Inverse and Ill-posed Problems : Theory and Applications.

Preface -- Denotations -- 1 Basic concepts and examples -- 1.1 On the definition of inverse and ill-posed problems -- 1.2 Examples of inverse and ill-posed problems -- 2 Ill-posed problems -- 2.1 Well-posed and ill-posed problems -- 2.2 On stability in different spaces -- 2.3 Quasi-solution. The Ivanov theorems -- 2.4 The Lavrentiev method -- 2.5 The Tikhonov regularization method -- 2.6 Gradient methods -- 2.7 An estimate of the convergence rate with respect to the objective functional -- 2.8 Conditional stability estimate and strong convergence of gradient methods applied to ill-posed problems -- 2.9 The pseudoinverse and the singular value decomposition of an operator -- 3 Ill-posed problems of linear algebra -- 3.1 Generalization of the concept of a solution. Pseudo-solutions -- 3.2 Regularization method -- 3.3 Criteria for choosing the regularization parameter -- 3.4 Iterative regularization algorithms -- 3.5 Singular value decomposition -- 3.6 The singular value decomposition algorithm and the Godunov method -- 3.7 The square root method -- 3.8 Exercises -- 4 Integral equations -- 4.1 Fredholm integral equations of the first kind -- 4.2 Regularization of linear Volterra integral equations of the first kind -- 4.3 Volterra operator equations with boundedly Lipschitz-continuous kernel -- 4.4 Local well-posedness and uniqueness on the whole -- 4.5 Well-posedness in a neighborhood of the exact solution -- 4.6 Regularization of nonlinear operator equations of the first kind -- 5 Integral geometry -- 5.1 The Radon problem -- 5.2 Reconstructing a function from its spherical means -- 5.3 Determining a function of a single variable from the values of its integrals. The problem of moments -- 5.4 Inverse kinematic problem of seismology -- 6 Inverse spectral and scattering problems.

6.1 Direct Sturm-Liouville problem on a finite interval -- 6.2 Inverse Sturm-Liouville problems on a finite interval -- 6.3 The Gelfand-Levitan method on a finite interval -- 6.4 Inverse scattering problems -- 6.5 Inverse scattering problems in the time domain -- 7 Linear problems for hyperbolic equations -- 7.1 Reconstruction of a function from its spherical means -- 7.2 The Cauchy problem for a hyperbolic equation with data on a time-like surface -- 7.3 The inverse thermoacoustic problem -- 7.4 Linearized multidimensional inverse problem for the wave equation -- 8 Linear problems for parabolic equations -- 8.1 On the formulation of inverse problems for parabolic equations and their relationship with the corresponding inverse problems for hyperbolic equations -- 8.2 Inverse problem of heat conduction with reverse time (retrospective inverse problem) -- 8.3 Inverse boundary-value problems and extension problems -- 8.4 Interior problems and problems of determining sources -- 9 Linear problems for elliptic equations -- 9.1 The uniqueness theorem and a conditional stability estimate on a plane -- 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 -- 9.3 Analysis of the direct initial boundary value problem for the Laplace equation -- 9.4 The extension problem for an equation with self-adjoint elliptic operator -- 10 Inverse coefficient problems for hyperbolic equations -- 10.1 Inverse problems for the equation utt = uxx - q(x)u + F(x,t) -- 10.2 Inverse problems of acoustics -- 10.3 Inverse problems of electrodynamics -- 10.4 Local solvability of multidimensional inverse problems -- 10.5 Method of the Neumann to Dirichlet maps in the half-space.

10.6 An approach to inverse problems of acoustics using geodesic lines -- 10.7 Two-dimensional analog of the Gelfand-Levitan-Krein equation -- 11 Inverse coefficient problems for parabolic and elliptic equations -- 11.1 Formulation of inverse coefficient problems for parabolic equations. Association with those for hyperbolic equations -- 11.2 Reducing to spectral inverse problems -- 11.3 Uniqueness theorems -- 11.4 An overdetermined inverse coefficient problem for the elliptic equation. Uniqueness theorem -- 11.5 An inverse problem in a semi-infinite cylinder -- Appendix A -- A.1 Spaces -- A.2 Operators -- A.3 Dual space and adjoint operator -- A.4 Elements of differential calculus in Banach spaces -- A.5 Functional spaces -- A. 6 Equations of mathematical physics -- Appendix B -- B.1 Supplementary exercises and control questions -- B.2 Supplementary references -- Epilogue -- Bibliography -- Index.