Gregory E. Fasshauer

On choosing “optimal” shape parameters for RBF 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.

Solving differential equations with radial basis functions: multilevel methods and smoothing

Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper.

Stable Evaluation of Gaussian Radial Basis Function Interpolants

We provide a new way to compute and evaluate Gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter, i.e., for “flat” kernels. This work is motivated by the fundamental ideas proposed earlier by Bengt Fornberg and his coworkers. However, following Mercers theorem, an

Newton iteration with multiquadrics for the solution of nonlinear PDEs

L_2(\mathbb{R}^d, \rho)

Using meshfree approximation for multi‐asset American options

-orthonormal expansion of the Gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by Fornberg, Larsson, and Flyer and that is applicable in arbitrary space dimensions

Minimal energy surfaces using parametric splines

d

Multistep approximation algorithms: Improved convergence rates through postconditioning with smoothing kernels

. In addition to obtaining an accurate approximation of the radial basis function interpolant (using many terms in the series expansion of the kernel), we also propose and investigate a highly accurate least-squares approximation based on early truncation of the kernel expansion.

On Dimension-independent Rates of Convergence for Function Approximation with Gaussian Kernels

Abstract Newton iteration is a standard tool for the numerical solution of nonlinear partial differential equations. We show how globally supported multiquadric radial basis functions can be used for this task. One of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquadric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm. Some experiments with Nash iteration are also included.

Approximation of stochastic partial differential equations by a kernel-based collocation method

Abstract We study the applicability of meshfree approximation schemes for the solution of multi‐asset American option problems. In particular, we consider a penalty method which allows us to remove the free and moving boundary by adding a small and continuous penalty term to the Black‐Scholes equation. Time discretization is achieved by a linearly implicit θ method. A comparison with results obtained recently by two of the authors using a linearly implicit finite difference method is included.

An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels

Abstract We explore the construction of parametric surfaces which interpolate prescribed 3D scattered data using spaces of parametric splines defined on a 2D triangulation. The method is based on minimizing certain natural energy expressions. Several examples involving filling holes and crowning surfaces are presented, and the role of the triangulation as a parameter is explored. The problem of creating closed surfaces is also addressed. This requires introducing spaces of splines on certain generalized triangulations.