Iterative Methods for Ill-Posed Problems: An Introduction

✍ Scribed by Anatoly B. Bakushinsky; Mihail Yu. Kokurin; Alexandra Smirnova

• Publisher: De Gruyter
• Year: 2010
• Tongue: English
• Leaves: 152
• Series: Inverse and Ill-Posed Problems Series; 54
• Category: Library

✦ Synopsis

Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions.

Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.

• Delivers a unified approach for the development of methods to solve real world problems
• Must have for solving problems for numerous problems in industry and research: gravimetry, tomography, nondestructive testing of materials, and airport security
• Excercises at the end of every chapter
  

✦ Table of Contents

Preface
Contents
1 The regularity condition. Newton's method
1.1 Preliminary results
1.2 Linearization procedure
1.3 Error analysis
Problems
2 The Gauss-Newton method
2.1 Motivation
2.2 Convergence rates
Problems
3 The gradient method
3.1 The gradient method for regular problems
3.2 Ill-posed case
Problems
4 Tikhonov's scheme
4.1 The Tikhonov functional
4.2 Properties of a minimizing sequence
4.3 Other types of convergence
4.4 Equations with noisy data
Problems
5 Tikhonov's scheme for linear equations
5.1 The main convergence result
5.2 Elements of spectral theory
5.3 Minimizing sequences for linear equations
5.4 A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides
5.5 The discrepancy principle
5.6 Approximation of a quasi-solution
Problems
6 The gradient scheme for linear equations
6.1 The technique of spectral analysis
6.2 A priori stopping rule
6.3 A posteriori stopping rule
Problems
7 Convergence rates for the approximation methods in the case of linear irregular equations
7.1 The source-type condition (STC)
7.2 STC for the gradient method
7.3 The saturation phenomena
7.4 Approximations in case of a perturbed STC
7.5 Accuracy of the estimates
Problems
8 Equations with a convex discrepancy functional by Tikhonov's method
8.1 Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional
8.2 An illustrative example
Problems
9 Iterative regularization principle
9.1 The idea of iterative regularization
9.2 The iteratively regularized gradient method
Problems
10 The iteratively regularized Gauss-Newton method
10.1 Convergence analysis
10.2 Further properties of IRGN iterations
10.3 A unified approach to the construction of iterative methods for irregular equations
10.4 The reverse connection control
Problems
11 The stable gradient method for irregular nonlinear equations
11.1 Solving an auxiliary finite dimensional problem by the gradient descent method
11.2 Investigation of a difference inequality
11.3 The case of noisy data
Problems
12 Relative computational efficiency of iteratively regularized methods
12.1 Generalized Gauss-Newton methods
12.2 A more restrictive source condition
12.3 Comparison to iteratively regularized gradient scheme
Problems
13 Numerical investigation of two-dimensional inverse gravimetry problem
13.1 Problem formulation
13.2 The algorithm
13.3 Simulations
Problems
14 Iteratively regularized methods for inverse problem in optical tomography
14.1 Statement of the problem
14.2 Simple example
14.3 Forward simulation
14.4 The inverse problem
14.5 Numerical results
Problems
15 Feigenbaum's universality equation
15.1 The universal constants
15.2 Ill-posedness
15.3 Numerical algorithm for 2 < z < 12
15.4 Regularized method for z > 13
Problems
16 Conclusion
References
Index