IPI
Identification of sound-soft 3D obstacles from phaseless data
Olha Ivanyshyn Rainer Kress
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
Thirty years and still counting
David Colton Rainer Kress
N/A
keywords:
IPI
Electrical impedance tomography using a point electrode inverse scheme for complete electrode data
Fabrice Delbary Rainer Kress
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.
keywords: boundary integral equations Newton scheme. Electrical impedance tomography point electrode model complete electrode model
IPI
Integral equations for inverse problems in corrosion detection from partial Cauchy data
Fioralba Cakoni Rainer Kress
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.
keywords: Inverse boundary value problem integral equations partial boundary measurements impedance boundary condition.

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