David Colton

A simple method for solving inverse scattering problems in the resonance region

This paper is concerned with the development of an inversion scheme for two-dimensional inverse scattering problems in the resonance region which does not use nonlinear optimization methods and is relatively independent of the geometry and physical properties of the scatterer. It is assumed that the far field pattern corresponding to observation angle and plane waves incident at angle is known for all . From this information, the support of the scattering obstacle is obtained by solving the integral equation where k is the wavenumber and is on a rectangular grid containing the scatterer. The support is found by noting that is unbounded as approaches the boundary of the scattering object from inside the scatterer. Numerical examples are given showing the practicality of this method.

The Linear Sampling Method in Inverse Electromagnetic Scattering

We survey the linear sampling method for solving the inverse scattering problem for time-harmonic electromagnetic waves at fixed frequency. We consider scattering by an obstacle as well as scattering by an inhomogeneous medium both in and . Included in our discussion is the use of regularization methods for ill-posed problems and numerical examples in both two and three dimensions.

Recent Developments in Inverse Acoustic Scattering Theory

We survey some of the highlights of inverse scattering theory as it has developed over the last 15 years, with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves. Included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects.

Qualitative Methods in Inverse Scattering Theory

A major problem in the use of ultrasound or microwaves for purposes of nondestructive testing or medical imaging is the computational complexity of solving the inverse scattering problem that arises in such applications. This is due to the fact that in order to achieve satisfactory resolution and sufficient penetration of the wave into the material it is often necessary to use frequencies in the resonance region. In this case the inverse scattering problem is not only improperly posed but also nonlinear and even in the case of two dimensions the time needed to solve such problems can be prohibitive. To date the time consuming nature of the problem has mainly been dealt with by the introduction of various innovative schemes that avoid the use of volume integral equations and instead rely on finite difference or finite element methods (cf. [5], [8]). However, for large scale problems (for example those involving imaging of the human body) the problem of computational complexity remains a serious problem for any practitioner. In this paper we would like to propose a different approach to this problem that, although still in its infancy, has the promise of providing rapid solutions to a number of inverse scattering problems of practical significance.

A simple method using Morozov's discrepancy principle for solving inverse scattering problems

This paper is a continuation of earlier research in which a simple inversion scheme was given for inverse scattering problems in the resonance region which is easy to implement and is relatively independent of the geometry and physical properties of the scatterer. The purpose of the paper is to give new and improved theorems establishing the mathematical basis of this method and to show how noisy data can be treated using Morozovs discrepancy principle where the regularization parameter is a function of an auxiliary parameter appearing in the inversion scheme.

A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region II

In a previous paper [SIAM J. Appl. Math. (1985), pp. 1039–1053] we presented a new method for determining the shape of an acoustically soft obstacle from a knowledge of the time-harmonic incident wave and the far field pattern of the scattered wave. The method given there was based on knowing the far field pattern for an interval of values of the square of the wave number k such that this interval contained the first eigenvalue

An introduction to inverse scattering and inverse spectral problems

\lambda _1

The linear sampling method for cracks

of the interior Dirichlet problem. The purpose of this paper is to extend the methods of our earlier paper to treat the case when the far field pattern is only known for a single value of the wave number such that

The Linear Sampling Method for Solving the Electromagnetic Inverse Scattering Problem

k^2

A linear sampling method for the detection of leukemia using microwaves

is not equal to