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.
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
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
Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case
Eliane Bécache Laurent Bourgeois Lucas Franceschini Jérémi Dardé
Inverse Problems & Imaging 2015, 9(4): 971-1002 doi: 10.3934/ipi.2015.9.971
In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical Lagrange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations.
keywords: heat/wave equation with lateral Cauchy data quasi-reversibility method let-set method inverse obstacle problem Backward heat equation mixed formulation. finite element method
Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems
Jérémi Dardé
Inverse Problems & Imaging 2016, 10(2): 379-407 doi: 10.3934/ipi.2016005
We study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method proves to be efficient even with highly corrupted data.
keywords: quasi-reversibility method. parabolic inverse problems Elliptic inverse problems
Stability estimates for Navier-Stokes equations and application to inverse problems
Mehdi Badra Fabien Caubet Jérémi Dardé
Discrete & Continuous Dynamical Systems - B 2016, 21(8): 2379-2407 doi: 10.3934/dcdsb.2016052
In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method.
keywords: Cauchy problem. Carleman inequalities Stability estimate inverse problems Navier-Stokes equations

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