Contents
Preface to the Second Edition Preface to the First Edition
Chapter 1 Scattered Data Approximation an Mtvariate 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.. Clough-Tocher Method 20
2.2.4 Powell-Sabin Method 21
. Boole and Coons Patches
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 Prory&bsp; 36
2.5.2 Convergence Theorem of Sibson Interpolation 39
2.5.3 Interpolation Convergence Theorem for Interpolation with Lipschitz Constant Diminishing Prory&bsp; 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.. MonotoniciyndBundedness 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-dric Methods 73
3.8 M uasi-interpolation for Higher Order Derivative Approximation 84
3.9 Stability for Derivative Approximation with FD and M 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 si-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.. Galerkin Method 171
6.2.4 Golberg Method 172
Bibliography 173
本书是应用数学与算学中有关曲面及多元函数插值、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曲面、Sibson方法或自然邻近法、Shepard方法、Kriging方法、薄板样条方法、径向基函数方法、运动二乘法、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.本书补充了作者近年来的新成果，包括M-拟插值对高阶导数的逼近和利用差商及M拟插值对高阶导数逼近的稳定分析。
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