Stable computations with flat radial basis functions using vector valued rational approximations
One commonly finds in applications of smooth radial basis functions (RBFs) that scaling the kernels so they are 'flat' leads to smaller discretization errors. However, the direct numerical approach for computing with flat RBFs (RBF Direct) is severely ill conditioned. We present an algorithm for bypassing this ill conditioning that is based on a new method for rational approximation (RA) of vector valued analytic functions with the property that all components of the vector share the same singularities. This new algorithm (RBF RA) is more accurate, robust, and easier to implement than the Contour Pade method, which is similarly based on vector valued rational approximation. In contrast to the stable RBF QR and RBF GA algorithms, which are based on finding a better conditioned base in the same RBF space, the new algorithm can be used with any type of smooth radial kernel, and it is also applicable to a wider range of tasks (including calculating Hermite type implicit RBF FD stencils) We present a series of numerical experiments demonstrating the effectiveness of this new method for computing RBF interpolants in the flat regime. We also demonstrate the flexibility of the method by using it to compute implicit RBF FD formulas in the flat regime and then using these for solving Poisson's equation in a 3 D spherical shell.
J. Comput. Phys.