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散乱数据拟合的模型、方法和理论(第二版)(英文版) 版权信息
- ISBN：9787030748553
- 条形码：9787030748553 ; 978-7-03-074855-3
- 装帧：一般胶版纸
- 册数：暂无
- 重量：暂无
- 所属分类：>
散乱数据拟合的模型、方法和理论(第二版)(英文版) 内容简介
本书是应用数学与计算数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*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.本书补充了作者近年来的新成果，包括MQ-拟插值对高阶导数的逼近和利用差商及MQ拟插值对高阶导数逼近的稳定性分析。
散乱数据拟合的模型、方法和理论(第二版)(英文版) 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
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
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散乱数据拟合的模型、方法和理论(第二版)(英文版) 节选
Chapter 1 Scattered Data Approximation and Multivariate Polynomial Interpolation 1.1 Motivation Problems Functions studied in mathematics have been widely used to model various practical problems. For example， one can utilize a linear function (a straight line) to describe the trajectory of light. The physics law says that light travels along a straight line in vacuum without the impact of a magnetic field. However， light is not often in vacuum. Besides light in fact is impacted by the magnetic field of the earth and the sun. Therefore， the description that light travels along a straight line is only an approximation. Such an approximation is well accepted because the trajectory of light in our daily life is so close to a straight line that the difference between the true trajectory and the straight line is negligible. Strictly speaking， two reasons make such an approximation acceptable. Firstly， the accurate description of the trajectory of light is almost impossible. Secondly， the straight line approximation to the trajectory of light， though simple， is sufficiently accurate. It is unnecessary to use nonlinear and complex functions to describe the trajectory of light. Such an example suggests two principles for general approximation problems. Firstly， it is better to choose a simple and useful function to approximate our target problems. Secondly， such an approximation should be “good” enough. Suppose we know the underlying problems can be described by certain functions， and sometimes with many measurements observations available， a natural question is how to model this problem approximately with functions. Precisely， how to find an approximation function 5(x) to approximate the underlying unknown function Suppose that the observations are function values ， if the approximation function s(x) is required to pass through the observed values at ， say ，then we call s(x) is an interpolation to f(x). If the difference between s(x) and f(x) is minimized under certain metric or measurement ， then s(x) is called an approximation to ， which is also often referred to as fitting in engineering. It is obvious that interpolation is a special case of approximation. Indeed， interpolation is the mathematical foundation of approximation. Sometimes， further constraints may be enforced for certain practical applications. For example， convexity and shape preserving is always a desired property in computer-aided design ， that is， the approximation function s(x) should have the similar convexity and shape with f(x). Such problems can be viewed as general fitting or approximation problems. The focus of this book is multivariate scattered data interpolation and approximation. By default， we believe the readers have already equipped with basic knowledge of one dimensional approximation. Otherwise， one can refer to textbooks on approximation theory and computer-aided design. For scattered data interpolation and approximation， the observations or measurements are generally irregular， not necessarily located on a mesh grid. For interpolation and approximation on mesh grids， one can use the tensor-product approaches. Besides， we also discuss the case when observations of measurements are linear functionals. In this book， we usually start with a physical model which requires scattered data approximation， hopefully， such models can stimulate readers’ interest and make the underlying motivation more accessible; and then we analyze the underlying problems mathematically in rigor; finally we discuss how to implement corresponding methods on computers. All introduced methods are programmable. Some of them are used in current popular software. Next， we shall introduce several examples. Through these examples， we shall illustrate where scattered data approximation problems come from and how to translate them to mathematical problems. 1.1.1 Problems from Applications In reservoir simulation， one would like to tell the oil distribution through the study of the permeability formation of oil. The distribution of permeability can be viewed as a function in M3. Certain samples can be obtained from dr ill-wells. The locations of drill-wells are usually scattered. Such a problem of reconstructing the permeability function is a scattered data interpolation problem. In weather forecast， a contour map of the temperature field can be obtained by interpolation of some data from meteorological observatories. Such observatories are scattered on the earth. This problem can also be viewed as a scattered data interpolation problem. If we further require the trends of temperature change， we may need further information like the velocity field of wind. This is a much more complicated scattered data approximation problem. In the problem of copying， imitation and archaeological restoration of paleontology， one often needs to reconstruct a new model through some measured values of the underlying object. Due to the complexity
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