Note publique d'information : 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.
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