Mathematical and Analytical Techniques with Applications to Engineering

Inverse Problems

Mathematical and Analytical Techniques with Applications to Engineering

Authors: Ramm, Alexander G.

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About this book

Inverse problems are the problems that consist of finding an unknown property of an object or a medium from the observation or a response of this object or a medium to a probing signal. Thus the theory of inverse problems yields a theoretical basis for remote sensing and non-destructive evaluation. For example, if an acoustic plane wave is scattered by an obstacle, and one observes the scattered field from the obstacle, or in some exterior region, then the inverse problem is to find the shape and material properties of the obstacle. Such problems are important in the identification of flying objects (airplanes, missiles etc.) objects immersed in water (submarines, fish) and in many other situations.

This book presents the theory of inverse spectral and scattering problems and of many other inverse problems for differential equations in an essentially self-contained way. An outline of the theory of ill-posed problems is given, because inverse problems are often ill-posed. There are many novel features in this book. The concept of property C, introduced by the author, is developed and used as the basic tool for a study of a wide variety of one- and multi-dimensional inverse problems, making the theory easier and shorter.

New results include

recovery of a potential from I-function and applications to classical and new inverse scattering and spectral problems,

study of inverse problems with"incomplete data",

study of some new inverse problems for parabolic and hyperbolic equations,

discussion of some non-overdetermined inverse problems,

a study of inverse problems arising in the theory of ground-penetrating radars,

development of DSM (dynamical systems method) for solving ill-posed nonlinear operator equations,

comparison of the Ramm's inversion method for solving fixed-energy inverse scattering problem with the method based on the Dirichlet-to-Neumann map,

derivation of the range of applicability and error estimates for Born's inversion,

a study of some integral geometry problems, including tomography,

inversion formulas for the spherical means,

proof of the invertibility of the steps in the Gel'fand-Levitan and Marchenko inversion procedures,

derivation of the inversion formulas and stability estimates for the multidimensional inverse scattering problems with fixed-energy noisy discrete data,

new uniqueness and stability results in obstacle inverse scattering,

formulation and a solution of an inverse problem of radiomeasurements,

methods for finding small inhomogeneities from surface scattering data.

Reviews

From the reviews of the first edition:

"The author’s attention is addressed … to inverse problems for scattering theory … . the reviewer emphasizes the author’s effort of simplifying both the results in the mathematical literature and his own ones as well as the effort of unifying the theory of the inverse problems illustrated. Moreover, the proofs of the various propositions are almost always inserted in the book so that we can say the volume is friendly to readers … . Finally, the proofs are well written … ." (Alfredo Lorenzi, Zentralblatt MATH, Vol. 1083, 2006)

“This book is devoted to the investigation of inverse problems for PDEs, in particular, to the theory of inverse spectral problems and inverse scattering problems. … Graduate students and researchers will find in this work extensive reference material including a collection of relevant classical results as well as several new developments in the area of inverse and ill-posed problems.” (Antonio C. G. Leitão, Mathematical Reviews, Issue 2012 k)


Table of contents (11 chapters)

Table of contents (11 chapters)
  • Introduction

    Pages 1-17

  • Methods of Solving Ill-Posed Problems

    Pages 19-89

  • One-Dimensional Inverse Scattering and Spectral Problems

    Pages 91-226

  • Inverse Obstacle Scattering

    Pages 227-253

  • Stability of the Solutions to 3D Inverse Scattering Problems with Fixed-Energy Data

    Pages 255-316

Buy this book

eBook 149,79 €
price for Spain (gross)
  • ISBN 978-0-387-23218-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
  • Institutional customers should get in touch with their account manager
Hardcover 207,99 €
price for Spain (gross)
Softcover 194,38 €
price for Spain (gross)
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Bibliographic Information

Bibliographic Information
Book Title
Inverse Problems
Book Subtitle
Mathematical and Analytical Techniques with Applications to Engineering
Authors
Series Title
Mathematical and Analytical Techniques with Applications to Engineering
Copyright
2005
Publisher
Springer US
Copyright Holder
Springer-Verlag US
eBook ISBN
978-0-387-23218-8
DOI
10.1007/b100958
Hardcover ISBN
978-0-387-23195-2
Softcover ISBN
978-1-4419-3585-4
Series ISSN
1559-7458
Edition Number
1
Number of Pages
XX, 442
Topics