Scattered data approximation分散数据逼近

2004-12  

Wendland, Holger  

336  

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Many practical applications require the reconstruction of a multivariate function from discrete, unstructured data. This book gives a self-contained, complete introduction into this subject. It concentrates on truly meshless methods such as radial basis functions, moving least squares, and partitions of unity. The book starts with an overview on typical applications of scattered data approximation, coming from surface reconstruction, fluid-structure interaction, and the numerical solution of partial differential equations. It then leads the reader from basic properties to the current state of research, addressing all important issues, such as existence, uniqueness, approximation properties, numerical stability, and efficient implementation. Each chapter ends with a section giving information on the historical background and hints for further reading. Complete proofs are included, making this perfectly suited for graduate courses on multivariate approximation and it can be used to support courses in computer aided geometric design, and meshless methods for partial differential equations.

Preface1 Applications and motivations 1.1 Surface reconstruction! 1.2 Fluid-structure interaction in aeroelasticity 1.3 Grid-free semi-Lagrangian advection 1.4 Learning from splines 1.5 Approximation and approximation orders 1.6 Notation 1.7 Notes and comments2 Haar spaces and multivariate polynomials　　2.1 The Mairhuber-Curtis theorem 2.2 Multivariate polynomials3 Local polynomial reproduction 　3.1 Definition and basic properties 　3.2 Norming sets 　3.3 Existence for regions with cone condition 　3.4 Notes and comments4 Moving least squares 　4.1 Definition and characterization　　4.2 Local polynomial reproduction by moving least squares　　4.3 Generalizations　　4.4 Notes and comments5 Auxiliary tools from analysis and measure theory 　 5.1 Bessel functions 　5.2 Fourier transform and approximation by convolution 　 5.3 Measure theory6 Positivie definite functions 6.1 Definition and basic properties 6.2 Boehner’s characterization 6.3 Radial functions 6.4 Functions, kernels, and other norms 6.5 Notes and comments7 Completely monotone functions 7.1 Definition and first characterization 7.2 The Bernstein-Hausdorff-Widder characterization 7.3 Schoenberg's characterization 7.4 Notes and comments8 Conditionally positive definite functions 8.1 Definition and basic properties 8.2 An analogue of Buchner’s characterization 8.3 Examples of generalized Fourier transform 8.4 Radial conditionally positive definite functions 8.5 Interpolation by conditionally positive definite functions 8.6 Notes and comments9 Compactly supported functions 9.1 General remarks 9.2 Dimension walk 9.3 Piecewise polynomial functions with local support 9.4 Compactly supported functions of minimal degree 9.5 Generalizations 9.6 Notes and comments10 Native spaces 10.1 Reproducing-kernel Hilbert spaces 10.2 Native spaces for positive definite kernels 10.3 Native spaces for conditionally positive definite kernels 10.4 Further characterizations of native spaces 10.5 Special cases of native spaces 10.6 An embedding theorem 10.7 Restriction and extension 10.8 Notes and comments11 Error estimates for radial basis function interpolation 11.1 Power function and first estimates 11.2 Error estimates in terms of the fill distance ……12 Stability13 Optimal recovery14 Data strutures15 Numerical methods16 Generalized interpolation17 Interpolation on spheres and other manifoldsReferencesIndex

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