Dynamical Systems Method and Applications: Theoretical Developments and Numerical Examples

Dynamical Systems Method and Applications: Theoretical Developments and Numerical Examples......Page 5
CONTENTS......Page 11
List of Figures......Page 19
List of Tables......Page 21
Preface......Page 23
Acknowledgments......Page 31
PART I......Page 33
1.1 What this book is about......Page 35
1.2 What the DSM (Dynamical Systems Method) is......Page 36
1.3 The scope of the DSM......Page 37
1.5 Motivations......Page 41
2.1 Basic definitions. Examples......Page 43
2.2 Variational regularization......Page 62
2.3 Quasi-solutions......Page 72
2.4 Iterative regularization......Page 76
2.5 Quasi-inversion......Page 79
2.6 Dynamical systems method (DSM)......Page 82
2.7 Variational regularization for nonlinear equations......Page 85
3.1 Every solvable well-posed problem can be solved by DSM......Page 89
3.2 DSM and Newton-type methods......Page 93
3.3 DSM and the modified Newton\'s method......Page 95
3.4 DSM and Gauss–Newton-type methods......Page 96
3.6 DSM and the simple iterations method......Page 97
3.7 DSM and minimization methods......Page 98
3.8 Ulm\'s method......Page 100
4.1 Equations with bounded operators......Page 103
4.2 Another approach......Page 113
4.3 Equations with unbounded operators......Page 119
4.4 Iterative methods......Page 120
4.5 Stable calculation of values of unbounded operators......Page 122
5.1 Basic nonlinear differential inequality......Page 125
5.2 An operator inequality......Page 129
5.3 A nonlinear inequality......Page 131
5.4 The Gronwall-type inequalities......Page 134
5.5 Another operator inequality......Page 135
5.6 A generalized version of the basic nonlinear inequality......Page 136
5.6.1 Formulations and results......Page 137
5.6.2 Applications......Page 143
5.7 Some nonlinear inequalities and applications......Page 146
5.7.1 Formulations and results......Page 147
5.7.2 Applications......Page 157
6.1 Auxiliary results......Page 165
6.2 Formulation of the results and proofs......Page 171
6.3 The case of noisy data......Page 174
7.1 Formulation of the problem. The results and proofs......Page 177
7.2 Noisy data......Page 180
7.3 Iterative solution......Page 182
7.4 Stability of the iterative solution......Page 185
8.1 Spectral assumption......Page 187
8.2 Existence of a solution to a nonlinear equation......Page 190
9.1 Well-posed problems......Page 193
9.2 Ill-posed problems......Page 195
9.3 Singular perturbation problem......Page 196
10.1 Well-posed problems......Page 201
10.2 Ill-posed problems......Page 204
11.1 Statement of the problem......Page 209
11.2 Ill-posed problems......Page 211
12.1 Formulation of the results......Page 213
12.2 Proofs......Page 220
13.1 Surjectivity of nonlinear maps......Page 227
13.2 When is a local homeomorphism a global one?......Page 228
14.1 Introduction......Page 233
14.2 Iterative solution of well-posed problems......Page 234
14.3 Iterative solution of ill-posed equations with monotone operator......Page 236
14.4 Iterative methods for solving nonlinear equations......Page 239
14.5 Ill-posed problems......Page 242
15.1 Stable numerical differentiation......Page 245
15.2 Stable differentiation of piecewise-smooth functions......Page 253
15.3 Simultaneous approximation of a function and its derivative by interpolation polynomials......Page 263
15.4 Other methods of stable differentiation......Page 269
15.5 DSM and stable differentiation......Page 273
15.6 Stable calculating singular integrals......Page 279
PART II......Page 285
16.1 An iterative scheme for solving linear operator equations......Page 287
16.2 DSM with fast decaying regularizing function......Page 291
17 DSM of gradient type for solving linear operator equations......Page 301
17.1.1 Exact data......Page 302
17.1.2 Noisy data fδ......Page 303
17.1.3 Discrepancy principle......Page 304
17.2.1 Systems with known spectral decomposition......Page 308
17.2.2 On the choice of t0......Page 311
18 DSM for solving linear equations with finite-rank operators......Page 313
18.1.1 Exact data......Page 314
18.1.2 Noisy data fδ......Page 315
18.1.3 Discrepancy principle......Page 316
18.1.4 An iterative scheme......Page 320
18.1.5 An iterative scheme with a stopping rule based on a discrepancy principle......Page 322
18.1.6 Computing uδ(tδ)......Page 325
19 A discrepancy principle for equations with monotone continuous operators......Page 327
19.1 Auxiliary results......Page 328
19.2 A discrepancy principle......Page 331
19.3 Applications......Page 333
20 DSM of Newton-type for solving operator equations with minimal smoothness assumptions......Page 339
20.1 DSM of Newton-type......Page 340
20.1.1 Inverse function theorem......Page 342
20.1.2 Convergence of the DSM......Page 344
20.1.3 The Newton method......Page 347
20.2 A justification of the DSM for global homeomorphisms......Page 352
20.3 DSM of Newton-type for solving nonlinear equations with monotone operators......Page 354
20.3.1 Existence of solution and a justification of the DSM for exact data......Page 355
20.3.2 Solving equations with monotone operators when the data are noisy......Page 361
20.4 Implicit Function Theorem and the DSM......Page 370
20.4.1 Example......Page 375
21 DSM of gradient type......Page 379
21.1 Auxiliary results......Page 381
21.2 DSM gradient method......Page 385
21.3 An iterative scheme......Page 391
22.1 DSM of simple iteration type......Page 405
22.1.1 Auxiliary results......Page 406
22.1.2 Main results......Page 411
22.2 An iterative scheme for solving equations with σ-inverse monotone operators......Page 420
22.2.1 Auxiliary results......Page 422
22.2.2 Main results......Page 429
23 DSM for solving nonlinear operator equations in Banach spaces......Page 441
23.1 Proofs......Page 444
23.2 The case of continuous F\'(u)......Page 450
PART III......Page 453
24.1.1 Numerical experiments with Hilbert matrix......Page 455
24.2 Numerical experiments with Fredholm integral equations of the first kind......Page 457
24.2.1 Numerical experiments for computing second derivative......Page 458
24.3 Numerical experiments with an image restoration problem......Page 461
24.4.1 Numerical experiments with an inverse problem for the heat equation......Page 464
24.5.1 The first approach......Page 467
24.5.2 The second approach......Page 469
25.1.1 An experiment with an operator defined on H = L2[0, 1]......Page 473
25.1.2 An experiment with an operator defined on a dense subset of H = L2[0, 1]......Page 478
25.2 DSM of gradient type......Page 480
25.3 DSM of simple iteration type......Page 482
26.1 Introduction......Page 487
26.2 Description of the method......Page 490
26.2.1 Noisy data......Page 500
26.2.2 Stopping rule......Page 501
26.2.3 The algorithm......Page 506
26.3.1 The parameters k, a0, d......Page 507
26.3.2 Experiments......Page 508
26.4 Conclusion......Page 521
A.l Contraction mapping principle......Page 523
A.2 Existence and uniqueness of the local solution to the Cauchy problem......Page 527
A.3 Derivatives of nonlinear mappings......Page 531
A.4 Implicit function theorem......Page 534
A.5 An existence theorem......Page 536
A.6 Continuity of solutions to operator equations with respect to a parameter......Page 538
A.7 Monotone operators in Banach spaces......Page 542
A.8 Existence of solutions to operator equations......Page 545
A.9 Compactness of embeddings......Page 549
Appendix B: Bibliographical notes......Page 553
References......Page 557
Index......Page 569