Inverse and Ill-posed Problems (eBook)

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Sergey I. Kabanikhin, Sobolev Institute of Mathematics, Novosibirsk, Russia.

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