Finite element methods for Maxwell's equations.

Scattered Data Approximation

Many radial basis function (RBF) methods contain a free shape parameter that plays an important role for the accuracy of the method. In most papers the authors end up choosing this shape parameter by trial and error or some other ad hoc means. The method of cross validation has long been used in the statistics literature, and the special case of leave-one-out cross validation forms the basis of the algorithm for choosing an optimal value of the shape parameter proposed by Rippa in the setting of scattered data interpolation with RBFs. We discuss extensions of this approach that can be applied in the setting of iterated approximate moving least squares approximation of function value data and for RBF pseudo-spectral methods for the solution of partial differential equations. The former method can be viewed as an efficient alternative to ridge regression or smoothing spline approximation, while the latter forms an extension of the classical polynomial pseudo-spectral approach. Numerical experiments illustrating the use of our algorithms are included.

/pdf/on-choosing-optimal-shape-parameters-for-rbf-approximation-4673pg6gmn.pdf

On choosing “optimal” shape parameters for RBF approximation

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

/pdf/the-exponentially-convergent-trapezoidal-rule-3kr0k99w3a.pdf

The Exponentially Convergent Trapezoidal Rule

Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations

-Spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries. Since radial basis function (RBF) approxima-ions permit this even for multivariate scattered data, there has been much recent interest in practical algorithms to compute these approximations effectively. Several types of RBFs feature a free parameter (e.g., c in the multiquadric (MQ) case @f(r) = @/r^2 + c^2). The limit of c -> ~ (increasingly flat basis functions) has not received much attention because it leads to a severely ill-conditioned problem. We present here an algorithm which avoids this difficulty, and which allows numerically stable computations of MQ RBF interpolants for all parameter values. We then find that the accuracy of the resulting approximations, in some cases, becomes orders of magnitude higher than was the case within the previously available parameter range. Our new method provides the first tool for the numerical exploration of MQ RBF interpolants in the limit of c -> ~. The method is in no way specific to MQ basis functions and can-without any change-be applied to many other cases as well.

Stable computation of multiquadric interpolants for all values of the shape parameter

Scattered node compact finite difference-type formulas generated from radial basis functions

RBF approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of PDEs. RBFs allow arbitrarily scattered data, generalize easily to several space dimensions, and can be spectrally accurate. However, accuracy degradations near boundaries in many cases severely limit the utility of this approach. With that as motivation, this study aims at gaining a better understanding of the properties of RBF approximations near the ends of an interval in 1-D and towards edges in 2-D.

Observations on the behavior of radial basis function approximations near boundaries

https://opensky.ucar.edu:/islandora/object/articles%3A18179/datastream/PDF/view

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

The Radial Basis Function (RBF) method is one of the primary tools for interpolating multidimensional scattered data. The methods' ability to handle arbitrarily scattered data, to easily generalize to several space dimensions, and to provide spectral accuracy have made it particularly popular in several different types of applications. Some of the more recent of these applications include cartography, neural networks, medical imaging, and the numerical solution of partial differential equations (PDEs). In this thesis we study three issues with the RBF method that have received very little attention in the literature. 
First, we focus on the behavior of RBF interpolants near boundaries. Like most interpolation methods, a common feature of the RBF method is how relatively inaccurate the interpolants are near boundaries. Such boundary induced errors can severely limit the utility of the RBF method for numerically solving certain PDEs. With that as motivation, we investigate the behavior of RBF interpolants near boundaries and propose the first practical techniques for ameliorating the errors there. 
We next focus on some numerical developments for the RBF method based on infinitely smooth RBFs. Most infinitely smooth RBFs feature a free “shape” parameter e such that, as the magnitude of e decreases, the RBFs become increasingly flat. While small values of e typically result in more accurate interpolants, the direct method of computing the interpolants suffers from severe numerical ill-conditioning as e → 0. Until recently, this ill-conditioning has severely limited the range of e that could be considered in the RBF method. We present a novel numerical approach that largely overcomes the numerical ill-conditioning and allows for the stable computation of RBF interpolants for all values of e, including the limiting e = 0 case. This new method provides the first tool for the numerical exploration of RBF interpolants as e → 0. 
The third focus of the thesis is on the behavior of RBF interpolants as e → 0. In most cases the interpolants converge to a finite degree multivariate polynomial interpolant as e → 0. However, in rare situations the interpolants may diverge. We investigate this phenomenon in great detail both numerically and analytically, and link it directly to the failure of a condition known as “polynomial unisolvency”. We also find that the Gaussian RBF is inherently different from the other standard infinitely smooth RBFs in that it appears to result in an interpolant that never diverges as e → 0. 
We conclude with a brief overview of two future research opportunities related to the topics of the thesis. The first involves using RBF interpolants to generate scattered-node finite difference formulas. The second involves using RBF interpolants to generate linear multistep methods for solving ordinary differential equations.

/pdf/radial-basis-function-interpolation-numerical-and-analytical-1p9fmksty8.pdf

Grady B. Wright

Papers

Stable computation of multiquadric interpolants for all values of the shape parameter

Scattered node compact finite difference-type formulas generated from radial basis functions

Observations on the behavior of radial basis function approximations near boundaries

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

Radial basis function interpolation: numerical and analytical developments