径向基函数逼近中的若干问题研究
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摘要
本文研究径向基函数逼近中的相关问题。
径向基函数方法是多元逼近中最流行的方法之一。它利用径向对称函数可以容易的将多元数据逼近问题转化为本质上是一元的逼近问题。径向基函数的研究是从径向基函数插值研究开始的。到现在为止,相关的理论研究已经比较完善[12,32,40,41,48,58,66,67],但是关于径向基函数空间基于小波分析的多尺度分解以及以径向基函数为核函数的针对多元散乱数据的拟插值仍然是两个没有很好解决的问题。本文的工作就是围绕这两个问题进行展开的。首先,本文从一般的Hilbert空间中的非稳定的MRA紧框架的充要条件出发,构造了径向基函数空间的非稳定的MRA紧框架。然后,在Wu[85]给出的拟插值的基础上,研究了加入形状参数后的误差分析,并重新构造了新的核函数,从而提高拟插值的逼近阶。本文的最后给出了拟插值的相关数值研究。
下面对全文的结构安排作一下简单介绍。
第一章是绪论。这一章中,我们简要的介绍了相关问题的背景,研究现状以及论文中的主要结果。
第二章是准备工作。我们介绍了论文中所需要的一些相关理论基础。
第三章我们将Chui以及Beutel等提出的MRA紧框架理论推广到一般内积空间上,给出了构造定义在一般区域上的函数Hilbert空间中的非稳定的MRA紧框架的充要条件。
第四章我们首先给出了径向基函数的本性Hilbert空间的相关理论,然后在Hilbert空间中的非稳定的MRA紧框架的充要条件的基础上,给出了径向基函数的本性Hilbert空间中的非稳定的MRA紧框架的构造方法。
第五章首先介绍了基于散乱数据的积分公式,然后在Wu[85]给出的多元空间中的拟插值公式的基础上,引入形状参数并给出了带参数的拟插值的误差估计。最后,我们构造新的径向基核函数,并给出了基于新的核函数的拟插值以及相关的误差估计。理论估计说明新的拟插值公式提高了逼近的阶数。
第六章我们给出了定义在星型区域上的函数的延拓的一般方法,并且对拟插值公式进行了数值研究,数值研究表明拟插值的收敛速度比理论估计的快,而且形状参数的选择有一定的规律。
The present Ph.D.dissertation is concerned with some problems in approximation theory with radial basis function.
Radial basis function methods is one of the most popular methods in approximation theory.The advantage of radial basis function methods is solving the multivariate problems with scattered data easily as much as that in one dimension.It begins with radial basis function interpolation,and till now,it's well-developed except two problems: one is decomposition of the native space of radial basis function based on multi-resolution analysis;the other is the quasi-interpolation for scattered data with radial kernels.The present Ph.D.dissertation is studying about these problems.First,we generalized the sufficient and necessary conditions for nonstationary MRA tight frame to the general Hilbert functions space,and constructed nonstationary MRA tight frame of native space of radial basis functions.Second,we study the influence of shape parameter in quasi-interpolation which is proposed in[85],and then constructed a new quasi-interpolation with a new radial kernel.The convergence rate of new quasi-interpolation is fast than that in[85].And last, we show a series of numerical examples with different kinds of kernels and parameters. Several aspects have been studied,including the best choice of shape parameter and the numerical convergence rate.
The whole dissretation is organized as follows.
Chapter One is preface.This chapter is devoted to the background of these problems, and the main results of this Ph.D.dissertation are also illustrated.
Chapter Two is preliminaries.We introduce some related knowledge which are needed in the dissertation.
Chapter Three generalized the theory for nonstationary MRA tight frame to general Hilbert functions space,and got the sufficient and necessary conditions for nonstationary MRA tight frame of it.
Chapter Four first introduced the basic theory of native space,and then constructed nonstationary MRA tight frame of native space of radial basis functions based on the results of Chapter Three,in other words,we gave the decomposition of the native space based on nonstationary MRA tight frame.
Chapter Five first introduced the integral formula based on scattered data and the quasi-interpolation for multivariate scattered data which proposed in[85].and then we study the influence of shape parameter and get the error estimation with it.At the end of this chapter,we constructed a new radial kernel and then got a new quasi-interpolation for scattered data,which was shown to provide higher approximation order than that in [85].
Chapter Six first gave a scheme to extend the function defined the so-called starlike compact domain to whole space with Hermitian extrapolation.Then we show a series of numerical examples with different kinds of kernels and parameters,and we study the best choice of shape parameter and the numerical convergence rate.
