How fast do radial basis function interpolants of analytic functions converge?

The question in the title is answered using tools of potential theory. Convergence and divergence rates of interpolants of analytic functions on the unit interval are analyzed. The starting point is a complex variable contour integral formula for the remainder in RBF interpolation. We study a generalized Runge phenomenon and explore how the location of centers and affects convergence. Special attention is given to Gaussian and inverse quadratic radial functions, but some of the results can be extended to other smooth basis functions. Among other things, we prove that, under mild conditions, inverse quadratic RBF interpolants of functions that are analytic inside the strip $|Im(z)| < (1/2\epsilon)$, where $\epsilon$ is the shape parameter, converge exponentially

A well-balanced meshless tsunami propagation and inundation model

We present a novel meshless tsunami propagation and inundation model. We discretize the nonlinear shallow-water equations using a well-balanced scheme relying on radial basis function based finite differences. The inundation model relies on radial basis function generated extrapolation from the wet points closest to the wet-dry interface into the dry region. Numerical results against standard one- and two-dimensional benchmarks are presented.Comment: 20 pages, 13 figure

Error saturation in Gaussian radial basis functions on a finite interval

AbstractRadial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ϕ(x;α,h)=exp(−α2(x/h)2): in the limit h→0 for fixed α, the error does not converge to zero, but rather to ES(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases ES(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α2)‖f‖∞ — huge compared to the O(2π/α2)exp(−π2/[4α2]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α≪1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π2/[4α2])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then αoptimum(δ)=1/−2log(δ/0.06)

On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

We investigate the influence of the shape parameter in the meshless Gaussian RBF finite difference method with irregular centres on the quality of the approximation of the Dirichlet problem for the Poisson equation with smooth solution. Numerical experiments show that the optimal shape parameter strongly depends on the problem, but insignificantly on the density of the centres. Therefore, we suggest a multilevel algorithm that effectively finds near-optimal shape parameter, which helps to significantly reduce the error. Comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the Gaussian RBF is provided

ANALYTICAL SOLUTION OF GLOBAL 2D DESCRIPTION OF SHIP GEOMETRY WITH DISCONTINUITIES USING COMPOSITION OF POLYNOMIAL RADIAL BASIS FUNCTIONS

One of the well-known problems in the curves and surfaces reconstruction theory regarding global analytic object description, besides the description of its curvature changes, inflexions and non-bijective parts, is the existence of oscillations near point discontinuities in the middle of the range and at the boundaries of the description. In the ship geometric modelling, ship hull form is usually described globally using parametric methods based on B-spline and NURB-spline, for they have general property of describing discontinuities. Nevertheless, they are not enabling direct, exact calculation of ship\u27s geometric properties, i.e. the calculation of the integrals for determining geometric and other geometry properties or the intersection with water surface. Because of above mentioned, the predominant way of computing geometric properties of the ships is still numerical computation using Simpson integration methods, which also dictates mesh based description of an observed geometry. Analytical solution of global 2D description for ship geometry with discontinuities will be shown in this paper, using the composition of polynomial RBFs, thus solving computational geometry problems, too

Variable Shape Parameter Strategies in Radial Basis Function Methods

The Radial Basis Function (RBF) method is an important tool in the interpolation of multidimensional scattered data. The method has several important properties. One is the ability to handle sparse and scattered data points. Another property is its ability to interpolate in more than one dimension. Furthermore, the Radial Basis Function method provides phenomenal accuracy which has made it very popular in many fields. Some examples of applications using the RBF method are numerical solutions to partial differential equations, image processing, and cartography. This thesis involves researching Radial Basis Functions using different shape parameter strategies. First, we introduce the Radial Basis Function method by stating its history and development in Chapter 1. Second, we explain how Radial Basis Functions work in Chapter 2. Chapter 3 compares RBF interpolation to polynomial interpolation. Chapters 4 and 5 introduce the idea of variable shape parameters. In these chapters we compare and analyze the variable shape parameters in one and two dimensions. In Chapter 6, we introduce the challenges in interpolations due to errors in boundary regions. Here, we try to reduce the error using different shape parameter strategies. Chapter 7 lists the conclusions resulting from the research