Inverse Problems

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.