Inverse Problems. Mathematical and analytical techniques with applications to engineering
type: Item
instance of: scholarly article
title: Inverse Problems. Mathematical and analytical techniques with applications to engineering
zbMATH Open document ID: 1083.35002
author: A. G. Ramm
published in: Mathematical and Analytical Techniques with Applications to Engineering
publication date: 2005-01-11T00:00:00Z
review text: The book begins with a list of inverse problems ranging from the classical spectral one to those related to scattering theory, tomography, acoustics, electrodynamics, image processing, heat diffusion, and so on. Most of them are discussed throughout the volume. The first part of the book is devoted to the theoretical treatment of inverse problems via variational methods for linear problems -- reading from the celebrated Tikhonov regularization procedure -, to quasisolutions and the Backus and Gilbert method. Then emphasis is laid on possibly noisy problems related to nonlinear -- possibly unbounded -- operators in a Banach space framework and to the discrepancy principle in a Hilbert space setting. A novelty in this treatment consists in the so-called dynamical approach of associating with the nonlinear equation \[ F(u):=B(u)-f=0 \tag{1} \] in a Hilbert space \(H\), a suitable evolution Cauchy problem \[ u'(t)=\Phi(t,u(t)),\qquad u(0)=u_0, \] admitting a unique solution \(u:[0,+\infty)\to H\) for which the limit \(u(+\infty)\) exists and solves equation (1), whenever \(F\) satisfies suitable assumptions. Of course, \(u(t)\) can be chosen, for large \(t\), as an approximation of the searched for solution \(u\) to (1). Such a method, which can be applied to both well posed and ill-posed problems, is endowed with several applications showing how \(\Phi\) can be chosen starting from \(F\). From Chapter 3 on the author begins the treatment of inverse problems related to elliptic equations in one or several dimensions. The starting point is the analysis of the celebrated one-dimensional inverse scattering and inverse spectral problems going back to Gelfand-Levitan and Marchenko. This chapter contains many original contributions by the author such as the so-called C-property. A particular attention is paid to the more difficult inverse scattering problem with fixed energy phase shifts related to the recovery of spherically symmetric potentials. Taking advantage of these on…
Mathematics Subject Classification ID: 35-01
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 81U40
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 78A46
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 34L25
Mathematics Subject Classification ID: 31B20
zbMATH DE Number: 2126325
zbMATH Keywords: inverse scattering problems
zbMATH Keywords: recovering potentials in Schrödinger equations
zbMATH Keywords: recovering unknown coefficients in hyperbolic and parabolic equations
zbMATH Keywords: inverse problems in potential theory
zbMATH Keywords: Pompeiu's problem
zbMATH Keywords: uniqueness
zbMATH Keywords: continuous dependence
zbMATH Keywords: non-uniqueness
zbMATH Keywords: C-property
reviewed by: Alfredo Lorenzi
MaRDI profile type: Publication
full work available at URL: b100958
OpenAlex ID: W4231148981
DOI: 10.1007/B100958
description: scientific article; zbMATH DE number 2126325