs: 0955-7997

Abstract

Abstract: Continuously differentiable radial basis functions ( C ∞ -RBFs) are the best method to solve numerically higher dimensional partial differential equations (PDEs). Among the reasons are: 1. An n-dimensional problem becomes a one-dimensional radial distance problem, 2. The convergence rate increases with the dimensionality, 3. Such RBFs possess spectral convergence.Finitely supported polynomial methods only converge at polynomial rates. C ∞ -RBFs have global support; the systems of equations may become computationally singular if the condition number exceeds the inverse machine epsilon, ε M . The solution to computational singularity is to decrease the effective ε M by either hardware or software methods. Computer scientists developed rapidly executable multi-precision packages.