## Meshfree Approximation Methods with MATLABMeshfree 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 |

451 | |

491 | |

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### Common terms and phrases

algorithm approach approximation order basic basis function interpolation bdyl bdyO Chapter Chebyshev coefficients collocation matrix collocation points compactly supported completely monotone Compute condition number conditionally positive definite convergence data points data sites defined definite and radial definite of order distance matrix dsites equations evaluation matrix evaluation points Example Fasshauer Figure fill distance Fourier transform Franke's function Halton points Hermite interpolation intentionally left blank interpolation matrix interpolation problem knot Laguerre-Gaussians least squares approximation linear system lines Matlab maxerr maximum error monotone functions moving least squares multilevel native space non-stationary non-symmetric norm obtain optimal partition of unity PDEs plot polynomial positive definite functions positive definite radial power function Program quasi-interpolant radial basis function RBF interpolant reproduction scattered data interpolation Schaback shape parameter solution sprintf stationary strictly conditionally positive strictly positive definite Table test function testfunction Theorem thin plate splines unit square values vector weight function Wendland Wendland 2005a zero