Iterative methods for ill-posed problems : an introduction

✍ Scribed by A B BakushinskiiМ†; M Iпё UпёЎ Kokurin; A B Smirnova

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 150
• Series: Inverse and ill-posed problems series, v. 54
• Category: Library

✦ Synopsis

Machine generated contents note: 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

✦ Table of Contents

Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 10
1.1 Preliminary results......Page 14
1.2 Linearization procedure......Page 15
1.3 Error analysis......Page 17
Problems......Page 19
2.1 Motivation......Page 23
2.2 Convergence rates......Page 25
Problems......Page 27
3.1 The gradient method for regular problems......Page 29
3.2 Ill-posed case......Page 31
Problems......Page 33
4.1 The Tikhonov functional......Page 36
4.2 Properties of a minimizing sequence......Page 37
4.3 Other types of convergence......Page 40
4.4 Equations with noisy data......Page 42
Problems......Page 43
5.1 The main convergence result......Page 45
5.2 Elements of spectral theory......Page 47
5.3 Minimizing sequences for linear equations......Page 48
5.4 A priori agreement between the regularization parameter and the error for equations with perturbed right-hand sides......Page 50
5.5 The discrepancy principle......Page 53
Problems......Page 56
6.1 The technique of spectral analysis......Page 58
6.2 A priori stopping rule......Page 61
6.3 A posteriori stopping rule......Page 62
Problems......Page 66
7.1 The source-type condition (STC)......Page 67
7.2 STC for the gradient method......Page 70
7.3 The saturation phenomena......Page 72
7.4 Approximations in case of a perturbed STC......Page 74
7.5 Accuracy of the estimates......Page 75
Problems......Page 76
8.1 Some difficulties associated with Tikhonov's method in case of a convex discrepancy functional......Page 77
8.2 An illustrative example......Page 78
Problems......Page 80
9.1 The idea of iterative regularization......Page 82
9.2 The iteratively regularized gradient method......Page 83
Problems......Page 87
10.1 Convergence analysis......Page 89
10.2 Further properties of IRGN iterations......Page 92
10.3 A unified approach to the construction of iterative methods for irregular equations......Page 96
10.4 The reverse connection control......Page 97
Problems......Page 101
11.1 Solving an auxiliary finite dimensional problem by the gradient descent method......Page 103
11.2 Investigation of a difference inequality......Page 107
11.3 The case of noisy data......Page 108
Problems......Page 110
12.1 Generalized Gauss-Newton methods......Page 111
12.2 A more restrictive source condition......Page 113
12.3 Comparison to iteratively regularized gradient scheme......Page 114
Problems......Page 115
13.1 Problem formulation......Page 116
13.2 The algorithm......Page 117
13.3 Simulations......Page 118
Problems......Page 122
14.1 Statement of the problem......Page 124
14.2 Simple example......Page 125
14.3 Forward simulation......Page 127
14.4 The inverse problem......Page 129
14.5 Numerical results......Page 132
Problems......Page 134
15.1 The universal constants......Page 136
15.3 Numerical algorithm for 2 <= z <= 12......Page 138
15.4 Regularized method for z >= 13......Page 140
Problems......Page 141
16 Conclusion......Page 143
References......Page 145
Index......Page 150