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Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering de Alexander G. Ramm

Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering: Mathematical and Analytical Techniques with Applications to Engineering

Autor Alexander G. Ramm
en Limba Engleză Paperback – 19 noi 2010
Inverse Problems is a monograph which contains a self-contained presentation of the theory of several major inverse problems and the closely related results from the theory of ill-posed problems. The book is aimed at a large audience which include graduate students and researchers in mathematical, physical, and engineering sciences and in the area of numerical analysis.
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Specificații

ISBN-13: 9781441935854
ISBN-10: 1441935851
Pagini: 464
Ilustrații: XX, 442 p.
Dimensiuni: 155 x 235 x 24 mm
Greutate: 0.64 kg
Ediția:Softcover reprint of hardcover 1st ed. 2005
Editura: Springer Us
Colecția Springer
Seria Mathematical and Analytical Techniques with Applications to Engineering

Locul publicării:New York, NY, United States

Public țintă

Research

Cuprins

Methods of Solving Ill-Posed Problems.- One-Dimensional Inverse Scattering and Spectral Problems.- Inverse Obstacle Scattering.- Stability of the Solutions to 3D Inverse Scattering Problems with Fixed-Energy Data.- Non-Uniqueness and Uniqueness Results.- Inverse Problems of Potential Theory and Other Inverse Source Problems.- Non-Overdetermined Inverse Problems.- Low-Frequency Inversion.- Wave Scattering by Small Bodies of Arbitrary Shapes.- The Pompeiu Problem.

Recenzii

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)

Textul de pe ultima copertă

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.

Caracteristici

The dynamical systems method (DSM) for solving operator equations, especially nonlinear and ill-posed Many novel results in the presentation of the theory of one-dimensional inverse problems Many novel results in the presentation of inverse obstacle scattering, including very weak assumptions on the smoothness of the obstacle, stability estimates for the solution, and an analysis of the earlier works A mathematically justified method, developed by the author for solving multidiemsional inverse potential scattering problem with fixed-energy data; error estimates A study of a non-overdetermined inverse problem Derivation of the analytical formulas for the S-matrix for eletromagnetic and acoustic wave scattering by small bodies of arbitrary shapes Discussion of the Pompeiu problem and the Schiffer's conjecture Includes supplementary material: sn.pub/extras