Preface to the Second Edition Preface to the First Edition Chapter 1 Scattered Data Approximation and Multivariate Polynomial Interpolation 1.1 Motivation Problems 1.1.1 Problems from Applications 1.1.2 Problems from Mathematics 1.2 Haar Condition for Interpolation 1.3 Multivariate Polynomial Interpolation for Scattered Data 1.3.1 Aitken Formula for Multivariate Interpolation 1.3.2 Newton Formula for Multivariate Polynomial Interpolation Chapter 2 Local Methods 2.1 Triangulation and Function Representation on a Triangle 2.2 Smooth Connection Methods Based on Triangulation 2.2.1 Linear Interpolation and Piecewise Linear Interpolation 2.2.2 Nine-Parameter Cubic Method 2.2.3 Clough-Tocher Method 2.2.4 Powell-Sabin Method 2.3 Boole and Coons Patches 2.4 Subdivision Methods for Scattered Data Approximation 2.4.1 Chaikin Method 2.4.2 Doo-Sabin Method 2.4.3 Four-Point Method 2.4.4 Butterfly Algorithm 2.5 Sibson Interpolation or Natural Proximity 2.5.1 Scattered Data Interpolation with Lipschitz Constant Diminishing Property 2.5.2 Convergence Theorem of Sibson Interpolation 2.5.3 Interpolation Convergence Theorem for Interpolation with Lipschitz Constant Diminishing Property 2.6 Shepard Method 2.6.1 Shepard Interpolation with Derivative Information 2.6.2 Generalization of Shepard Method Chapter 3 Global Methods 3.1 Random Function Preliminary 3.2 Kriging Method 3.2.1 Inverse of Univariate Markov Type Correlation Matrix 3.2.2 The Solution to Kriging Problem with Univariate Gaussian Iype Correlation Matrix 3.2.3 Monotonicity and Boundedness of Kriging Interpolation Operator 3.2.4 Condition Number of Correlation Matrix 3.3 Universal Kriging 3.4 Co-Kriging 3.4.1 Nugget Effect of Interpolation Operator 3.4.2 Application of Co-Kriging on Hermite Interpolation 3.5 Interpolation for Generalized Linear Functionals 3.6 Splines 3.7 Multi-Quadric Methods 3.8 MQ Quasi-interpolation for Higher Order Derivative Approximation 3.9 Stability for Derivative Approximation with FD and M 3.10 Radial Basis Functions 3.10.1 Radial Basis Function Interpolation 3.10.2 Existence of Radial Basis Function Interpolation
Chapter 4 Theory on Radial Basis Function Interpolation 4.1 Convergence and Convergence Rate 4.1.1 Quasi-Interpolation, Strang-Fix Condition and Shift Invariant Space 4.2 Convergence Results for Scattered Data Radial Basis Function Interpolation 4.2.1 Error Estimation 4.2.2 Construction of Admissible Vectors 4.3 Positive Definite Radial Basis Functions 4.4 Bochner Theory for Radial Basis Functions 4.5 Radial Functions and Strang-Fix Conditions Chapter 5 Other Scattered Data Interpolation Methods 5.1 Moving Least Squares 5.1.1 Least Squares 5.1.2 Moving Least Squares 5.1.3 Interpolating Moving Least Squares Methods 5.1.4 Divide and Conquer on General Domain 5.2 Convergence Analysis of Shepard Methods 5.2.1 Convergence Analysis for the Shepard Method 5.3 Implicit Splines 5.3.1 Other Scattered Data Interpolation Methods 5.4 Partition of Unity Chapter 6 Scatter Data Interpolation for Numerical Solutions of PDEs 5.5 R-function 6.1 Generalized Functional Interpolations and Numerical Methods for PDEs 6.2 Other Multivariate Approximation Methods for PDEs 6.2.1 Least Squares Methods 6.2.2 Collocation 6.2.3 Galerkin Method 6.2.4 Golberg Method Bibliography
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