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A two dimensional inverse scattering problem in which an acoustic plane wave is incident on a cylindrical object with the index of refraction varying only in two spatial directions is discussed. First, a modified method of lines procedure is developed to and the scattering amplitudes of the wave scattered from the object and compared with exact scattering amplitudes for some exactly solvable examples to verify its performance. Then, an inversion procedure is developed that uses the scattering amplitude for finding the object's profile. For that, Reese T. Prosser's procedure for the inversion in three dimensions is modified to our two dimensional case. Then, the back-scattered data is used to find both the Born approximation and Reese T. Prosser's first order approximation to find inversion results. Since the procedure requires the data to be available both for on-shell and off-shell values of the wave number, an interpolation method is developed to recover the needed data and inversion results are compared with exact values for a solid cylinder, a hollow cylinder and a shifted cylinder with constant index of refraction.
Limited successes in the solution of the inverse problems have been achieved only in one-dimensional cases (Gelfand-Levitan and layer striping [sic] methods are among the most notable). These methods are generally unstable numerically since the procedures used to calculate the index of refraction are ill-conditioned. We present a method for the solution of inverse problems for the one dimensional Helmholtz equation. The scheme is based on a combination of the standard Riccati equation for the impedance function with a new trace formula for the derivative of the index of refraction, and can be viewed as a frequency domain version of the layer-stripping approach.
Inverse scattering theory is a major theme in applied mathematics, with applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting challenges in the development of efficient inversion algorithms. A further complication is that anisotropic materials cannot be uniquely determined from given scattering data. In the first edition of Inverse Scattering Theory and Transmission Eigenvalues, the authors discussed methods for determining the support of inhomogeneous media from measured far field data and the role of transmission eigenvalue problems in the mathematical development of these methods. In this second edition, three new chapters describe recent developments in inverse scattering theory. In particular, the authors explore the use of modified background media in the nondestructive testing of materials and methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data. They also examine nonscattering wave numbers—a subset of transmission eigenvalues—using techniques taken from the theory of free boundary value problems for elliptic partial differential equations and discuss the dualism of scattering poles and transmission eigenvalues that has led to new methods for the numerical computation of scattering poles. This book will be of interest to research mathematicians and engineers and physicists working on problems in target identification. It will also be useful to advanced graduate students in many areas of applied mathematics.
Limited successes in the solution of the inverse problems have been achieved only in one-dimensional cases (Gelfand-Levitan and layer striping [sic] methods are among the most notable). These methods are generally unstable numerically since the procedures used to calculate the index of refraction are ill-conditioned. We present a method for the solution of inverse problems for the one dimensional Helmholtz equation. The scheme is based on a combination of the standard Riccati equation for the impedance function with a new trace formula for the derivative of the index of refraction, and can be viewed as a frequency domain version of the layer-stripping approach.
A careful exposition of a research field of current interest. This includes a brief survey of the subject and an introduction to recent developments and unsolved problems.
The purpose of this text is to present the theory and mathematics of inverse scattering, in a simple way, to the many researchers and professionals who use it in their everyday research. While applications range across a broad spectrum of disciplines, examples in this text will focus primarly, but not exclusively, on acoustics. The text will be especially valuable for those applied workers who would like to delve more deeply into the fundamentally mathematical character of the subject matter.Practitioners in this field comprise applied physicists, engineers, and technologists, whereas the theory is almost entirely in the domain of abstract mathematics. This gulf between the two, if bridged, can only lead to improvement in the level of scholarship in this highly important discipline. This is the book's primary focus.
Inverse Problems in Scattering and Imaging is a collection of lectures from a NATO Advanced Research Workshop that integrates the expertise of physicists and mathematicians in different areas with a common interest in inverse problems. Covering a range of subjects from new developments on the applied mathematics/mathematical physics side to many areas of application, the book achieves a blend of research, review, and tutorial contributions. It is of interest to researchers in the areas of applied mathematics and mathematical physics as well as those working in areas where inverse problems can be applied.
Over the last twenty years, the growing availability of computing power has had an enormous impact on the classical fields of direct and inverse scattering. The study of inverse scattering, in particular, has developed rapidly with the ability to perform computational simulations of scattering processes and led to remarkable advances in a range of
Thus, a smooth scatterer is reconstructed very accurately from a limited amount of available data. The scheme has the asymptotic cost O(n2), where n is the number of features to be recovered (as do classical layer-stripping algorithms), and is stable with respect to perturbations of the scattering data. The performance of the algorithm is illustrated with several numerical examples."