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E-raamat: Linear Inverse Problems: The Maximum Entropy Connection [World Scientific e-raamat]

(Iesa, Venezuela), (Univ Metropolitana, Venezuela)
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Linear Inverse Problems: The Maximum Entropy ConnectionSuurem pilt
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World Scientific e-raamatudRohkem infot World Scientific e-raamatute kohta
Raamatu kodulehekülg: http://www.worldscientific.com/worldscibooks/10.1142/8061#t=toc
The book describes a useful tool for solving linear inverse problems subject to convex constraints. The method of maximum entropy in the mean automatically takes care of the constraints. It consists of a technique for transforming a large dimensional inverse problem into a small dimensional non-linear variational problem.A variety of mathematical aspects of the maximum entropy method are explored as well.
Preface v
List of Figures
xv
List of Tables
xxi
1 Introduction
1(4)
2 A collection of linear inverse problems
5(20)
2.1 A battle horse for numerical computations
5(1)
2.2 Linear equations with errors in the data
6(2)
2.3 Linear equations with convex constraints
8(2)
2.4 Inversion of Laplace transforms from finite number of data points
10(1)
2.5 Fourier reconstruction from partial data
11(1)
2.6 More on the non-continuity of the inverse
12(1)
2.7 Transportation problems and reconstruction from marginals
13(2)
2.8 CAT
15(5)
2.9 Abstract spline interpolation
20(1)
2.10 Bibliographical comments and references
21(4)
3 The basics about linear inverse problems
25(12)
3.1 Problem statements
25(5)
3.2 Quasi solutions and variational methods
30(1)
3.3 Regularization and approximate solutions
31(4)
3.4 Appendix
35(1)
3.5 Bibliographical comments and references
36(1)
4 Regularization in Hilbert spaces: Deterministic and stochastic approaches
37(16)
4.1 Basics
37(3)
4.2 Tikhonov's regularization scheme
40(4)
4.3 Spectral cutoffs
44(2)
4.4 Gaussian regularization of inverse problems
46(2)
4.5 Bayesian methods
48(1)
4.6 The method of maximum likelihood
49(2)
4.7 Bibliographical comments and references
51(2)
5 Maxentropic approach to linear inverse problems
53(34)
5.1 Heuristic preliminaries
53(5)
5.2 Some properties of the entropy functionals
58(1)
5.3 The direct approach to the entropic maximization problem
59(3)
5.4 A more detailed analysis
62(2)
5.5 Convergence of maxentropic estimates
64(3)
5.6 Maxentropic reconstruction in the presence of noise
67(3)
5.7 Maxentropic reconstruction of signal and noise
70(2)
5.8 Maximum entropy according to Dacunha-Castelle and Gamboa. Comparison with Jaynes' classical approach
72(7)
5.8.1 Basic results
72(5)
5.8.2 Jaynes' and Dacunha and Gamboa's approaches
77(2)
5.9 MEM under translation
79(1)
5.10 Maxent reconstructions under increase of data
80(2)
5.11 Bibliographical comments and references
82(5)
6 Finite dimensional problems
87(28)
6.1 Two classical methods of solution
87(3)
6.2 Continuous time iteration schemes
90(1)
6.3 Incorporation of convex constraints
91(7)
6.3.1 Basics and comments
91(4)
6.3.2 Optimization with differentiable non-degenerate equality constraints
95(2)
6.3.3 Optimization with differentiate, non-degenerate inequality constraints
97(1)
6.4 The method of projections in continuous time
98(1)
6.5 Maxentropic approaches
99(13)
6.5.1 Linear systems with band constraints
100(2)
6.5.2 Linear system with Euclidean norm constraints
102(2)
6.5.3 Linear systems with non-Euclidean norm constraints
104(1)
6.5.4 Linear systems with solutions in unbounded convex sets
105(4)
6.5.5 Linear equations without constraints
109(3)
6.6 Linear systems with measurement noise
112(1)
6.7 Bibliographical comments and references
113(2)
7 Some simple numerical examples and moment problems
115(54)
7.1 The density of the Earth
115(10)
7.1.1 Solution by the standard L2 [ 0, 1] techniques
116(1)
7.1.2 Piecewise approximations in L2([ 0, 1])
117(1)
7.1.3 Linear programming approach
118(2)
7.1.4 Maxentropic reconstructions: Influence of a priori data
120(2)
7.1.5 Maxentropic reconstructions: Effect of the noise
122(3)
7.2 A test case
125(16)
7.2.1 Standard L2[ 0, 1] technique
126(1)
7.2.2 Discretized L2[ 0, 1] approach
127(1)
7.2.3 Maxentropic reconstructions: Influence of a priori data
128(3)
7.2.4 Reconstruction by means of cubic splines
131(4)
7.2.5 Fourier versus cubic splines
135(6)
7.3 Standard maxentropic reconstruction
141(5)
7.3.1 Existence and stability
144(2)
7.3.2 Some convergence issues
146(1)
7.4 Some remarks on moment problems
146(6)
7.4.1 Some remarks about the Hamburger and Stieltjes moment problems
149(3)
7.5 Moment problems in Hilbert spaces
152(2)
7.6 Reconstruction of transition probabilities
154(2)
7.7 Probabilistic approach to Hausdorff's moment problem
156(2)
7.8 The very basics about cubic splines
158(1)
7.9 Determination of risk measures from market price of risk
159(5)
7.9.1 Basic aspects of the problem
159(2)
7.9.2 Problem statement
161(1)
7.9.3 The maxentropic solution
