Details

Subject(s)

• Sampling (Statistics) [Browse]
• Inverse problems (Differential equations)—Numerical solutions [Browse]

Related name

• Nashed, M. Zuhair [Browse]

Series

• Mathematical Surveys and Monographs [More in this series]
• Mathematical Surveys and Monographs ; v.274 [More in this series]

Summary note

The goal of this book is to introduce the reader to methodologies in recovery problems for objects, such as functions and signals, from partial or indirect information. The recovery of objects from a set of data demands key solvers of inverse and sampling problems. Until recently, connections between the mathematical areas of inverse problems and sampling were rather tenuous. However, advances in several areas of mathematical research have revealed deep common threads between them, which proves that there is a serious need for a unifying description of the underlying mathematical ideas and concepts. Freeden and Nashed present an integrated approach to resolution methodologies from the perspective of both these areas.Researchers in sampling theory will benefit from learning about inverse problems and regularization methods, while specialists in inverse problems will gain a better understanding of the point of view of sampling concepts. This book requires some basic knowledge of functional analysis, Fourier theory, geometric number theory, constructive approximation, and special function theory. By avoiding extreme technicalities and elaborate proof techniques, it is an accessible resource for students and researchers not only from applied mathematics, but also from all branches of engineering and science.

Source of description

Description based on: online resource; title from pdf title page (ProQuest Ebook Central, viewed on January 2, 2024).

Contents

• Front Cover
• Contents
• Preface
• Part 1. Introductory remarks
• Chapter 1. Constituents of the univariate antenna problem
• 1.1. Problem induced regularization
• 1.2. Problem induced sampling
• 1.3. Recovery as common thread
• 1.4. Organization of the work
• Part 2. Regularization tools
• Chapter 2. Functional and Fourier analytic auxiliaries
• 2.1. Functional analytic tools
• 2.2. Fourier asymptotics and limit relations
• Part 3. Regularization methodologies
• Chapter 3. Matricial methodologies of resolution
• 3.1. Least squares method and pseudoinverse
• 3.2. Singular value decomposition
• 3.3. Regularization methods
• Chapter 4. Compact operator methodologies of resolution
• 4.1. Least squares method and pseudoinverse
• 4.2. Singular value decomposition
• 4.3. Regularization methods
• 4.4. Approximation and data error
• Chapter 5. Example realizations light: univariate differentiation
• 5.1. Regularization by change of space and/or topology
• 5.2. Singular value decomposition
• 5.3. Tikhonov regularization
• 5.4. Multiscale regularization
• 5.5. Finite difference schemes
• 5.6. Fredholm integral equation with strong noise
• 5.7. Fredholm integral equation with weak noise
• Chapter 6. Reconstruction and regularization methods
• 6.1. Concept of regularization revisited
• 6.2. Use of a priori bounds
• 6.3. Tikhonov regularization
• 6.4. Characterization of regularizers
• 6.5. Reconstruction optimality
• 6.6. Order optimality of regularizing filters
• 6.7. Pseudoinverses in reproducing kernel Hilbert spaces
• 6.8. Iterative methods as regularization schemes
• 6.9. Projection methods
• 6.10. Multiscale spectral regularization
• 6.11. Multiscale space mollifier schemes
• 6.12. Backus-Gilbert method
• 6.13. Stochastic regularization methods
• 6.14. Nonlinear ill-posed problems.
• 6.15. Methodologies and dilemmas
• Part 4. Regularization examples
• Chapter 7. Regularization methodologies in geotechnology
• 7.1. Potential theoretic auxiliaries
• 7.2. Relevant inverse potential problems
• 7.3. Satellite gravitational gradiometry (SGG)
• 7.4. Inverse gravimetry (IG)
• 7.5. Inverse magnetometry (IM)
• Part 5. Sampling tools
• Chapter 8. Lattice point and special function theoretic auxiliaries
• 8.1. Geometric number theory
• 8.2. Periodization and Poisson summation formula
• 8.3. Metaharmonic theory and Bessel function asymptotics
• Part 6. Sampling methodologies
• Chapter 9. Sampling over continuously connected pointsets
• 9.1. Univariate Fourier theory based Shannon sampling
• 9.2. Reproducing kernel induced sampling concepts
• 9.3. Sampling sets in reproducing kernel spaces
• 9.4. Sampling induced reproducing kernel space framework
• 9.5. Sampling in shift-invariant spaces
• Chapter 10. Sampling over discretely given pointsets
• 10.1. Dirac mass
• 10.2. Infinite networks and relative Hilbert spaces
• 10.3. Point masses of finite reproducing kernel Hilbert space norm
• 10.4. Infinite square matrices
• 10.5. Point masses and concrete models
• 10.6. Point mass samples and interpolation
• Chapter 11. Polyharmonic finite bandwidth sampling
• 11.1. Multivariate integral formulas to boundary conditions
• 11.2. Sampling to Dirichlet conditions
• 11.3. Finite bandwidth splines
• Chapter 12. Polyharmonic infinite bandwidth sampling
• 12.1. Integral formulas over Euclidean spaces
• 12.2. Asymptotically defined reproducing kernel Hilbert space
• 12.3. Thin plate spline concept
• 12.4. Cardinal spline concept
• 12.5. Relationship to the multivariate Shannon kernel
• Chapter 13. Polymetaharmonic finite bandwidth sampling
• 13.1. Univariate lattice point theory based Shannon sampling.
• 13.2. Multivariate lattice functions and summation formulas
• 13.3. Bivariate lattice point identities and Shannon-type sampling
• 13.4. Multivariate Gauss-Weierstrass integral transform
• 13.5. Multivariate Gauss-Weierstrass summability Shannon sampling
• 13.6. Multivariate Shannon-type nonsummability sampling
• 13.7. Gauss-Weierstrass wavelet sampling
• Chapter 14. Polymetaharmonic infinite bandwidth sampling
• 14.1. Univariate lattice point theory based sampling
• 14.2. Multivariate summation and lattice point identities
• 14.3. Multivariate lattice sampling
• Part 7. Sampling examples
• Chapter 15. Sampling methodologies in technology
• 15.1. Exponential reproduction inversion in gravimetry
• 15.2. Slepian spline inversion in antenna technology
• Part 8. Concluding remarks
• Chapter 16. Recovery as interconnecting whole
• 16.1. Regularization as constituting methodology
• 16.2. Sampling as constituting methodology
• 16.3. Recovery as common thread
• List of symbols
• Bibliography
• Index
• Back Cover.

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ISBN

1-4704-7451-4

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