径向基函数方法是多元逼近中最流行的方法之一。它利用径向对称函数可以容易的将多元数据逼近问题转化为本质上是一元的逼近问题。径向基函数的研究是从径向基函数插值研究开始的。到现在为止,相关的理论研究已经比较完善[12,32,40,41,48,58,66,67],但是关于径向基函数空间基于小波分析的多尺度分解以及以径向基函数为核函数的针对多元散乱数据的拟插值仍然是两个没有很好解决的问题。本文的工作就是围绕这两个问题进行展开的。首先,本文从一般的Hilbert空间中的非稳定的MRA紧框架的充要条件出发,构造了径向基函数空间的非稳定的MRA紧框架。然后,在Wu[85]给出的拟插值的基础上,研究了加入形状参数后的误差分析,并重新构造了新的核函数,从而提高拟插值的逼近阶。本文的最后给出了拟插值的相关数值研究。
下面对全文的结构安排作一下简单介绍。
第一章是绪论。这一章中,我们简要的介绍了相关问题的背景,研究现状以及论文中的主要结果。
第二章是准备工作。我们介绍了论文中所需要的一些相关理论基础。
第三章我们将Chui以及Beutel等提出的MRA紧框架理论推广到一般内积空间上,给出了构造定义在一般区域上的函数Hilbert空间中的非稳定的MRA紧框架的充要条件。
第四章我们首先给出了径向基函数的本性Hilbert空间的相关理论,然后在Hilbert空间中的非稳定的MRA紧框架的充要条件的基础上,给出了径向基函数的本性Hilbert空间中的非稳定的MRA紧框架的构造方法。
第五章首先介绍了基于散乱数据的积分公式,然后在Wu[85]给出的多元空间中的拟插值公式的基础上,引入形状参数并给出了带参数的拟插值的误差估计。最后,我们构造新的径向基核函数,并给出了基于新的核函数的拟插值以及相关的误差估计。理论估计说明新的拟插值公式提高了逼近的阶数。
第六章我们给出了定义在星型区域上的函数的延拓的一般方法,并且对拟插值公式进行了数值研究,数值研究表明拟插值的收敛速度比理论估计的快,而且形状参数的选择有一定的规律。
The present Ph.D.dissertation is concerned with some problems in approximation theory with radial basis function.
Radial basis function methods is one of the most popular methods in approximation theory.The advantage of radial basis function methods is solving the multivariate problems with scattered data easily as much as that in one dimension.It begins with radial basis function interpolation,and till now,it's well-developed except two problems: one is decomposition of the native space of radial basis function based on multi-resolution analysis;the other is the quasi-interpolation for scattered data with radial kernels.The present Ph.D.dissertation is studying about these problems.First,we generalized the sufficient and necessary conditions for nonstationary MRA tight frame to the general Hilbert functions space,and constructed nonstationary MRA tight frame of native space of radial basis functions.Second,we study the influence of shape parameter in quasi-interpolation which is proposed in[85],and then constructed a new quasi-interpolation with a new radial kernel.The convergence rate of new quasi-interpolation is fast than that in[85].And last, we show a series of numerical examples with different kinds of kernels and parameters. Several aspects have been studied,including the best choice of shape parameter and the numerical convergence rate.
The whole dissretation is organized as follows.
Chapter One is preface.This chapter is devoted to the background of these problems, and the main results of this Ph.D.dissertation are also illustrated.
Chapter Two is preliminaries.We introduce some related knowledge which are needed in the dissertation.
Chapter Three generalized the theory for nonstationary MRA tight frame to general Hilbert functions space,and got the sufficient and necessary conditions for nonstationary MRA tight frame of it.
Chapter Four first introduced the basic theory of native space,and then constructed nonstationary MRA tight frame of native space of radial basis functions based on the results of Chapter Three,in other words,we gave the decomposition of the native space based on nonstationary MRA tight frame.
Chapter Five first introduced the integral formula based on scattered data and the quasi-interpolation for multivariate scattered data which proposed in[85].and then we study the influence of shape parameter and get the error estimation with it.At the end of this chapter,we constructed a new radial kernel and then got a new quasi-interpolation for scattered data,which was shown to provide higher approximation order than that in [85].
Chapter Six first gave a scheme to extend the function defined the so-called starlike compact domain to whole space with Hermitian extrapolation.Then we show a series of numerical examples with different kinds of kernels and parameters,and we study the best choice of shape parameter and the numerical convergence rate.
引文
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