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Inverse Problems & Imaging

2009 , Volume 3 , Issue 2

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Preface
Fioralba Cakoni, Houssem Haddar and Michele Piana
2009, 3(2): i-i doi: 10.3934/ipi.2009.3.2i +[Abstract](51) +[PDF](34.3KB)
Abstract:
This special issue is dedicated to Professors David Colton and Rainer Kress in honor of their contribution and leadership in the area of direct and inverse scattering theory for more then 30 years. The papers in this special issue were solicited from the invited speakers at the International Conference on Inverse Scattering Problems organized in honor of the 65th birthdays of David Colton and Rainer Kress held in the seaside resort of Sestry Levante, Italy, May 8-10, 2008.
    As organizers of this conference and close collaborators of Professors Colton and Kress, we are very honored to have had the opportunity to facilitate this special scientific and social event. It was a particular occasion that gathered together long term colleagues, collaborators, former students and friends of Professors Colton and Kress. And now it gives us particular pleasure to be guest editors of this special issue of Inverse Problems and Imaging which is a collection of original research papers in the area of scattering theory and inverse problems. Much of the work presented here has been directly or indirectly influenced by these two scientists, offering the reader a glimpse of their significant impact in this research area.
   We would like to thank all of those who have contributed a paper for this special issue. A special thanks goes to the Editor in Chief of Inverse Problems and Imaging, Lassi Päivärinta, for supporting and facilitating this publication. We would also like to thank all the participants of the Sestri Levante Conference who made such a successful, stimulating and pleasant event possible. Last (but definitely not least!) we would like to thank the sponsors of the conference: the European Office of Aerospace Research and Development of the United States Air Force Office of Scientific Research, the University of Genova, the University of Verona, the Istituto Nazionale di Alta Matematica - Gruppo Nazionale di Calcolo Scientifico, the University of Göttingen, the University of Delaware and INRIA Center of Saclay Ile de France.
Thirty years and still counting
David Colton and Rainer Kress
2009, 3(2): 151-153 doi: 10.3934/ipi.2009.3.151 +[Abstract](75) +[PDF](49.6KB)
Abstract:
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On the existence of transmission eigenvalues
Andreas Kirsch
2009, 3(2): 155-172 doi: 10.3934/ipi.2009.3.155 +[Abstract](253) +[PDF](214.7KB)
Abstract:
The investigation of the far field operator and the Factorization Method in inverse scattering theory leads naturally to the study of corresponding interior transmission eigenvalue problems. In contrast to the classical Dirichlet- or Neumann eigenvalue problem for $-\Delta$ in bounded domains these interior transmiision eigenvalue problem fail to be selfadjoint. In general, existence of eigenvalues is an open problem. In this paper we prove existence of eigenvalues for the scalar Helmholtz equation (isotropic and anisotropic cases) and Maxwell's equations under the condition that the contrast of the scattering medium is large enough.
An inverse fluid--solid interaction problem
Peter Monk and Virginia Selgas
2009, 3(2): 173-198 doi: 10.3934/ipi.2009.3.173 +[Abstract](197) +[PDF](546.1KB)
Abstract:
This paper is devoted to studying the Linear Sampling Method (LSM) applied to the inverse problem for the fluid-solid interaction problem of determining the shape of the solid from far field measurements of the fluid pressure field. We provide a simplified proof of the uniqueness problem in this case, an analysis of the appropriate interior transmission problem, and the existence of a solution to the LSM that can be used as an indicator for the shape of the solid. The analysis of uniqueness rests on a new technical result concerning regularity. Finally we present some numerical results for the method.
On a Hybrid method for shape reconstruction of a buried object in an elastostatic half plane
Roman Chapko
2009, 3(2): 199-210 doi: 10.3934/ipi.2009.3.199 +[Abstract](63) +[PDF](250.8KB)
Abstract:
We apply the hybrid method for determining the shape of a bounded object in the elastostatic half plane from given Cauchy data on the boundary. The identifiability of the shape is investigated. For the integral representation of the function and the traction on the boundaries, the Green's function approach based on Kelvin's fundamental solution is used. The approximation of the integral operators with various singularities is made by trigonometrical and sinc quadratures. The presented numerical experiments exhibit the feasibility of the hybrid method for the system of differential equations and its stability in the case of noisy data.
