Lýsing:
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.
Annað
- Höfundur: Holger Wendland
- Útgáfa:1
- Útgáfudagur: 13-12-2004
- Hægt að prenta út 5 bls.
- Hægt að afrita 5 bls.
- Format:Page Fidelity
- ISBN 13: 9780511261893
- Print ISBN: 9780521843355
- ISBN 10: 0511261896
Efnisyfirlit
- Half-title
- Series-title
- Title
- Copyright
- Contents
- Preface
- 1 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 comments
- 2 Haar spaces and multivariate polynomials
- 2.1 The Mairhuber?Curtis theorem
- 2.2 Multivariate polynomials
- 3 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 comments
- 4 Moving least squares
- 4.1 Definition and characterization
- 4.2 Local polynomial reproduction by moving least squares
- 4.3 Generalizations
- 4.4 Notes and comments
- 5 Auxiliary tools from analysis and measure theory
- 5.1 Bessel functions
- 5.2 Fourier transform and approximation by convolution
- 5.3 Measure theory
- 6 Positive definite functions
- 6.1 Definition and basic properties
- 6.2 Bochner?s characterization
- 6.3 Radial functions
- 6.5 Notes and comments
- 7 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 comments
- 8 Conditionally positive definite functions
- 8.1 Definition and basic properties
- 8.2 An analogue of Bochner?s characterization
- 8.3 Examples of generalized Fourier transforms
- 8.4 Radial conditionally positive definite functions
- 8.5 Interpolation by conditionally positive definite functions
- 8.6 Notes and comments
- 9 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 comments
- 10 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 comments
- 11 Error estimates for radial basis function interpolation
- 11.1 Power function and first estimates
- 11.2 Error estimates in terms of the fill distance
- 11.3 Estimates for popular basis functions
- 11.4 Spectral convergence for Gaussians and (inverse) multiquadrics
- 11.5 Improved error estimates
- 11.6 Sobolev bounds for functions with scattered zeros
- 11.7 Notes and comments
- 12 Stability
- 12.1 Trade-off principle
- 12.2 Lower bounds for Lambdamin
- 12.3 Change of basis
- 12.4 Notes and comments
- 13 Optimal recovery
- 13.1 Minimal properties of radial basis functions
- 13.2 Abstract optimal recovery
- 13.3 Notes and comments
- 14 Data structures
- 14.1 The fixed-grid method
- 14.2 kd-Trees
- 14.3 bd-Trees
- 14.4 Range trees
- 14.5 Notes and comments
- 15 Numerical methods
- 15.1 Fast multipole methods
- 15.2 Approximation of Lagrange functions
- 15.3 Alternating projections
- 15.4 Partition of unity
- 15.5 Multilevel methods
- 15.6 A greedy algorithm
- 15.7 Concluding remarks
- 15.8 Notes and comments
- 16 Generalized interpolation
- 16.1 Optimal recovery in Hilbert spaces
- 16.2 Hermite?Birkhoff interpolation
- 16.3 Solving PDEs by collocation
- 16.4 Notes and comments
- 17 Interpolation on spheres and other manifolds
- 17.1 Spherical harmonics
- 17.2 Positive definite functions on the sphere
- 17.3 Error estimates
- 17.4 Interpolation on compact manifolds
- 17.5 Notes and comments
- References
- Index
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- Gerð : 208
- Höfundur : 16357
- Útgáfuár : 2004
- Leyfi : 379