简介
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.
目录
Introduction p. 1
Radial basis function interpolation in MATLAB p. 17
Positive definite functions p. 27
Examples of strictly positive definite radial functions p. 37
Completely monotone and multiply monotone functions p. 47
Scattered data interpolation with polynomial precision p. 53
Conditionally positive definite functions p. 63
Examples of conditionally positive definite functions p. 67
Conditionally positive definite radial functions p. 73
Miscellaneous theory : other norms and scattered data fitting on manifolds p. 79
Compactly supported radial basis functions p. 85
Interpolation with compactly supported RBFs in MATLAB p. 95
Reproducing Kernel Hilbert spaces and native spaces for strictly positive definite functions p. 103
The power function and native space error estimates p. 111
Refined and improved error bounds p. 125
Stability and trade-off principles p. 135
Numerical evidence for approximation order results p. 141
The optimally of RBF interpolation p. 159
Least squares RBF approximation with MATLAB p. 165
Theory for least squares approximation p. 177
Adaptive least squares approximation p. 181
Moving least squares approximation p. 191
Examples of MLS generating functions p. 205
MLS approximation with MATLAB p. 211
Error bounds for moving least squares approximation p. 225
Approximate moving least squares approximation p. 229
Numerical experiments for approximate MLS approximation p. 237
Fast fourier transforms p. 243
Partition of unity methods p. 249
Approximation of point cloud data in 3D p. 255
Fixed level residual iteration p. 265
Multilevel iteration p. 277
Adaptive iteration p. 291
Improving the condition number of the interpolation matrix p. 303
Other efficient numerical methods p. 321
Generalized hermite interpolation p. 333
RBF hermite interpolation in MATLAB p. 339
Solving elliptic partial differential equations via RBF collocation p. 345
Non-symmetric RBF collocation in MATLAB p. 353
Symmetric RBF collocation in MATLAB p. 365
Collocation with CSRBFs in MATLAB p. 375
Using radial basis functions in pseudospectral mode p. 387
RBF-PS methods in MATLAB p. 401
RBF Galerkin methods p. 419
RBF Galerkin methods in MATLAB p. 423
Useful facts from discrete mathematics p. 427
Useful facts from analysis p. 431
Additional computer programs p. 435
Catalog of RBFs with derivatives p. 443
Radial basis function interpolation in MATLAB p. 17
Positive definite functions p. 27
Examples of strictly positive definite radial functions p. 37
Completely monotone and multiply monotone functions p. 47
Scattered data interpolation with polynomial precision p. 53
Conditionally positive definite functions p. 63
Examples of conditionally positive definite functions p. 67
Conditionally positive definite radial functions p. 73
Miscellaneous theory : other norms and scattered data fitting on manifolds p. 79
Compactly supported radial basis functions p. 85
Interpolation with compactly supported RBFs in MATLAB p. 95
Reproducing Kernel Hilbert spaces and native spaces for strictly positive definite functions p. 103
The power function and native space error estimates p. 111
Refined and improved error bounds p. 125
Stability and trade-off principles p. 135
Numerical evidence for approximation order results p. 141
The optimally of RBF interpolation p. 159
Least squares RBF approximation with MATLAB p. 165
Theory for least squares approximation p. 177
Adaptive least squares approximation p. 181
Moving least squares approximation p. 191
Examples of MLS generating functions p. 205
MLS approximation with MATLAB p. 211
Error bounds for moving least squares approximation p. 225
Approximate moving least squares approximation p. 229
Numerical experiments for approximate MLS approximation p. 237
Fast fourier transforms p. 243
Partition of unity methods p. 249
Approximation of point cloud data in 3D p. 255
Fixed level residual iteration p. 265
Multilevel iteration p. 277
Adaptive iteration p. 291
Improving the condition number of the interpolation matrix p. 303
Other efficient numerical methods p. 321
Generalized hermite interpolation p. 333
RBF hermite interpolation in MATLAB p. 339
Solving elliptic partial differential equations via RBF collocation p. 345
Non-symmetric RBF collocation in MATLAB p. 353
Symmetric RBF collocation in MATLAB p. 365
Collocation with CSRBFs in MATLAB p. 375
Using radial basis functions in pseudospectral mode p. 387
RBF-PS methods in MATLAB p. 401
RBF Galerkin methods p. 419
RBF Galerkin methods in MATLAB p. 423
Useful facts from discrete mathematics p. 427
Useful facts from analysis p. 431
Additional computer programs p. 435
Catalog of RBFs with derivatives p. 443
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