On the ill-conditioned nature of C∞ RBF strong collocation

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, εM. The solution to computational singularity is to decrease the effective εM by either hardware or software methods. Computer scientists developed rapidly executable multi-precision packages.