Linear Inverse Problems

The following sections are included:

• Preface

• Contents

• List of Figures

• List of Tables

In this volume we examine some basic aspects of linear inverse problems, examine some standard methods of solutions, and present the method of maximum entropy in the mean (MEM for short) as a powerful alternative method to solve constrained linear inverse problems. We shall see that the method takes care of the constraints in a natural way, and competes reasonably with the standard methods. As we shall eventually see below, MEM includes some standard methods as particular cases. We shall illustrate the range of applications with examples from many different fields…

The following sections are included:

• A battle horse for numerical computations

• Linear equations with errors in the data

• Linear equations with convex constraints

• Inversion of Laplace transforms from finite number of data points

• Fourier reconstruction from partial data

• More on the non-continuity of the inverse

• Transportation problems and reconstruction from marginals

• CAT

• Abstract spline interpolation

• Bibliographical comments and references

• References

The following sections are included:

• Problem statements

• Quasi solutions and variational methods

• Regularization and approximate solutions

• Appendix

• Bibliographical comments and references

• References

The following sections are included:

• Basics

• Tikhonov's regularization scheme

• Spectral cutoffs

• Gaussian regularization of inverse problems

• Bayesian methods

• The method of maximum likelihood

• Bibliographical comments and references

• References

The following sections are included:

• Heuristic preliminaries

• Some properties of the entropy functionals

• The direct approach to the entropic maximization problem

• A more detailed analysis

• Convergence of maxentropic estimates

• Maxentropic reconstruction in the presence of noise

• Maxentropic reconstruction of signal and noise

• Maximum entropy according to Dacunha-Castelle and Gamboa. Comparison with Jaynes' classical approach
  • Basic results
  • Jaynes' and Dacunha and Gamboa's approaches

• MEM under translation

• Maxent reconstructions under increase of data

• Bibliographical comments and references

• References

The following sections are included:

• Two classical methods of solution

• Continuous time iteration schemes

• Incorporation of convex constraints
  • Basics and comments
  • Optimization with differentiable non-degenerate equality constraints
  • Optimization with differentiable, non-degenerate inequality constraints

• The method of projections in continuous time

• Maxentropic approaches
  • Linear systems with band constraints
  • Linear system with Euclidean norm constraints
  • Linear systems with non-Euclidean norm constraints
  • Linear systems with solutions in unbounded convex sets
  • Linear equations without constraints
    • Gaussian reference measure
    • The Laplace reference measure

• Linear systems with measurement noise

• Bibliographical comments and references

• References

The following sections are included:

• The density of the Earth
  • Solution by the standard L₂[0, 1] techniques
  • Piecewise approximations in L₂([0, 1])
  • Linear programming approach
  • Maxentropic reconstructions: Influence of a priori data
  • Maxentropic reconstructions: Effect of the noise

• A test case
  • Standard L₂[0, 1] technique
  • Discretized L₂[0, 1] approach
  • Maxentropic reconstructions: Influence of a priori data
  • Reconstruction by means of cubic splines
  • Fourier versus cubic splines

• Standard maxentropic reconstruction
  • Existence and stability
  • Some convergence issues

• Some remarks on moment problems
  • Some remarks about the Hamburger and Stieltjes moment problems

• Moment problems in Hilbert spaces

• Reconstruction of transition probabilities

• Probabilistic approach to Hausdorff's moment problem

• The very basics about cubic splines

• Determination of risk measures from market price of risk
  • Basic aspects of the problem
    • Examples of distortion functions
  • Problem statement
    • Problem statement
    • Problem discretization
  • The maxentropic solution
  • Description of numerical results
    • A consistency test

• Bibliographical comments and references

• References

The following sections are included:

• A simple integral equation
  • The random function approach
  • The random measure approach: Gaussian measures
  • The random measure approach: Compound Poisson measures
  • The random measure approach: Gaussian fields
  • Closing remarks

• A simple example: Inversion of a Fourier transform given a few coefficients

• Maxentropic regularization for problems in Hilbert spaces
  • Gaussian measures
  • Exponential measures
  • Degenerate measures in Hilbert spaces and spectral cut off regularization
  • Conclusions

• Bibliographical comments and references

• Reference

The following sections are included:

• Generalities

• Reconstruction from marginals

• A curious impossibility result and its counterpart
  • The bad news
  • The good news

• The Hilbert space set up for the tomographic problem
  • More on nonuniquenes of reconstructions

• The Russian Twist

• Why does it work

• Reconstructions using (classical) entropic, penalized methods in Hilbert space

• Some maxentropic computations

• Maxentropic approach to reconstruction from marginals in the discrete case
  • Reconstruction from marginals by maximum entropy on the mean
  • Reconstruction from marginals using the standard maximum entropy method

• Transportation and linear programming problems

• Bibliographical comments and references

• References

The following sections are included:

• Motivation

• Basics about Laplace transforms

• The inverse Laplace transform is not continuous

• A method of inversion
  • Expansion in sine functions
  • Expansion in Legendre polynomials
  • Expansion in Laguerre polynomials

• From Laplace transforms to moment problems

• Standard maxentropic approach to the Laplace inversion problem

• Maxentropic approach in function space: The Gaussian case

• Maxentropic linear splines

• Connection with the complex interpolation problem

• Numerical examples

• Bibliographical comments and references

• References

The following sections are included:

• Preliminaries

• Example 1

• Example 2

• Example 3

• Example 4

• Example 5

• Example 6

• Reference

The following sections are included:

• Problem setup
  • List of questions for you to answer

• Answers to the questions
  • Introductory comments
  • Answers

• Bibliographical comments and references

• References

The following sections are included:

• Appendix A Basic topology

• Appendix B Basic measure theory and probability

• Appendix C Banach spaces

• Appendix D Further properties of entropy functionals

• Appendix E Software user guide