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Inverse and Ill-posed Problems Theory and Applications

:	Sergey I. Kabanikhin
:	Inverse and Ill-posed Problems Theory and Applications
:	Walter de Gruyter GmbH& Co.KG
:	9783110224016
:	Inverse and Ill-Posed Problems SeriesISSN
:	1
:	CHF 161.70
:

:	Allgemeines, Lexika
:	English

:	475
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

< >The text demonstrates the methods for proving the existence (if at all) and finding of inverse and ill-posed problems solutions in linear algebra, integral and operator equations, integral geometry, spectral inverse problems, and inverse scattering problems. It is given comprehensive background material for linear ill-posed problems and for coefficient inverse problems for hyperbolic, parabolic, and elliptic equations. A lot of examples for inverse problems from physics, geophysics, biology, medicine, and other areas of application of mathematics are included.

< >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