Inverse and ill-posed problems. Theory and applications.
type: Item
instance of: scholarly article
title: Inverse and ill-posed problems. Theory and applications.
zbMATH Open document ID: 1247.65077
DOI: 10.1515/9783110224016
published in: Inverse and Ill-Posed Problems Series
publication date: 2011-08-11T00:00:00Z
review text: This monograph provides a well-structured and comprehensive survey of various types of inverse problems. In the sequel we summarize some of the key features of this book. The first chapter is devoted to a classification of inverse problems (inverse boundary value problems, inverse source problems, coefficient problems, reverse time problems), and classes of well-posed problems and their ill-posed counterparts are presented. Simple examples are also considered to present first ideas of ill-posedness. Chapter 2 provides a brief introduction to the basic theory of ill-posed problems and their regularization. Topics include but are not restricted to the modulus of continuity, quasi-solutions, Lavrentiev and Tikhonov regularization, general regularization schemes, parameter choice strategies for Tikhonov regularization to solve perturbed equations (including operator perturbations), gradient methods, the conjugate gradient method, the pseudoinverse, and the singular value decomposition. The section on gradient methods contains a comprehensive convergence theory. In Chapter 3, the fundamental concepts for solving ill-conditioned finite-dimensional linear equations are presented. This includes pseudo-solutions, Lavrentiev and Tikhonov regularization, residual principles, the singular value decomposition and Godunov's method. In Chapter 4, integral equations are considered. Lavrentiev regularization for integral equations of the first kind is considered first, both for symmetric Fredholm equations with Hilbert-Schmidt kernels and Volterra equations with smooth kernels. Then for a class of nonlinear Volterra integral equations of the second kind, existence, uniqueness and conditional stability of solutions is considered. Such equations arise, e.g., after some reformulation of a coefficient problem for the wave equation. The chapter concludes with the regularization of related nonlinear Volterra equations of the first kind. Here, Lavrentiev's method and a difference scheme ar…
reviewed by: Robert Plato
Mathematics Subject Classification ID: 65J22
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 45Q05
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65F22
Mathematics Subject Classification ID: 65M30
Mathematics Subject Classification ID: 65M32
Mathematics Subject Classification ID: 65N20
Mathematics Subject Classification ID: 65N21
Mathematics Subject Classification ID: 80A23
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 35Q61
Mathematics Subject Classification ID: 44A12
Mathematics Subject Classification ID: 65R10
Mathematics Subject Classification ID: 65R30
Mathematics Subject Classification ID: 65R32
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 45-02
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 34A55
zbMATH DE Number: 5938009
zbMATH Keywords: inverse problem
zbMATH Keywords: ill-posed problem
zbMATH Keywords: quasi-solution
zbMATH Keywords: pseudo-solution
zbMATH Keywords: regularization method
zbMATH Keywords: parameter choice strategy
zbMATH Keywords: residual principle
zbMATH Keywords: Tikhonov method
zbMATH Keywords: Lavrentiev method
zbMATH Keywords: Landweber iteration
zbMATH Keywords: singular value decomposition
zbMATH Keywords: Godunov method
zbMATH Keywords: gradient method
zbMATH Keywords: steepest descent method
zbMATH Keywords: minimum error method
zbMATH Keywords: conjugate gradient method
zbMATH Keywords: Newton's method
zbMATH Keywords: pseudoinverse
zbMATH Keywords: conditional stability
zbMATH Keywords: Fredholm integral equation of the first kind
zbMATH Keywords: Volterra integral equation of the first kind
zbMATH Keywords: Sturm-Liouville problem
zbMATH Keywords: inverse Sturm-Liouville problem
zbMATH Keywords: inverse scattering problem
zbMATH Keywords: Gelfand-Levitan method
zbMATH Keywords: Radon transform
zbMATH Keywords: Jost function
zbMATH Keywords: wave equation
zbMATH Keywords: parabolic equation
zbMATH Keywords: heat equation
zbMATH Keywords: reverse time
zbMATH Keywords: Fourier series
zbMATH Keywords: Lattès-Lions method
zbMATH Keywords: elliptic problem
zbMATH Keywords: hyperbolic equation
zbMATH Keywords: acoustic equation
zbMATH Keywords: Maxwell's equations
zbMATH Keywords: Neumann to Dirichlet map
zbMATH Keywords: Gelfand-Levitan-Krein equation

Inverse and ill-posed problems. Theory and applications. (zbMATH Keywords)
zbMATH Keywords: modulus of continuity

Inverse and ill-posed problems. Theory and applications. (zbMATH Keywords)
zbMATH Keywords: monograph

Inverse and ill-posed problems. Theory and applications. (zbMATH Keywords)
zbMATH Keywords: Hilbert-Schmidt-kernels

Inverse and ill-posed problems. Theory and applications. (zbMATH Keywords)
zbMATH Keywords: nonlinear Volterra integral equation of the second kind

Inverse and ill-posed problems. Theory and applications. (zbMATH Keywords)
zbMATH Keywords: moment problem

Inverse and ill-posed problems. Theory and applications. (zbMATH Keywords)
zbMATH Keywords: eigenvalue

Inverse and ill-posed problems. Theory and applications. (zbMATH Keywords)
zbMATH Keywords: eigenfunction

Inverse and ill-posed problems. Theory and applications. (zbMATH Keywords)
zbMATH Keywords: Cauchy problem

Inverse and ill-posed problems. Theory and applications. (author)
author: Sergey I. Kabanikhin

Inverse and ill-posed problems. Theory and applications. (MaRDI profile type)
MaRDI profile type: Publication

Inverse and ill-posed problems. Theory and applications. (full work available at URL)
full work available at URL: https://doi.org/10.1515/9783110224016

Inverse and ill-posed problems. Theory and applications. (OpenAlex ID)
OpenAlex ID: W4298293704