Linear Inverse Problems The Maximum Entropy Connection 1st Edition by Gzyl Henryk ISBN 981433877X 9789814338776 by Henryk Gzyl, Yurayh Velásquez 9789814338776, 981433877X instant download after payment.

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ISBN 10: 981433877X 
ISBN 13: 9789814338776
Author: Gzyl Henryk

Linear Inverse Problems The Maximum Entropy Connection 1st Table of contents:

1. Introduction Linear Inverse Problems
2. A collection of linear inverse problems
2.1 A battle horse for numerical computations
2.2 Linear equations with errors in the data
2.3 Linear equations with convex constraints
2.4 Inversion of Laplace transforms from finite number of data points
2.5 Fourier reconstruction from partial data
2.6 More on the non-continuity of the inverse
2.7 Transportation problems and reconstruction from marginals
2.8 CAT
2.9 Abstract spline interpolation
2.10 Bibliographical comments and references
References
3. The basics about linear inverse problems
3.1 Problemstatements
3.2 Quasi solutions and variational methods
3.3 Regularization and approximate solutions
3.4 Appendix
3.5 Bibliographical comments and references
References
4. Regularization in Hilbert spaces: Deterministic and stochastic approaches
4.1 Basics
4.2 Tikhonov’s regularization scheme
4.3 Spectral cutoffs
4.4 Gaussian regularization of inverse problems
4.5 Bayesianmethods
4.6 The method ofmaximumlikelihood
4.7 Bibliographical comments and references
References
5. Maxentropic approach to linear inverse problems
5.1 Heuristic preliminaries
5.2 Some properties of the entropy functionals
5.3 The direct approach to the entropic maximization problem
5.4 Amore detailed analysis
5.5 Convergence ofmaxentropic estimates
5.6 Maxentropic reconstruction in the presence of noise
5.7 Maxentropic reconstruction of signal and noise
5.8 Maximum entropy according to Dacunha-Castelle and Gamboa. Comparison with Jaynes’ classical ap
5.8.1 Basic results
5.8.2 Jaynes’ and Dacunha and Gamboa’s approaches
5.9 MEM under translation
5.10 Maxent reconstructions under increase of data
5.11 Bibliographical comments and references
References
6. Finite dimensional problems
6.1 Two classical methods of solution
6.2 Continuous time iteration schemes
6.3 Incorporation of convex constraints
6.3.1 Basics and comments
6.3.2 Optimization with differentiable non-degenerate equality constraints
6.3.3 Optimization with differentiable, non-degenerate inequality constraints
6.4 The method of projections in continuous time
6.5 Maxentropic approaches
6.5.1 Linear systems with band constraints
6.5.2 Linear system with Euclidean norm constraints
6.5.3 Linear systems with non-Euclidean norm constraints
6.5.4 Linear systems with solutions in unbounded convex sets
6.5.5 Linear equations without constraints
6.6 Linear systems with measurement noise
6.7 Bibliographical comments and references
References
7. Some simple numerical examples and moment problems
7.1 The density of the Earth
7.1.1 Solution by the standard L2[0, 1] techniques
7.1.2 Piecewise approximations in L2([0, 1])
7.1.3 Linear programming approach
7.1.4 Maxentropic reconstructions: Influence of a priori data
7.1.5 Maxentropic reconstructions: Effect of the noise
7.2 A test case
7.2.1 Standard L2[0,1] technique
7.2.2 Discretized L2[0,1] approach
7.2.3 Maxentropic reconstructions: Influence of a priori data
7.2.4 Reconstruction by means of cubic splines
7.2.5 Fourier versus cubic splines
7.3 Standard maxentropic reconstruction
7.3.1 Existence and stability
7.3.2 Some convergence issues
7.4 Some remarks on moment problems
7.4.1 Some remarks about the Hamburger and Stieltjes moment problems
7.5 Moment problems in Hilbert spaces
7.6 Reconstruction of transition probabilities
7.7 Probabilistic approach to Hausdorff’s moment problem
7.8 The very basics about cubic splines
7.9 Determination of risk measures from market price of risk
7.9.1 Basic aspects of the problem
7.9.1.1 Examples of distortion functions
7.9.2 Problemstatement
7.9.2.1 Problem statement
7.9.2.2 Problem discretization
7.9.3 Themaxentropic solution
7.9.4 Description of numerical results
7.9.4.1 A consistency test
7.10 Bibliographical comments and references
References
8. Some infinite dimensional problems
8.1 A simple integral equation
8.1.1 The random function approach
8.1.2 The random measure approach: Gaussian measures
8.1.3 The random measure approach: Compound Poissonmeasures
8.1.4 The random measure approach: Gaussian fields
8.1.5 Closing remarks
8.2 A simple example: Inversion of a Fourier transform given a few coefficients
8.3 Maxentropic regularization for problems in Hilbert spaces
8.3.1 Gaussianmeasures
8.3.2 Exponentialmeasures
8.3.3 Degenerate measures in Hilbert spaces and spectral cut off regularization
8.3.4 Conclusions
8.4 Bibliographical comments and references
Reference
9. Tomography, reconstruction from marginals and transportation problems
9.1 Generalities
9.2 Reconstruction frommarginals
9.3 A curious impossibility result and its counterpart
9.3.1 The bad news
9.3.2 The good news
9.4 The Hilbert space set up for the tomographic problem
9.4.1 More on nonuniquenes of reconstructions
9.5 The Russian Twist
9.6 Why does it work
9.7 Reconstructions using (classical) entropic, penalized methods in Hilbert space
9.8 Some maxentropic computations
9.9 Maxentropic approach to reconstruction from marginals in the discrete case
9.9.1 Reconstruction from marginals by maximum entropy on the mean
9.9.2 Reconstruction from marginals using the standard maximumentropymethod
9.10 Transportation and linear programming problems
9.11 Bibliographical comments and references
References
10. Numerical inversion of Laplace transforms
10.1 Motivation
10.2 Basics about Laplace transforms
10.3 The inverse Laplace transform is not continuous
10.4 Amethod of inversion
10.4.1 Expansion in sine functions
10.4.2 Expansion in Legendre polynomials
10.4.3 Expansion in Laguerre polynomials
10.5 From Laplace transforms to moment problems
10.6 Standard maxentropic approach to the Laplace inversion problem
10.7 Maxentropic approach in function space: The Gaussian case
10.8 Maxentropic linear splines
10.9 Connection with the complex interpolation problem
10.10 Numerical examples
10.11 Bibliographical comments and references
References
11. Maxentropic characterization of probability distributions
11.1 Preliminaries
11.2 Example 1
11.3 Example 2
11.4 Example 3
11.5 Example 4
11.6 Example 5
11.7 Example 6
Reference
12. Is an image worth a thousand words?
12.1 Problem setup
12.1.1 List of questions for you to answer
12.2 Answers to the questions
12.2.1 Introductory comments
12.2.2 Answers
12.3 Bibliographical comments and references

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Tags: Gzyl Henryk, Inverse, Problems