Stable computations with flat radial basis functions using vector-valued rational approximations

Abstract

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-Padé 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 3D spherical shell.

27 pages, lots of figures, Matlab code included

Cited by in corpus (10)

• Hyperviscosity-Based Stabilization for Radial Basis Function-Finite Difference (RBF-FD) Discretizations of Advection-Diffusion Equations
• A stabilized radial basis-finite difference (RBF-FD) method with hybrid kernels
• RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for Solving PDEs on Surfaces
• Monomial augmentation guidelines for RBF-FD from accuracy vs. computational time perspective
• An improved radial basis-pseudospectral method with hybrid Gaussian-cubic kernels
• A new variable shape parameter strategy for RBF approximation using neural networks
• GP-MOOD: A positive-preserving high-order finite volume method for hyperbolic conservation laws
• A stable algorithm for divergence-free radial basis functions in the flat limit
• Towards stability of radial basis function based cubature formulas
• Towards stability results for global radial basis function based quadrature formulas