Roland Potthast

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.

Point Sources and Multipoles in Inverse Scattering Theory

INTRODUCTION AND TOOLS A Survey About Inverse Scattering Theory Basic Definitions and Tools DIRECT SCATTERING PROBLEMS Acoustic Obstacle Scattering The Inhomogeneous Acoustic Medium Electromagnetic Scattering by a Perfect Conductor The Electromagnetic Inhomogeneous Medium Scattering by Orthotropic Media Anisotropic Electromagnetic Media UNIQUENESS AND STABILITY IN INVERSE SCATTERING Acoustic Scattering Electromagnetic Scattering THE CASE OF FINITE DATA Finite Data in Inverse Acoustic Scattering Inverse Electromagnetic Scattering THE POINT-SOURCE METHOD AND APPLICATIONS Reconstruction Of Acoustic Scatters Inverse Electromagnetic Scattering Reconstruction of the Boundary Values of Scattered Fields Convergence of a Regularized Newton Method SINGULAR SOURCES AND SHAPE RECONSTRUCTION Acoustic Scattering Electromagnetic Scattering LINEAR SAMPLING METHODS The Original Linear Sampling Method Spectral Theory and a Modified Linear Sampling Method Shape Reconstruction for Orthotropic Media Anisotropic Electromagnetic Media REFERENCES INDEX

A survey on sampling and probe methods for inverse problems

A solid or semisolid propellant comprising grains of propellant or propellant components bonded together so as to create voids within the propellant volume, said grains bonded together with sufficient strength to substantially delay the fluidization of the propellant by the onset of Taylor unstable burning, said propellant having a rapid burn rate below that associated with Taylor unstable burn. In another embodiment, the grains are held within and the voids are filled with viscous fluid binder such as a petroleum oil, said binder functioning to hinder Taylor unsatable burning and yet permit very rapid burning within the propellant volume.

Frechet differentiability of boundary integral operators in inverse acoustic scattering

Using integral equation methods to solve the time-harmonic acoustic scattering problem with Dirichlet boundary conditions, it is possible to reduce the solution of the scattering problem to the solution of a boundary integral equation of the second kind. We show the Frechet differentiability of the boundary integral operators which occur. We then use this to prove the Frechet differentiability of the scattered field with respect to the boundary. Finally we characterize the Frechet derivative of the scattered field by a boundary value problem with Dirichlet conditions, in an analogous way to that used by Firsch.

A fast new method to solve inverse scattering problems

In this paper we present a new method for solving inverse obstacle scattering problems where the far field for many incident plane waves is known. The main idea of the method is to approximate point sources by a superposition of incident plane waves, to use a characterization of the obstacle in terms of the location of these point sources and to translate and rotate the approximating function. We will describe the method and provide numerical examples.

Stability estimates and reconstructions in inverse acoustic scattering using singular sources

Abstract The problem of stability for the reconstruction of the scattered field us in the exterior of a scatterer D from far field data u∞ and for the reconstruction of a sound-soft or sound-hard scatterer D from the far field pattern u ∞ ( x ,d), x ,d∈Ω for all incident plane waves is investigated. In particular, we show how stability estimates are linked to the behaviour of the scattered field Φs for incident point-sources and to special minimum norm solutions for the Herglotz wave operator. The results can also be used to formulate a method of singular sources for the reconstruction of ∂D which is new in inverse scattering. The reconstruction method is described and numerical examples are provided.

THE NO RESPONSE TEST—A SAMPLING METHOD FOR INVERSE SCATTERING PROBLEMS ∗

We describe a novel technique,which we call the no response test,to locate the support of a scatterer from knowledge of a far field pattern of a scattered acoustic wave. The method uses a set of sampling surfaces and a special test response to detect the support of a scatterer without a priori knowledge of the physical properties of the scatterer. Specifically,the method does not depend on information about whether the scatterer is penetrable or impenetrable nor does it depend on any knowledge of the nature of the scatterer (absorbing,reflecting,etc.). In contrast to previous sampling algorithms,the techniques described here enable one to locate obstacles or inhomogeneities from the far field pattern of only one incident field—the no response test is a one- wave method. We investigate the theoretical basis for the no response test and derive a one-wave uniqueness proof for a region containing the scatterer. We show how to find the object within this region. We demonstrate the applicability of the method by reconstructing sound-soft,sound-hard, impedance,and inhomogeneous medium scatterers in two dimensions from one wave with full and limited aperture far-field data.

Frechet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain

Using integral equation methods to solve the time harmonic acoustic scattering problem with Neumann boundary condition it is possible to reduce the solution of the scattering problem to the solution of a boundary integral equation of the second kind. We show the Frechet differentiability of the boundary integral operators which occur. They are considered in dependence of the boundary as integral operators in the spaces of continuous functions. Then we use this to prove the Frechet differentiability of the scattered fields. Finally we characterize the Frechet derivatives of the scattered fields by a suitable boundary value problem.

Reconstruction of a current distribution from its magnetic field

We consider the inverse problem of reconstructing a current distribution from measurements of its magnetic field. Uniqueness issues and simulations for the reconstruction are studied. Given the magnetic field on a surface surrounding the current distribution we show that a projection of the current distribution can be reconstructed uniquely. In addition, we derive some properties of directed current distributions that reflect the properties and difficulties of the reconstruction. A Tikhonov-projection scheme complemented by an artefact-correction algorithm is employed to reconstruct the current distribution within a cuboid. By numerical examples in three dimensions we show that for measurement errors up to 1% we can detect areas of low-current density within the cuboid.

Domain derivatives in electromagnetic scattering

Within the integral equation approach we study the dependence of the solution to the electromagnetic scattering problem from a perfect conductor with respect to the obstacle. We study the differentiability properties of strongly singular and vector valued boundary integral operators in Holder spaces. We prove that the solution to the scattering problem depends infinitely differentiable on the boundary of the obstacle. We give a characterization of the first derivative as a solution to a boundary value problem.