The inverse acoustic obstacle scattering problem and its interior dual
Brian Sleeman
2009, 3(2): 211-229 doi: 10.3934/ipi.2009.3.211 +[Abstract](96) +[PDF](234.3KB)
Abstract:
This paper addresses possible connections between two classical inverse problems arising in wave propagation. The first is the problem of extracting geometrical information about an unknown bounded domain from a knowledge its eigen-frequencies. The chosen method of investigation being the high frequency asymptotics of the associated counting function. The second problem is the inverse obstacle scattering problem. That is the determination of an unknown obstacle from far field data. This problem is investigated through the high frequency asymptotics of the associated scattering phase. It turns out that there is a remarkable similarity between the asymptotic expansions in each of these problems. We discuss a number of ideas and techniques along the way including representations of the scattering matrix and the Kirchoff approximation. We also show how to solve scattering problems for polygonal obstacles. Whether there is a deep physical connection between interior and exterior scattering problems remains a challenging area of research.
Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering
Giovanni Bozza, Massimo Brignone, Matteo Pastorino, Andrea Randazzo and Michele Piana
2009, 3(2): 231-241 doi: 10.3934/ipi.2009.3.231 +[Abstract](86) +[PDF](7611.7KB)
Abstract:
In this paper a new formulation of the Linear Sampling Method, called the no-Sampling Linear Sampling Method, is applied to the imaging and detection of unknown scatterers located inside an inhomogeneous background. Namely, by following a previous work by Colton and Monk, a modified far--field equation is used, which allows one to use line current sources and nearfield measurements. The Green's function of the inhomogeneous background is numerically computed and used as the right hand side of the modified farfield equation. The proposed method is then applied to two different scenarios: the detection of breast tumors and the imaging of cracks inside a dielectric slab. A numerical analysis of the method capabilities is performed when the model parameters are not exactly known.
Vector ellipsoidal harmonics and neuronal current decomposition in the brain
George Dassios and Michalis N. Tsampas
2009, 3(2): 243-257 doi: 10.3934/ipi.2009.3.243 +[Abstract](94) +[PDF](300.6KB)
Abstract:
Vector ellipsoidal harmonics are introduced here for the first time and their analytic peculiarities, as well as their limitations, are analyzed. A novelty of these vectorial base functions is that we need to introduce two different inner products in order to obtain orthogonality on the surface of any ellipsoid. Furthermore, in contrast to the vector spherical harmonics which are independent of the radial variable, the vector ellipsoidal harmonics can not be defined uniformly over a family of confocal ellipsoids. An expansion theorem is proved which secures completeness of the vectorial harmonics as well as a non-trivial algorithm that determines the coefficients of the expansion. Then, these new functions are used to prove that, for the realistic ellipsoidal model of the human head, there exists a component of the neuronal current that is invisible by the electroencephalographic measurements while it is detectable through magnetoencephalographic measurements in the exterior of the head. Furthermore, in contrast to the case of the sphere, where no part of the current contributes both to the electric potential and to the magnetic field, we prove here that, in the case of the ellipsoid, there is a part of the current that influences the electroencephalographic as well as the magnetoencephalographic recordings.
A time-domain probe method for three-dimensional rough surface reconstructions
Corinna Burkard and Roland Potthast
2009, 3(2): 259-274 doi: 10.3934/ipi.2009.3.259 +[Abstract](81) +[PDF](436.9KB)
Abstract:
The task of this paper is to develop a Time-Domain Probe Method for the reconstruction of impenetrable scatterers. The basic idea of the method is to use pulses in the time domain and the time-dependent response of the scatterer to reconstruct its location and shape. The method is based on the basic causality principle of time-dependent scattering. The method is independent of the boundary condition and is applicable for limited aperture scattering data.
   In particular, we discuss the reconstruction of the shape of a rough surface in three dimensions from time-domain measurements of the scattered field. In practise, measurement data is collected where the incident field is given by a pulse. We formulate the time-domain field reconstruction problem equivalently via frequency-domain integral equations or via a retarded boundary integral equation based on results of Bamberger, Ha-Duong, Lubich. In contrast to pure frequency domain methods here we use a time-domain characterization of the unknown shape for its reconstruction.
   Our paper will describe the Time-Domain Probe Method and relate it to previous frequency-domain approaches on sampling and probe methods by Colton, Kirsch, Ikehata, Potthast, Luke, Sylvester et al. The approach significantly extends recent work of Chandler-Wilde and Lines (2005) and Luke and Potthast (2006) on the time-domain point source method. We provide a complete convergence analysis for the method for the rough surface scattering case and provide numerical simulations and examples.
