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Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering (Mathematical and Analytical Techniques with Applications to)by Alexander G. Ramm
Synopses & ReviewsPublisher Comments: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 nondestructive 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 selfcontained way. An outline of the theory of illposed problems is given, because inverse problems are often illposed. 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 multidimensional inverse problems, making the theory easier and shorter. New results include recovery of a potential from Ifunction 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 nonoverdetermined inverse problems, a study of inverse problems arising in the theory of groundpenetrating radars, development of DSM (dynamical systems method) for solving illposed nonlinear operator equations, comparison of the Ramm's inversion method for solving fixedenergy inverse scattering problem with the method based on the DirichlettoNeumann 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'fandLevitan and Marchenko inversion procedures, derivation of the inversion formulas and stability estimates for the multidimensional inverse scattering problems with fixedenergy 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.
Synopsis:Inverse Problems is a monograph which contains a selfcontained presentation of the theory of several major inverse problems and the closely related results from the theory of illposed 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.
Table of ContentsIntroduction. Methods of soliving illposed problems. Onedimensional inverse scattering and spectral problems. Inverse obstacle scattering. Stability of the solution to 3D Inverse scattering problems with fixedenergy data. Nonuniqueness and uniqueness results. Inverse problems of potential theory and other inverse source problems. Nonoverdetermined inverse problems. Lowfrequency inversions. Wave scattering by small bodies of arbitrary shapes. The Pompeiu problem. Bilbliographical Notes. References. Index.
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