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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.
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.
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."
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.
This classic book provides a rigorous treatment of the Riesz?Fredholm theory of compact operators in dual systems, followed by a derivation of the jump relations and mapping properties of scalar and vector potentials in spaces of continuous and H?lder continuous functions. These results are then used to study scattering problems for the Helmholtz and Maxwell equations. Readers will benefit from a full discussion of the mapping properties of scalar and vector potentials in spaces of continuous and H?lder continuous functions, an in-depth treatment of the use of boundary integral equations to solve scattering problems for acoustic and electromagnetic waves, and an introduction to inverse scattering theory with an emphasis on the ill-posedness and nonlinearity of the inverse scattering problem.
Inverse Problems in Scattering exposes some of the mathematics which has been developed in attempts to solve the one-dimensional inverse scattering problem. Layered media are treated in Chapters 1--6 and quantum mechanical models in Chapters 7--10. Thus, Chapters 2 and 6 show the connections between matrix theory, Schur's lemma in complex analysis, the Levinson--Durbin algorithm, filter theory, moment problems and orthogonal polynomials. The chapters devoted to the simplest inverse scattering problems in quantum mechanics show how the Gel'fand--Levitan and Marchenko equations arose. The introduction to this problem is an excursion through the inverse problem related to a finite difference version of Schrödinger's equation. One of the basic problems in inverse quantum scattering is to determine what conditions must be imposed on the scattering data to ensure that they correspond to a regular potential, which involves Lebesque integrable functions, which are introduced in Chapter 9.
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
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.