Full identification of acoustic sources with multiple frequencies and boundary measurements
Carlos J. S. Alves, Nuno F. M. Martins and Nilson C. Roberty
2009, 3(2): 275-294 doi: 10.3934/ipi.2009.3.275 +[Abstract](90) +[PDF](3559.9KB)
Abstract:
In this paper we study the identification of acoustic sources in a domain $\Omega$ from boundary data. With a single frequency, we show that identification is possible if, besides the boundary data, considerable information regarding the type of the source is considered. For the general case, we present an identification result using multiple frequencies and boundary measurements. We show that for compactly supported sources in $\Omega$, the completion of Cauchy data has at most one solution and thus for this type of sources, identification is possible using variable frequencies and incomplete boundary measurements. A numerical method based on the reciprocity functional is proposed and tested for several numerical examples. For compact sources, a data completion method is proposed and tested in order to apply the previous method.
Reconstruction of perfectly conducting rough surfaces by the use of inhomogeneous surface impedance modeling
Emre Kiliç, Mehmet Çayören, Ali Yapar and Íbrahim Akduman
2009, 3(2): 295-307 doi: 10.3934/ipi.2009.3.295 +[Abstract](94) +[PDF](829.2KB)
Abstract:
A new method for the reconstruction of one dimensional profile of a perfectly conducting rough surface is presented. The method is based on the equivalent representation of the rough surface by means of an inhomogeneous impedance plane, whose surface impedance is recovered through the analytical continuation of the measured data. The equivalent problem allows one to calculate the total field in the whole space above the unknown surface. The use of boundary condition that the total electric field vanishes on the unknown surface enables to reduce the problem to the solution of a non-linear equation in terms of the unknown surface function. The non-linear equation is solved iteratively via Newton method and regularization in the least square sense is also applied. The effectiveness of the method has been demonstrated with several numerical simulations.
Unique continuation property for the elasticity with general residual stress
Gunther Uhlmann and Jenn-Nan Wang
2009, 3(2): 309-317 doi: 10.3934/ipi.2009.3.309 +[Abstract](72) +[PDF](148.9KB)
Abstract:
We prove the unique continuation property for the isotropic elasticity system with arbitrarily large residual stress. This work improves the result obtained in [10] where the residual stress is assumed to be small.
Recovering an obstacle using integral equations
William Rundell
2009, 3(2): 319-332 doi: 10.3934/ipi.2009.3.319 +[Abstract](69) +[PDF](239.6KB)
Abstract:
We consider the inverse problem of recovering the shape, location and surface properties of an object where the surrounding medium is both conductive and homogeneous and we measure Cauchy data on an accessible part of the exterior boundary. It is assumed that the physical situation is modelled by harmonic functions and the boundary condition on the obstacle is one of Dirichlet type. The purpose of this paper is to answer some of the questions raised in a recent paper that introduced a nonlinear integral equation approach for the solution of this type of problem.
An estimate for the free Helmholtz equation that scales
John Sylvester
2009, 3(2): 333-351 doi: 10.3934/ipi.2009.3.333 +[Abstract](82) +[PDF](234.7KB)
Abstract:
Wavelength plays a distinguished role in classical electromagnetic and acoustic scattering. Most significant features of the far field patterns radiated by a collection of sources or scatterers are related to their sizes and relative distances, measured in wavelengths. These significant features are reflected in the invariance of the Helmholtz equation with respect to translation, and its homogeneous scaling with respect to dilations. The weighted norms that were first developed to capture the correct decay properties of waves in Rn do not scale homogeneously and are not invariant with respect to translation. Lp estimates scale homogeneously and commute with translations and rotations. However, their scaling properties give estimates with a weaker dependence on wavenumber (for bounded sources and scatterers with support that extends over many wavelengths). We introduce some norms and estimates that commute with translations and scale homogeneously under dilations, while retaining the same sharp dependence on wavelength for extended sources as that of the weighted estimates.
A Newton method for reconstructing non star-shaped domains in electrical impedance tomography
Helmut Harbrecht and Thorsten Hohage
2009, 3(2): 353-371 doi: 10.3934/ipi.2009.3.353 +[Abstract](83) +[PDF](902.3KB)
Abstract:
We study the reconstruction of the shape of a perfectly conducting inclusion in three dimensional electrical impedance tomography (EIT) using a regularized Newton method. This method involves a least squares penalty in the form of an additional nonlinear operator to cope with the non-uniqueness of general parametrizations of the unknown boundary. We provide a convergence result for this method in the general framework of nonlinear ill-posed operator equations. Moreover, we discuss the evaluation of the forward operator in EIT, its derivative, and the adjoint of the derivative using a wavelet based boundary element method. Numerical examples illustrate the performance of our method.

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