Author 
Kabanikhin, S. I.

Series 
Inverse and illposed problems series ; 55 

Inverse and illposed problems series ;
v. 55.

Subject 
Inverse problems (Differential equations)


Boundary value problems  Improperly posed problems.

Description 
1 online resource (xv, 475 pages) : illustrations. 
Bibliography Note 
Includes bibliographical references and index. 
Contents 
Preface; Denotations; 1 Basic concepts and examples; 1.1 On the definition of inverse and illposed problems; 1.2 Examples of inverse and illposed problems; 2 Illposed problems; 2.1 Wellposed and illposed problems; 2.2 On stability in different spaces; 2.3 Quasisolution. 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 illposed problems. 

2.9 The pseudoinverse and the singular value decomposition of an operator3 Illposed problems of linear algebra; 3.1 Generalization of the concept of a solution. Pseudosolutions; 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 Lipschitzcontinuous kernel4.4 Local wellposedness and uniqueness on the whole; 4.5 Wellposedness 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 SturmLiouville problem on a finite interval6.2 Inverse SturmLiouville problems on a finite interval; 6.3 The GelfandLevitan 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 timelike 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 equations8.2 Inverse problem of heat conduction with reverse time (retrospective inverse problem); 8.3 Inverse boundaryvalue 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. 
Note 
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. 
Summary 
The text demonstrates the methods for proving the existence (if et all) and finding of inverse and illposed 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 illposed 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. 
Note 
Print version record. 
ISBN 
9783110224016 (electronic bk.) 

3110224011 (electronic bk.) 

3110224003 

9783110224009 

9783110224009 (alk. paper) 
OCLC # 
772845127 
Additional Format 
Print version: Kabanikhin, S I. Inverse and Illposed Problems : Theory and Applications. Berlin : De Gruyter, ©2011 9783110224009 
