# Meshfree Approximation Methods with MATLAB

World Scientific, 2007 - Technology & Engineering - 500 pages
Meshfree approximation methods are a relatively new area of research, and there are only a few books covering it at present. Whereas other works focus almost entirely on theoretical aspects or applications in the engineering field, this book provides the salient theoretical results needed for a basic understanding of meshfree approximation methods.The emphasis here is on a hands-on approach that includes MATLAB routines for all basic operations. Meshfree approximation methods, such as radial basis function and moving least squares method, are discussed from a scattered data approximation and partial differential equations point of view. A good balance is supplied between the necessary theory and implementation in terms of many MATLAB programs, with examples and applications to illustrate key points. Used as class notes for graduate courses at Northwestern University, Illinois Institute of Technology, and Vanderbilt University, this book will appeal to both mathematics and engineering graduate students.

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### Contents

 Introduction 1 Radial Basis Function Interpolation in MATLAB 17 Positive Definite Functions 27 Examples of Strictly Positive Definite Radial Functions 37 Completely Monotone and Multiply Monotone Functions 47 Scattered Data Interpolation with Polynomial Precision 53 Conditionally Positive Definite Functions 63 Conditionally Positive Definite Radial Functions 73
 Numerical Experiments for Approximate MLS Approximation 237 Fast Fourier Transforms 243 Partition of Unity Methods 249 Approximation of Point Cloud Data in 3D 255 Fixed Level Residual Iteration 265 Multilevel Iteration 277 Adaptive Iteration 291 Improving the Condition Number of the Interpolation Matrix 303

 Other Norms and Scattered Data Fitting 79 Compactly Supported Radial Basis Functions 85 Interpolation with Compactly Supported RBFs in Matlab 95 Reproducing Kernel Hilbert Spaces and Native Spaces 103 The Power Function and Native Space Error Estimates 111 Refined and Improved Error Bounds 125 Stability and TradeOff Principles 135 Numerical Evidence for Approximation Order Results 141 The Optimality of RBF Interpolation 159 Least Squares RBF Approximation with Matlab 165 Theory for Least Squares Approximation 177 Moving Least Squares Approximation 191 Examples of MLS Generating Functions 205 MLS Approximation with Matlab 211 Error Bounds for Moving Least Squares Approximation 225
 Other Efficient Numerical Methods 321 Generalized Hermite Interpolation 333 RBF Hermite Interpolation in Matlab 339 Solving Elliptic Partial Differential Equations via RBF Collocation 345 NonSymmetric RBF Collocation in Matlab 353 Symmetric RBF Collocation in MATLAB 365 Collocation with CSRBFs in Matlab 375 Using Radial Basis Functions in Pseudospectral Mode 387 RBFPS Methods in Matlab 401 RBF Galerkin Methods 419 Appendix A Useful Facts from Discrete Mathematics 427 Additional Computer Programs 435 Bibliography 451 Index 491 Copyright