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### Open Access Journals

IPI

The inverse problem for time-harmonic acoustic wave scattering to recover a sound-soft obstacle from a
given incident field and the far field pattern of the scattered field is considered. We split this problem into two
subproblems; first to reconstruct the shape from the modulus of the data and this is followed by
employing the full far field
pattern in a few measurement points to find the location of the obstacle.
We extend a nonlinear integral equation approach for shape reconstruction from the modulus of the far field
data [6] to the three-dimensional case. It is known, see [13], that the location
of the obstacle cannot be reconstructed from only the modulus of the far field pattern since it is invariant under
translations. However, employing the underlying invariance relation
and using only few far field measurements in the backscattering direction we
propose a novel approach for the localization of the obstacle.
The efficient implementation of the method is described and the feasibility of the approach is illustrated by numerical
examples.

keywords:
Acoustic Scattering
,
3D.
,
Inverse
Problem
,
Modulus of Far Field
,
Nonlinear Integral Equations

IPI

For the two dimensional inverse electrical impedance
problem in the case of piecewise constant
conductivities
with the currents injected at adjacent point electrodes
and the resulting voltages measured between the remaining electrodes,
in [3] the authors proposed
a nonlinear integral equation approach that
extends a method that has been suggested
by Kress and Rundell [10] for
the case of perfectly conducting inclusions.
As the main motivation for using a point electrode method
we emphasized on numerical difficulties arising in a corresponding
approach by Eckel and Kress [4, 5] for the complete electrode
model. Therefore, the purpose of the current paper is
to illustrate that the inverse scheme based on
point electrodes can be successfully employed
when synthetic data from the complete electrode
model are used.

IPI

We consider the inverse problem to recover a part $\Gamma_c$
of the boundary of a simply connected planar
domain
$D$ from a pair of Cauchy data of a harmonic function $u$ in $D$ on the remaining part
$\partial D\setminus \Gamma_c$ when $u$ satisfies a homogeneous impedance boundary
condition on $\Gamma_c$. Our approach extends a method that has been suggested
by Kress and Rundell [17] for recovering the interior boundary curve of
a doubly connected planar domain from a pair of Cauchy data on the exterior boundary
curve and is based on a system of nonlinear integral equations. As a byproduct,
these integral equations can also be used for the problem to extend incomplete Cauchy data and to solve
the inverse problem to recover an impedance profile on a known boundary curve.
We present
the mathematical foundation of the method and illustrate
its feasibility by numerical examples.

## Year of publication

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