Minisymposia Abstracts
Speaker:  Laurent Bourgeois ENSTA, Paris, France 
Title:  Imaging an elastic waveguide with the linear sampling method by using the Lamb modes 
Abstract:  In the context of elasticity in the harmonic regime, we consider the
problem of identication of some obstacles in a 2D or 3D waveguide by using
the far elds produced by the propagating guided modes (called Lamb modes
in 2D). The main issue is that contrary to the acoustic case, it is not possible
to expand the displacement eld with respect to the transverse modes as
it may be done for the acoustic eld. To cope with this issue, we use Fraser's
biorthogonality relationship between two hybrid variables that mix displacement
and stress components, which enables us expand those hybrid variables
on the transverse modes. We develop the formalism of the Linear Sampling
Method in such framework and show through 2D numerical experiments that
the identication of Dirichlet obstacles and cracks is ecient with only a few
Lamb modes.

Speaker:  Bojan Guzina University of Minnesota, USA 
Title:  Multifrequency reconstruction of scatters via the linear sampling method 
Abstract: 
This paper investigates the multifrequency reconstruction of soundsoft
and penetrable obstacles via the linear sampling method involving either
farfield or nearfield observations of the scattered field. On establishing a suitable
approximate solution to the linear sampling equation and making an assumption
of continuous frequency sweep, two possible choices for a cumulative
multifrequency indicator function of the scatterer's support are examined. The
first alternative, termed the "serial" indicator, is taken as a natural extension of
its monochromatic companion in the sense that its computation entails spacefrequency (as opposed to space) L^{2}norm of a solution to the linear sampling
equation. Under a set of assumptions that include experimental observations
down to zero frequency and compact frequency support of the wavelet used to
illuminate the obstacle, this indicator function is further related to its time domain
counterpart. As a second possibility, the socalled "parallel" indicator
is alternatively proposed as an L^{2}norm, in the frequency domain, of the
monochromatic indicator function. On the basis of a perturbation analysis
which demonstrates that the monochromatic solution of the linear sampling
equation behaves as O(k^{2} − k^{2}_{*}^{−m}), m ≤ 1 in the neighborhood of an isolated
eigenvalue, k^{2}_{*}, of the associated interior (Dirichlet or transmission) problem,
it is found that the "serial" indicator is unable to distinguish the interior from
the exterior of a scatterer in situations when the prescribed frequency band traverses
at least one such eigenvalue. In contrast the "parallel" indicator is, due
to its particular structure, shown to be insensitive to the presence of pertinent
interior eigenvalues (unknown beforehand and typically belonging to a countable
set), and thus to be robust in a generic scattering configuration. A set
of numerical results, including both "fine" and "coarse" frequency sampling, is
included to illustrate the performance of the competing (multifrequency) indicator
functions, demonstrating behavior that is consistent with the theoretical
results. The utility of multifrequency obstacle illumination is further examined
in the context of limitedaperture and noisepolluted observations.

Speaker:  Marc Bonnet Ecole Polytechnique, Paris, France 
Title:  3D timedomain topological sensitivity for wavebased crack imaging in elastic solids 
Abstract: 
The concept of topological sensitivity (TS)
quantifies the perturbation induced to a given cost functional by the
nucleation of an infinitesimal flaw in a reference defectfree body,
and may serve as a flaw indicator function. In this communication, the
TS is derived for threedimensional crack identification exploiting
overdetermined transient elastodynamic boundary data. This entails in
particular the derivation of the relevant polarization tensor, here
given for infinitesimal trial cracks in homogeneous or bimaterial
elastic bodies. Simple and efficient adjointstate based formulations
are used for computational efficiency, allowing to compute the TS
field for arbitrarily shaped elastic solids. The latter is then used
as an indicator function for the spatial location of the sought
crack(s). This approach, which allows a qualitative reconstruction of
cracks in terms of their location but also their orientation
(utilizing the fact that the polarization tensor depends on the normal
to the trial small crack), is implemented within a conventional FEM
platform (the Cast3m general purpose code developed by the French
Atomic Energy Commission (CEA)). A standard Newmark
unconditionallystable timemarching scheme is used for simulating
data, and for computing the free and adjoint solutions used in the
evaluation of the TS field. Extensive 3D timedomain numerical
experiments for the detection of cracks buried either in a homogeneous
pipelike structure or on the interface between two sandwiched plates
highlight its usefulness and performance. Tentative links with other
qualitative identification approaches are discussed. 
Speaker:  Michele Piana
University of Genova, Italy 
Title:  The inhomogeneous Lippmann–Schwinger equation and breast cancer detection using microwaves 
Abstract:  The problem of detecting breast cancer using microwaves
is described by means of the inhomogeneous Lippmann–Schwinger equation and
an hybrid reconstruction method is formulated for quantitatively evaluating
the refractive index of the cancerous tissue.

Speaker:  Drossos Gintides Technical University of Athens, Greece 
Title:  The inverse transmission eigenvalue problem for spherically symmetric index of refraction 
Abstract:  In this talk we will present some new results about the inverse transmission eigenvalue problem for a spherically symmetric
index of refraction ρ > 0. We will show that if ρ satisfies some
integral bound, then it can be uniquely recovered from the transmission eigenvalues
for which the corresponding eigenfunctions are spherically symmetric improving existing results. We
also consider the associated problem for the Schrödinger operator.
Finally, we will present some estimates concerning the existence of complex eigenvalues.

Speaker:  Roland Griesmaier University of Mainz, Germany 
Title:  Iterative and noniterative methods for the reconstruction of wires and thin tubes 
Abstract:  We consider the inverse problem of reconstructing thin wires and tubular conductivity inhomogeneities inside some threedimensional body from
measurements of electrostatic currents and potentials on its boundary. An
asymptotic analysis of the forward problem gives an asymptotic representation formula for the perturbation of the electrostatic potential on the boundary of the body caused by such inclusions. This representation is used to
derive an iterative Newtontype reconstruction method to recover the position and the shape of the wires or tubes using one measurement of the
boundary potential corresponding to some appropriate boundary current.
We discuss the performance of this algorithm, present numerical examples,
and compare those to results obtained by a noniterative qualitative reconstruction method developed earlier for this inverse problem.

Speaker:  Anne Cossonniere Ecole Polytechnique, Paris, France 
Title:  The interior transmission eigenvalue problem: A surface integral equation approach for piecewise constant index 
Abstract: 
We consider the socalled transmission eigenvalue problem in the
scalar case for media with piecewise constant index n. The problem is formulated
using a surface integral equation approach and Fredholm properties of the
obtained problem is analysed in suitable boundary Sobolev spaces for the trace
of the solution and the trace of its normal derivative, namely H^{−1/2} × H^{−3/2}.
We shall discuss in particular the cases where n − 1 changes sign. We also
show how this formulation can be used to numerically compute the transmission
eigenvalues, the main difficulty being the compactness of the underlying
boundary operator in usual trace spaces H^{1/2} × H^{−1/2}. The numerical part is
a joint work with F. Collino and M'B Fares. The proposed method also applies
to 3D Maxwell.
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