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
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.
ISBN
1-4704-7451-4
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