IPI
A quasi-reversibility approach to solve the inverse obstacle problem
Laurent Bourgeois Jérémi Dardé
Inverse Problems & Imaging 2010, 4(3): 351-377 doi: 10.3934/ipi.2010.4.351
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 $2D$.
keywords: inverse obstacle problem method of quasi-reversibility level set method.
IPI
Fine-tuning electrode information in electrical impedance tomography
Jérémi Dardé Harri Hakula Nuutti Hyvönen Stratos Staboulis
Inverse Problems & Imaging 2012, 6(3): 399-421 doi: 10.3934/ipi.2012.6.399
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.
keywords: Electrical impedance tomography Fréchet derivative domain derivative complete electrode model. model inaccuracies output least squares
IPI
The "exterior approach" to solve the inverse obstacle problem for the Stokes system
Laurent Bourgeois Jérémi Dardé
Inverse Problems & Imaging 2014, 8(1): 23-51 doi: 10.3934/ipi.2014.8.23
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.
keywords: quasi-reversibility method level set method inverse obstacle problem. Stokes system finite element method

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