Inverse and ill-posed problems
Subtitle: theory and applications
By: Sergey I. Kabanikhin
Contributors: Walter de Gruyter & Co
Series: Inverse and ill-posed problems series -- 55
Publisher: De Gruyter
Published: 2011
Place of publication: Berlin
Language: eng
Format: [electronic resource] :
Number of pages: 475
ISBN-13: 9783110224016
Subjects: Boundary value problems; Inverse problems (Differential equations); MATHEMATICS / Differential Equations / General; Improperly posed problems
Description: The text demonstrates the methods for proving the existence (if et 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.
Notes: 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.

Includes bibliographical references and index.

Also available in print.

Description based on print version record.
Table of contents: title: 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.; title: 2.9 The pseudoinverse and the singular value decomposition of an operator3 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.; title: 4.3 Volterra operator equations with boundedly Lipschitz-continuous kernel4.4 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.; title: 6.1 Direct Sturm-Liouville problem on a finite interval6.2 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.; title: 8.1 On the formulation of inverse problems for parabolic equations and their relationship with the corresponding inverse problems for hyperbolic equations8.2 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.