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### Open Access Journals

IPI

We introduce a new approach based on the coupling of the method of quasi-reversibility and a simple level set method in order to solve the inverse obstacle problem with Dirichlet boundary condition. We provide a theoretical justification of our approach and illustrate its feasibility with the help of numerical experiments in $2D$.

IPI

Electrical impedance tomography is a noninvasive imaging technique for
recovering the admittivity distribution inside a body from boundary
measurements of current and voltage.
In practice, impedance tomography suffers from inaccurate modelling
of the measurement setting: The exact electrode locations and the
shape of the imaged object are not necessarily known precisely.
In this work, we tackle the problem with imperfect electrode information
by introducing the Fréchet derivative of the boundary measurement
map of impedance tomography with respect to the electrode shapes and
locations. This enables us to include the fine-tuning of the information on
the electrode positions as a part of a Newton-type output least squares
reconstruction algorithm; we demonstrate that this approach is
feasible via a two-dimensional numerical example based on simulated data.
The impedance tomography
measurements are modelled by the complete electrode model,
which is in good agreement with real-life electrode measurements.

IPI

We apply an ``exterior approach" based on the coupling of a method of quasi-reversibility and of a level set method in order to recover a fixed obstacle immersed in a Stokes flow from boundary measurements.
Concerning the method of quasi-reversibility, two new mixed formulations are introduced in order to solve the ill-posed Cauchy problems for the Stokes system by using some classical conforming finite elements. We provide some proofs for the convergence of the quasi-reversibility methods on the one hand and of the level set method on the other hand.
Some numerical experiments in $2D$ show the efficiency of the two mixed formulations and of the exterior approach based on one of them.

IPI

We consider an inverse obstacle problem for the acoustic transient wave equation. More precisely, we wish to reconstruct an obstacle characterized by a Dirichlet boundary condition from lateral Cauchy data given on a subpart of the boundary of the domain and over a finite interval of time. We first give a proof of uniqueness for that problem and then propose an "exterior approach" based on a mixed formulation of quasi-reversibility and a level set method in order to actually solve the problem. Some 2D numerical experiments are provided to show that our approach is effective.

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