Inverse and ill-posed problems : theory and applications

Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 6
Denotations......Page 10
Contents......Page 14
1 Basic concepts and examples......Page 18
B.1 Supplementary exercises and control questions......Page 0
1.2 Examples of inverse and ill-posed problems......Page 26
2 Ill-posed problems......Page 39
2.1 Well-posed and ill-posed problems......Page 41
2.2 On stability in different spaces......Page 42
2.3 Quasi-solution. The Ivanov theorems......Page 45
2.4 The Lavrentiev method......Page 48
2.5 The Tikhonov regularization method......Page 51
2.6 Gradient methods......Page 59
2.7 An estimate of the convergence rate with respect to the objective functional......Page 66
2.8 Conditional stability estimate and strong convergence of gradient methods applied to ill-posed problems......Page 70
2.9 The pseudoinverse and the singular value decomposition of an operator......Page 79
3 Ill-posed problems of linear algebra......Page 85
3.1 Generalization of the concept of a solution. Pseudo-solutions......Page 87
3.2 Regularization method......Page 89
3.3 Criteria for choosing the regularization parameter......Page 94
3.5 Singular value decomposition......Page 96
3.6 The singular value decomposition algorithm and the Godunov method......Page 104
3.7 The square root method......Page 108
3.8 Exercises......Page 109
4 Integral equations......Page 115
4.2 Regularization of linear Volterra integral equations of the first kind......Page 121
4.3 Volterra operator equations with boundedly Lipschitz-continuous kernel......Page 128
4.4 Local well-posedness and uniqueness on the whole......Page 133
4.5 Well-posedness in a neighborhood of the exact solution......Page 135
4.6 Regularization of nonlinear operator equations of the first kind......Page 139
5 Integral geometry......Page 146
5.1 The Radon problem......Page 147
5.2 Reconstructing a function from its spherical means......Page 155
5.3 Determining a function of a single variable from the values of its integrals. The problem of moments......Page 156
5.4 Inverse kinematic problem of seismology......Page 161
6 Inverse spectral and scattering problems......Page 171
6.1 Direct Sturm-Liouville problem on a finite interval......Page 173
6.2 Inverse Sturm-Liouville problems on a finite interval......Page 180
6.3 The Gelfand-Levitan method on a finite interval......Page 183
6.4 Inverse scattering problems......Page 189
6.5 Inverse scattering problems in the time domain......Page 197
7 Linear problems for hyperbolic equations......Page 204
7.2 The Cauchy problem for a hyperbolic equation with data on a time-like surface......Page 207
7.3 The inverse thermoacoustic problem......Page 209
7.4 Linearized multidimensional inverse problem for the wave equation......Page 210
8 Linear problems for parabolic equations......Page 226
8.2 Inverse problem of heat conduction with reverse time (retrospective inverse problem)......Page 231
8.3 Inverse boundary-value problems and extension problems......Page 244
8.4 Interior problems and problems of determining sources......Page 245
9 Linear problems for elliptic equations......Page 250
9.1 The uniqueness theorem and a conditional stability estimate on a plane......Page 251
9.2 Formulation of the initial boundary value problem for the Laplace equation in the form of an inverse problem. Reduction to an operator equation......Page 255
9.3 Analysis of the direct initial boundary value problem for the Laplace equation......Page 256
9.4 The extension problem for an equation with self-adjoint elliptic operator......Page 261
10 Inverse coefficient problems for hyperbolic equations......Page 266
10.2 Inverse problems of acoustics......Page 289
10.3 Inverse problems of electrodynamics......Page 303
10.4 Local solvability of multidimensional inverse problems......Page 311
10.5 Method of the Neumann to Dirichlet maps in the half-space......Page 319
10.6 An approach to inverse problems of acoustics using geodesic lines......Page 323
10.7 Two-dimensional analog of the Gelfand-Levitan-Krein equation......Page 332
11 Inverse coefficient problems for parabolic and elliptic equations......Page 336
11.2 Reducing to spectral inverse problems......Page 338
11.3 Uniqueness theorems......Page 340
11.4 An overdetermined inverse coefficient problem for the elliptic equation. Uniqueness theorem......Page 344
11.5 An inverse problem in a semi-infinite cylinder......Page 345
Appendix A......Page 348
A.2 Operators......Page 367
A.3 Dual space and adjoint operator......Page 388
A.4 Elements of differential calculus in Banach spaces......Page 399
A.5 Functional spaces......Page 402
A.6 Equations of mathematical physics......Page 417
Appendix B......Page 428
B.2 Supplementary references......Page 430
Epilogue......Page 448
Bibliography......Page 450
Index......Page 474