Meshfree Approximation Methods with MATLAB

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World Scientific, Jan 1, 2007 - Mathematics - 500 pages
2 Reviews
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 result 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
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