We are concerned with the inverse scattering problem for the full line Schrödinger operator −∂ ² _x + q( x) with a steplike potential q a priori known on . Assuming is known and short range, we show that the unknown part of q can be recovered by



where

Magneto-acousto-electric tomography (MAET), also known as the Lorentz force or Hall effect tomography, is a novel hybrid modality designed to be a high-resolution alternative to the unstable electrical impedance tomography. In this paper, we analyze the existing mathematical models of this method, and propose a general procedure for solving the inverse problem associated with the MAET. It consists in applying to the data one of the algorithms of thermo-acoustic tomography, followed by solving the Neumann problem for the Laplace equation and the Poisson equation. For the particular case when the region of interest is a cube, we present an explicit series solution resulting in a fast reconstruction algorithm. As we show, both analytically and numerically, the MAET is a stable technique yielding high-resolution images even in the presence of significant noise in the data.

This paper is concerned with the inverse scattering problem of reconstructing the support of a periodic inhomogeneous medium from knowledge of the scattered field measured on a straight line above and below the periodic structure. A linear sampling method is proposed to reconstruct the support of the periodic inhomogeneous medium based on a linear operator equation. The mathematical analysis of the sampling method is developed and numerical examples are given showing the practicality of the reconstruction algorithm.

We propose a direct reconstruction method for the Calderon problem based on inversion of the Born series. We characterize the convergence, stability and approximation error of the method and illustrate its use in numerical reconstructions.

We study the convergence of a class of accelerated perturbation-resilient block-iterative projection methods for solving systems of linear equations. We prove convergence to a fixed point of an operator even in the presence of summable perturbations of the iterates, irrespective of the consistency of the linear system. For a consistent system, the limit point is a solution of the system. In the inconsistent case, the symmetric version of our method converges to a weighted least-squares solution. Perturbation resilience is utilized to approximate the minimum of a convex functional subject to the equations. A main contribution, as compared to previously published approaches to achieving similar aims, is a more than an order of magnitude speed-up, as demonstrated by applying the methods to problems of image reconstruction from projections. In addition, the accelerated algorithms are illustrated to be better, in a strict sense provided by the method of statistical hypothesis testing,...

The full text of this article is available in the PDF provided.

We present a new approach to construction of recursion operators for multidimensional integrable systems which have a Lax-type representation in terms of a pair of commuting vector fields. It is illustrated by the examples of the Manakov–Santini system which is a hyperbolic system in N dependent and ( N + 4) independent variables, where N is an arbitrary natural number, the six-dimensional generalization of the first heavenly equation, the modified heavenly equation and the dispersionless Hirota equation.

Dirichlet and transmission eigenvalues have important applications in qualitative methods in inverse scattering. Motivated by the fact that these eigenvalues can be obtained from scattering data, we propose a new eigenvalue method using multiple frequency data (EM ²F). The method detects eigenvalues and builds indicator functions to reconstruct the support of the target. Numerical reconstruction is quite satisfactory. Dirichlet or transmission eigenvalues are estimated. Furthermore, reconstruction of D and estimation of eigenvalues can be combined together to distinguish between the sound-soft obstacle and non-absorbing inhomogeneous medium.