162(1)
7.9.4 Description of numerical results
163(1)
7.10 Bibliographical comments and references
164(5)
8 Some infinite dimensional problems
169(16)
8.1 A simple integral equation
169(9)
8.1.1 The random function approach
170(3)
8.1.2 The random measure approach: Gaussian measures
173(1)
8.1.3 The random measure approach: Compound Poisson measures
174(2)
8.1.4 The random measure approach: Gaussian fields
176(1)
8.1.5 Closing remarks
177(1)
8.2 A simple example: Inversion of a Fourier transform given a few coefficients
178(1)
8.3 Maxentropic regularization for problems in Hilbert spaces
179(5)
8.3.1 Gaussian measures
179(3)
8.3.2 Exponential measures
182(1)
8.3.3 Degenerate measures in Hilbert spaces and spectral cut off regularization
183(1)
8.3.4 Conclusions
184(1)
8.4 Bibliographical comments and references
184(1)
9 Tomography, reconstruction from marginals and transportation problems
185(30)
9.1 Generalities
185(2)
9.2 Reconstruction from marginals
187(1)
9.3 A curious impossibility result and its counterpart
188(4)
9.3.1 The bad news
188(2)
9.3.2 The good news
190(2)
9.4 The Hilbert space set up for the tomographic problem
192(2)
9.4.1 More on nonuniquenes of reconstructions
194(1)
9.5 The Russian Twist
194(1)
9.6 Why does it work
195(3)
9.7 Reconstructions using (classical) entropic, penalized methods in Hilbert space
198(3)
9.8 Some maxentropic computations
201(2)
9.9 Maxentropic approach to reconstruction from marginals in the discrete case
203(6)
9.9.1 Reconstruction from marginals by maximum entropy on the mean
204(3)
9.9.2 Reconstruction from marginals using the standard maximum entropy method
207(2)
9.10 Transportation and linear programming problems
209(2)
9.11 Bibliographical comments and references
211(4)
10 Numerical inversion of Laplace transforms
215(26)
10.1 Motivation
215(1)
10.2 Basics about Laplace transforms
216(2)
10.3 The inverse Laplace transform is not continuous
218(1)
10.4 A method of inversion
218(4)
10.4.1 Expansion in sine functions
219(1)
10.4.2 Expansion in Legendre polynomials
220(1)
10.4.3 Expansion in Laguerre polynomials
221(1)
10.5 From Laplace transforms to moment problems
222(1)
10.6 Standard maxentropic approach to the Laplace inversion problem
223(2)
10.7 Maxentropic approach in function space: The Gaussian case
225(2)
10.8 Maxentropic linear splines
227(2)
10.9 Connection with the complex interpolation problem
229(1)
10.10 Numerical examples
230(6)
10.11 Bibliographical comments and references
236(5)
11 Maxentropic characterization of probability distributions
241(8)
11.1 Preliminaries
241(2)
11.2 Example 1
243(1)
11.3 Example 2
244(1)
11.4 Example 3
245(1)
11.5 Example 4
245(1)
11.6 Example 5
246(1)
11.7 Example 6
246(3)
12 Is an image worth a thousand words?
249(12)
12.1 Problem setup
249(2)
12.1.1 List of questions for you to answer
251(1)
12.2 Answers to the questions
251(7)
12.2.1 Introductory comments
251(1)
12.2.2 Answers
251(7)
12.3 Bibliographical comments and references
258(3)
Appendix A Basic topology
261(4)
Appendix B Basic measure theory and probability
265(14)
B.1 Some results from measure theory and integration
265(7)
B.2 Some probabilistic jargon
272(3)
B.3 Brief description of the Kolmogorov extension theorem
275(1)
B.4 Basic facts about Gaussian process in Hilbert spaces
276(3)
Appendix C Banach spaces
279(14)
C.1 Basic stuff
279(2)
C.2 Continuous linear operator on Banach spaces
281(2)
C.3 Duality in Banach spaces
283(6)
C.4 Operators on Hilbert spaces. Singular values decompositions
289(1)
C.5 Some convexity theory
290(3)
Appendix D Further properties of entropy functionals
293(20)
D.1 Properties of entropy functionals
293(4)
D.2 A probabilistic connection
297(4)
D.3 Extended definition of entropy
301(1)
D.4 Exponetial families and geometry in the space of probabilities
302(8)
D.4.1 The geometry on the set of positive vectors
304(2)
D.4.2 Lifting curves from G+ to G and parallel transport
306(1)
D.4.3 From geodesies to Kullback's divergence
307(1)
D.4.4 Coordinates on P
308(2)
D.5 Bibliographical comments and references
310(3)
Appendix E Software user guide
313
E.1 Installation procedure
313(3)
E.2 Quick start guide
316
E.2.1 Moment problems with MEM
317(1)
E.2.2 Moment problems with SME
318(1)
E.2.3 Moment problems with Quadratic Programming
318(1)
E.2.4 Transition probabilities problem with MEM
319(1)
E.2.5 Transition probabilities problem with SME
320(1)
E.2.6 Transition probabilities problem with Quadratic Programming
320(1)
E.2.7 Reconstruction from Marginals with MEM
320(1)
E.2.8 Reconstruction from Marginals with SME
321(1)
E.2.9 Reconstruction from Marginals with Quadratic Programming
321(1)
E.2.10 A generic problem in the form Ax = y, with MEM
322(1)
E.2.11 A generic problem in the form Ax = y, with SME
323(1)
E.2.12 A generic problem in the form Ax = y, with Quadratic Programming
323(1)
E.2.13 The results windows
323(1)
E.2.14 Messages that will appear
324(2)
E.2.15 Comments
326
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