An inverse scattering problem is considered for arbitrarily shaped cylindrical objects that have inhomogeneous impedance boundaries and are buried in arbitrarily shaped cylindrical dielectrics. Given the shapes of the impedance object and the dielectric, the inverse problem consists of reconstructing the inhomogeneous boundary impedance from a measured far field pattern for an incident time-harmonic plane wave. Extending the approach suggested by Akduman and Kress [Direct and inverse scattering problems for inhomogeneous impedance cylinders of arbitrary shape. Radio Sci. 38 (2003), pp. 1055-1064] for an impedance cylinder in an homogeneous background medium, both the direct and the inverse scattering problem are solved via boundary integral equations. For the inverse problem, representing the scattered field as a potential leads to severely ill-posed linear integral equations of the first kind for the densities. For their stable numerical solution Tikhonov regularization is employed. Knowing the scattered field, the boundary impedance function can be obtained from the boundary condition either by direct evaluation or by a least squares approach. We provide a mathematical foundation of the inverse method and illustrate its feasibility by numerical examples.

In this paper, we present a method for the solution of a location and shape reconstruction problem related to sound-soft obstacles buried in arbitrarily shaped penetrable cylinders in two dimensions. The direct problem considered here is to obtain the scattered near/far-field in the case of a single time-harmonic acoustic plane wave insonification. The aim of the inverse problem is to find the shape as well as the location of the buried scatterer from the limited/full aperture measurements of the scattered field via Newton iterations. Both for the direct and inverse scattering problems, different potential (layer) approaches are used to derive a system of boundary integral equations. The integral equations are evaluated by using Nyström and collocation methods. Moreover, in order to obtain a stable solution of the first kind of Fredholm-type integral equations Tikhonov regularization is employed. We test the applicability and the effectiveness of the inversion algorithm also with noisy limited far-field data and obtain satisfactory numerical results as illustrated in the last section.

The aim of this paper is to introduce inversion-based engineering applications and to investigate some of the important ones from mathematical point of view. To do this we employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems. More specifically, we first study location, shape, and boundary parameter reconstruction algorithms for the inaccessible targets in acoustics. The inverse problems for the time-dependent differential equations of isotropic and anisotropic elasticity are reviewed in the following section of the paper. These problems were the objects of the study by many authors in the last several decades. The physical interpretations for almost all of these problems are given, and the geophysical applications for some of them are described. In our last section, an introduction with many links into the literature is given for modern algorithms which combine techniques from classical inverse problems with stochastic tools into ensemble methods both for data assimilation as well as for forecasting.

A new second order Newton method for reconstructing the shape of a sound soft scatterer from the measured far-field pattern for scattering of time harmonic plane waves is presented. This method extends a hybrid between regularized Newton iterations and decomposition methods that has been suggested and analyzed in a number of papers by 16, 17] and has some features in common with the second degree method for ill-posed nonlinear problems as considered by Hettlich and Rundell [8]. The main idea of our iterative method is to use Huygen's principle, i.e., represent the scattered field as a single-layer potential. Given an approximation for the boundary of the scatterer, this leads to an ill-posed integral equation of the first kind that is solved via Tikhonov regularization. Then, in a second order Taylor expansion, the sound soft boundary condition is employed to update the boundary approximation. In an iterative procedure, these two steps are alternated until some stopping criterium is satisfied. We describe the method in detail and illustrate its feasibility through examples with exact and noisy data.

The aim of our study is the numerical computation of the wakefield and energy loss per unit length for relativistic, short (<10 ns) proton bunches interacting with an electron cloud inside the beam pipe. We present analytical expressions for the energy loss in the impulse kick approximation. For the simulation of the wakefields a 2D self-consistent, electrostatic particle-in-cell (PIC) code is employed. Results for the energy loss and for the wakefields are presented for the parameter scope of the CERN LHC and SPS. For selected parameters the results are compared to a three-dimensional (3D) electromagnetic PIC code.