书名：散乱数据拟合的模型、方法和理论（第二版）（英文版）
定价：98.0
ISBN：9787030748553
作者：吴宗敏
版次：1
出版时间：2023-03

内容提要：
本书是应用数学与计算数学中有关…面及多元函数插值、逼近、拟合的入门书籍,从多种物理背景、原理出发,导出相应的散乱数据拟合的数学模型及计算方法,进而逐个进行深入的理论分析。书中介绍了多元散乱数据拟合的一般方法,包括多元散乱数据多项式插值、基于三角剖分的插值方法、Boole和与Coons…面、Sibson方法或自然邻近法、Shepard方法、Kriging方法、薄板样条方法、MQ拟插值法、径向基函数方法、运动最小二乘法、隐函数样条方法、R函数法等；同时还特别介绍了近年来国际上越来越热并在无网格微分方程数值解方面有诸多应用的径向基函数方法及其相关理论。



目录：
Contents
Preface to the Second Edition Preface to the First Edition
Chapter 1　Scattered Data Approximation and Multivariate Polynomial Interpolation　1
1.1　Motivation Problems　1
1.1.1　Problems from Applications　2
1.1.2　Problems from Mathematics　3
1.2　Haar Condition for Interpolation　4
1.3　Multivariate Polynomial Interpolation for Scattered Data　6
1.3.1　Aitken Formula for Multivariate Interpolation　8
1.3.2　Newton Formula for Multivariate Polynomial Interpolation　8
Chapter 2　Local Methods　10
2.1　Triangulation and Function Representation on a　Triangle　10
2.2　Smooth Connection Methods Based on Triangulation　17
2.2.1　Linear Interpolation and Piecewise Linear Interpolation　17
2.2.2　Nine-Parameter Cubic Method　18
2.2.3　Clough-Tocher Method　20
2.2.4　Powell-Sabin Method　21
2.3　Boole and Coons Patches　23
2.4　Subdivision Methods for Scattered Data Approximation　26
2.4.1　Chaikin Method　27
2.4.2　Doo-Sabin Method　29
2.4.3　Four-Point Method　30
2.4.4　Butterfly Algorithm　32
2.5　Sibson Interpolation or Natural Proximity　33
2.5.1　Scattered Data Interpolation with Lipschitz Constant Diminishing Property　36
2.5.2　Convergence Theorem of Sibson Interpolation　39
2.5.3　Interpolation Convergence Theorem for Interpolation with Lipschitz Constant Diminishing Property　39
2.6　Shepard Method　40
2.6.1　Shepard Interpolation with Derivative Information　42
2.6.2　Generalization of Shepard Method　43
Chapter 3　Global Methods　44
3.1　Random Function Preliminary　44
3.2　Kriging Method　48
3.2.1　Inverse of Univariate Markov Type Correlation Matrix　51
3.2.2　The Solution to Kriging Problem with Univariate Gaussian Type
Correlation Matrix　52
3.2.3　Monotonicity and Boundedness of Kriging Interpolation Operator　53
3.2.4　Condition Number of Correlation Matrix　53
3.3　Universal Kriging　54
3.4　Co-Kriging　58
3.4.1　Nugget Effect of Interpolation Operator　60
3.4.2　Application of Co-Kriging on Hermite Interpolation　61
3.5　Interpolation for Generalized Linear Functionals　62
3.6　Splines　66
3.7　Multi-Quadric Methods　73
3.8　MQ Quasi-interpolation for Higher Order Derivative Approximation　84
3.9　Stability for Derivative Approximation with FD and MQ　89
3.10　Radial Basis Functions　94
3.10.1　Radial Basis Function Interpolation　95
3.10.2　Existence of Radial Basis Function Interpolation　95
Chapter 4　Theory on Radial Basis Function Interpolation　99
4.1　Convergence and Convergence Rate　99
4.1.1　Quasi-Interpolation, Strang-Fix Condition and Shift Invariant Space　99
4.2　Convergence Results for Scattered Data Radial Basis Function　Interpolation　104
4.2.1　Error Estimation　108
4.2.2　Construction of Admissible Vectors　109
4.3　Positive Definite Radial Basis Functions　112
4.4　Bodmer Theory for Radial Basis Functions　119
4.5　Radial Functions and Strang-Fix Conditions　126
Chapter 5　Other Scattered Data Interpolation Methods　139
5.1　Moving Least Squares　139
5.1.1　Least Squares　139
5.1.2　Moving Least Squares　140
5.1.3　Interpolating Moving Least Squares Methods　141
5.1.4　Divide and Conquer on General Domain　146
5.2　Convergence Analysis of Shepard Methods　147
5.2.1　Convergence Analysis for the Shepard Method　148
5.3　Implicit Splines　154
5.3.1　Other Scattered Data Interpolation Methods　157
5.4　Partition of Unity　158
5.5　R-function　159
Chapter 6　Scatter Data Interpolation for Numerical Solutions of PDEs　161
6.1　Generalized Functional Interpolations and Numerical Methods for PDEs　161
6.2　Other Multivariate Approximation Methods for PDEs　168
6.2.1　Least Squares Methods　169
6.2.2　Collocation　170
6.2.3　Galerkin Method　171
6.2.4　Golberg Method　172
Bibliography　173