Lin-Tian Luh

Lin-Tian Luh

In this paper we explore the influence of the shape parameter in the gaussian function on error estimates and present a set of criteria for its optimal choice.

Lin-Tian Luh

In this paper we present a set of criteria for the choice of the shape parameter c contained in multiquadrics.

Lin-Tian Luh

In this paper we present a set of criteria for the choice of the shape parameter c contained in multiquadrics.

Lin-Tian Luh

There is a constant c contained in the famous radial basis function shifted surface spline. It's called shape parameter. RBF people only know that this constant is very influential, while its optimal choice is unknown. This paper presents criteria of its optimal choice.

Lin-Tian Luh

A New Error Bound for shifted surface spline interpolation is presented. This error bound probably is the most powerful one up to now.

Lin-Tian Luh

A new error bound which is better than the current exponential-type error bound is presented in this paper.

Lin-Tian Luh

In this paper we apply the newly born choice theory of the shape parameters contained in the smooth radial basis functions to solve Poisson equations. Some people complain that Luh's choice theory, based on harmonic analysis, is mathematically complicated and applies only to function interpolations. Here we aim at presenting an easily accessible approach to solving differential equations with the choice theory which proves to be successful, not only by its easy accessibility, but also by its striking accuracy and efficiency.

Lin-Tian Luh

This article explores the influence of evenly spaced data points on radial-basis-function interpolation.

Lin-Tian Luh

An error bound for Gaussian Interpolation which is better than the current exponential-type error bound is presented.

Lin-Tian Luh

In this paper we present criteria for the choice of the shape parameter c contained in the famous radial function multiquadric. It may be of interest to RBF people and all people using radial basis functions to do approximation.

Lin-Tian Luh

In this paper we present criteria for the optimal choice of the shape parameter c contained in the famous radial function multiquadrics.

Lin-Tian Luh

It's well known that in the high-level error bound for multiquadric interpolation there is a crucial constant lambda lying between 0 and 1 which connot be calculated or even approximated. The purpose of this paper is to answer this question.

Lin-Tian Luh

In the 1990's exponential-type error bounds appeared in the theory of radial basis functions. This kind of error bounds is very powerful. However it only measures the difference between the approximant and approximand. Mathematicians and engineers often need to know the matching of the derivatives when dealing with partial differential equations. In this paper we extend this kind of error bounds to a form which measures the difference between the derivatives of the approximant and approximand.

Lin-Tian Luh

In this paper we point out that the commonly used error bound in the theory of radial basis functions contains an important error. It doesn't apply for derivatives. Thus one should be very careful when dealing with differential equations.

Lin-Tian Luh

It's well-known that there is a very powerful error bound for Gaussians put forward by Madych and Nelson in 1992. It's of the form$% | f(x)-s(x)| \leq (Cd)^{\frac{c}{d}}\left\Vert f\right\Vert_{h}$ where $C,c$ are constants, $h$ is the Gaussian function, $% s$ is the interpolating function, and d is called fill distance which, roughly speaking, measures the spacing of the points at which interpolation occurs. This error bound gets small very fast as $d\to 0$. The constants $C$ and $c$ are very sensitive. A slight change of them will result in a huge change of the error bound. The number $c$ can be calculated as shown in [9]. However, $C$ cannot be calculated, or even approximated. This is a famous question in the theory of radial basis functions. The purpose of this paper is to answer this question.

Lin-Tian Luh

In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions is extremely important. One of the reasons is that its error bound will converge very fast. However there is hitherto no such function which can be expressed in a simple form. This is a famous question. The purpose of this paper is to answer this question.

Lin-Tian Luh

In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions are extremely important for some reasons. First, its computational properties will be very good since it's compactly supported. Second, its error bound will converge very fast since it's infinitely differentiable. However there is hitherto no such functions which can be expressed in a simple form. This is a famous question. The purpose of this paper is to answer this question.

Lin-Tian Luh

Radial function interpolation of scattered data is a frequently used method for multivariate data fitting. One of the most frequently used radial functions is called shifted surface spline, introduced by Dyn, Levin and Rippa in \cite{Dy1} for $R^{2}$. Then it's extended to $R^{n}$ for $n\geq 1$. Many articles have studied its properties, as can be seen in \cite{Bu,Du,Dy2,Po,Ri,Yo1,Yo2,Yo3,Yo4}. When dealing with this function, the most commonly used error bounds are the one raised by Wu and Schaback in \cite{WS}, and the one raised by Madych and Nelson in \cite{MN2}. Both are $O(d^{l})$ as $d\to 0$, where $l$ is a positive integer and $d$ is the fill-distance. In this paper we present an improved error bound which is $O(ω^{1/d})$ as $d\to 0$, where $0<ω<1$ is a constant which can be accurately calculated.