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Lin-Tian Luh

Publications and source records attributed to Lin-Tian Luh.

18 recordsLinked to original sources
arXiv2019

Solving Poisson equations by the MN-curve approach

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.

math.NA↗
arXiv2010

The Shape Parameter in the Shifted Surface Spline

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.

math.NA↗
arXiv2010

The Mystery of the Shape Parameter II

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.

math.NA↗
arXiv2010

The Mystery of the Shape Parameter

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

math.NA↗
arXiv2006

An Extension of the Exponential-type Error Bounds for Multiquadric and Gaussian Interpolations

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.

math.NA↗
arXiv2006

On Wu and Schaback's Error Bound

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.

math.NA↗
arXiv2006

On the High-Level Error Bound for Gaussian Interpolation

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.

math.NA↗
arXiv2006

A New Radial Function

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.

math.NA↗
arXiv2006

A Smooth and Compactly Supported Radial Function

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.

math.NA↗
arXiv2006

The high-level error bound for shifted surface spline interpolation

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 $O(\omega^{1/d})$ as $d\to 0$, where $0<\omega <1$ is a constant which can be accurately calculated.

math.NA↗