Book contents
- Frontmatter
- Contents
- Preface
- 1 Applications and motivations
- 2 Haar spaces and multivariate polynomials
- 3 Local polynomial reproduction
- 4 Moving least squares
- 5 Auxiliary tools from analysis and measure theory
- 6 Positive definite functions
- 7 Completely monotone functions
- 8 Conditionally positive definite functions
- 9 Compactly supported functions
- 10 Native spaces
- 11 Error estimates for radial basis function interpolation
- 12 Stability
- 13 Optimal recovery
- 14 Data structures
- 15 Numerical methods
- 16 Generalized interpolation
- 17 Interpolation on spheres and other manifolds
- References
- Index
Preface
Published online by Cambridge University Press: 22 February 2010
- Frontmatter
- Contents
- Preface
- 1 Applications and motivations
- 2 Haar spaces and multivariate polynomials
- 3 Local polynomial reproduction
- 4 Moving least squares
- 5 Auxiliary tools from analysis and measure theory
- 6 Positive definite functions
- 7 Completely monotone functions
- 8 Conditionally positive definite functions
- 9 Compactly supported functions
- 10 Native spaces
- 11 Error estimates for radial basis function interpolation
- 12 Stability
- 13 Optimal recovery
- 14 Data structures
- 15 Numerical methods
- 16 Generalized interpolation
- 17 Interpolation on spheres and other manifolds
- References
- Index
Summary
Scattered data approximation is a recent, fast growing research area. It deals with the problem of reconstructing an unknown function from given scattered data. Naturally, it has many applications, such as terrain modeling, surface reconstruction, fluid-structure interaction, the numerical solution of partial differential equations, kernel learning, and parameter estimation, to name a few. Moreover, these applications come from such different fields as applied mathematics, computer science, geology, biology, engineering, and even business studies.
This book is designed to give a thorough, self-contained introduction to the field of multivariate scattered data approximation without neglecting the most recent results.
Having the above-mentioned applications in mind, it immediately follows that any competing method has to be capable of dealing with a very large number of data points in an arbitrary number of space dimensions, which might bear no regularity at all and which might even change position with time.
Hence, in my personal opinion a true scattered data method has to be meshless. This is an assumption that might be challenged but it will be the fundamental assumption throughout this book. Consequently, certain methods, that generally require a mesh, such as those using wavelets, multivariate splines, finite elements, box splines, etc. are immediately ruled out. This does not at all mean that such methods cannot sometimes be used successfully in the context of scattered data approximation; on the contrary, it just explains why these methods are not discussed in this book.
- Type
- Chapter
- Information
- Scattered Data Approximation , pp. ix - xPublisher: Cambridge University PressPrint publication year: 2004