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Linear inverse problems : the maximum entropy connection (with CD-ROM)

Author: Henryk Gzyl; Yurayh Velásquez
Publisher: New Jersey : World Scientific, 2011.
Series: Series on advances in mathematics for applied sciences, v. 83.
Edition/Format:   Book : CD for computer : EnglishView all editions and formats
Database:WorldCat
Summary:

Describes a useful tool for solving linear inverse problems subject to convex constraints. This book consists of a technique for transforming a large dimensional inverse problem into a small  Read more...

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Document Type: Book
All Authors / Contributors: Henryk Gzyl; Yurayh Velásquez
ISBN: 9789814338776 981433877X
OCLC Number: 681502962
Description: xix, 326 p. : ill.; 24 cm. + 1CD-ROM (4 3/4in.).
Contents: Machine generated contents note: 1.Introduction --
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 --
3.The basics about linear inverse problems --
3.1.Problem statements --
3.2.Quasi solutions and variational methods --
3.3.Regularization and approximate solutions --
3.4.Appendix --
3.5.Bibliographical comments and 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.Bayesian methods --
4.6.The method of maximum likelihood --
4.7.Bibliographical comments and 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.A more detailed analysis --
5.5.Convergence of maxentropic 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 approach --
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 --
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 differentiate, 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 --
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 --
7.9.2.Problem statement --
7.9.3.The maxentropic solution --
7.9.4.Description of numerical results --
7.10.Bibliographical comments and 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 Poisson measures --
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.Gaussian measures --
8.3.2.Exponential measures --
8.3.3.Degenerate measures in Hilbert spaces and spectral cut off regularization --
8.3.4.Conclusions --
8.4.Bibliographical comments and references --
9.Tomography, reconstruction from marginals and transportation problems --
9.1.Generalities --
9.2.Reconstruction from marginals --
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 maximum entropy method --
9.10.Transportation and linear programming problems --
9.11.Bibliographical comments and 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.A method 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 --
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 --
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 --
Appendix A Basic topology --
Appendix B Basic measure theory and probability --
B.1.Some results from measure theory and integration --
B.2.Some probabilistic jargon --
B.3.Brief description of the Kolmogorov extension theorem --
B.4.Basic facts about Gaussian process in Hilbert spaces --
Appendix C Banach spaces --
C.1.Basic stuff --
C.2.Continuous linear operator on Banach spaces --
C.3.Duality in Banach spaces --
C.4.Operators on Hilbert spaces. Singular values decompositions --
C.5.Some convexity theory --
Appendix D Further properties of entropy functionals --
D.1.Properties of entropy functionals --
D.2.A probabilistic connection --
D.3.Extended definition of entropy --
D.4.Exponetial families and geometry in the space of probabilities --
D.4.1.The geometry on the set of positive vectors --
D.4.2.Lifting curves from G+ to G and parallel transport --
D.4.3.From geodesies to Kullback's divergence --
D.4.4.Coordinates on P --
D.5.Bibliographical comments and references --
Appendix E Software user guide --
E.1.Installation procedure --
E.2.Quick start guide --
E.2.1.Moment problems with MEM --
E.2.2.Moment problems with SME --
E.2.3.Moment problems with Quadratic Programming --
E.2.4.Transition probabilities problem with MEM --
E.2.5.Transition probabilities problem with SME --
E.2.6.Transition probabilities problem with Quadratic Programming --
E.2.7.Reconstruction from Marginals with MEM --
E.2.8.Reconstruction from Marginals with SME --
E.2.9.Reconstruction from Marginals with Quadratic Programming --
E.2.10.A generic problem in the form Ax = y, with MEM --
E.2.11.A generic problem in the form Ax = y, with SME --
E.2.12.A generic problem in the form Ax = y, with Quadratic Programming --
E.2.13.The results windows --
E.2.14.Messages that will appear.
Series Title: Series on advances in mathematics for applied sciences, v. 83.
Responsibility: Henryk Gzyl & Yurayh Velásquez.

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This book serves as lecture notes and is suitable for a graduate-level course on inverse problems. It covers a broad subject, gives explanations in sufficient detail, and helps readers learn to apply Read more...

 
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schema:description"Machine generated contents note: 1.Introduction -- 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 -- 3.The basics about linear inverse problems -- 3.1.Problem statements -- 3.2.Quasi solutions and variational methods -- 3.3.Regularization and approximate solutions -- 3.4.Appendix -- 3.5.Bibliographical comments and 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.Bayesian methods -- 4.6.The method of maximum likelihood -- 4.7.Bibliographical comments and 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.A more detailed analysis -- 5.5.Convergence of maxentropic 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 approach -- 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 -- 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 differentiate, 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 -- 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 -- 7.9.2.Problem statement -- 7.9.3.The maxentropic solution -- 7.9.4.Description of numerical results -- 7.10.Bibliographical comments and 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 Poisson measures -- 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.Gaussian measures -- 8.3.2.Exponential measures -- 8.3.3.Degenerate measures in Hilbert spaces and spectral cut off regularization -- 8.3.4.Conclusions -- 8.4.Bibliographical comments and references -- 9.Tomography, reconstruction from marginals and transportation problems -- 9.1.Generalities -- 9.2.Reconstruction from marginals -- 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 maximum entropy method -- 9.10.Transportation and linear programming problems -- 9.11.Bibliographical comments and 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.A method 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 -- 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 -- 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 -- Appendix A Basic topology -- Appendix B Basic measure theory and probability -- B.1.Some results from measure theory and integration -- B.2.Some probabilistic jargon -- B.3.Brief description of the Kolmogorov extension theorem -- B.4.Basic facts about Gaussian process in Hilbert spaces -- Appendix C Banach spaces -- C.1.Basic stuff -- C.2.Continuous linear operator on Banach spaces -- C.3.Duality in Banach spaces -- C.4.Operators on Hilbert spaces. Singular values decompositions -- C.5.Some convexity theory -- Appendix D Further properties of entropy functionals -- D.1.Properties of entropy functionals -- D.2.A probabilistic connection -- D.3.Extended definition of entropy -- D.4.Exponetial families and geometry in the space of probabilities -- D.4.1.The geometry on the set of positive vectors -- D.4.2.Lifting curves from G+ to G and parallel transport -- D.4.3.From geodesies to Kullback's divergence -- D.4.4.Coordinates on P -- D.5.Bibliographical comments and references -- Appendix E Software user guide -- E.1.Installation procedure -- E.2.Quick start guide -- E.2.1.Moment problems with MEM -- E.2.2.Moment problems with SME -- E.2.3.Moment problems with Quadratic Programming -- E.2.4.Transition probabilities problem with MEM -- E.2.5.Transition probabilities problem with SME -- E.2.6.Transition probabilities problem with Quadratic Programming -- E.2.7.Reconstruction from Marginals with MEM -- E.2.8.Reconstruction from Marginals with SME -- E.2.9.Reconstruction from Marginals with Quadratic Programming -- E.2.10.A generic problem in the form Ax = y, with MEM -- E.2.11.A generic problem in the form Ax = y, with SME -- E.2.12.A generic problem in the form Ax = y, with Quadratic Programming -- E.2.13.The results windows -- E.2.14.Messages that will appear."@en
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