Quantification of the unique continuation property for the heat equation
Laurent Bourgeois
Mathematical Control & Related Fields 2017, 7(3): 347-367 doi: 10.3934/mcrf.2017012

In this paper we prove a logarithmic stability estimate in the whole domain for the solution to the heat equation with a source term and lateral Cauchy data. We also prove its optimality up to the exponent of the logarithm and show an application to the identification of the initial condition and to the convergence rate of the quasi-reversibility method.

keywords: Heat equation lateral Cauchy data stability estimate Carleman estimate distance function
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
Identification of generalized impedance boundary conditions in inverse scattering problems
Laurent Bourgeois Houssem Haddar
Inverse Problems & Imaging 2010, 4(1): 19-38 doi: 10.3934/ipi.2010.4.19
In the context of scattering problems in the harmonic regime, we consider the problem of identification of some Generalized Impedance Boundary Conditions (GIBC) at the boundary of an object (which is supposed to be known) from far field measurements associated with a single incident plane wave at a fixed frequency. The GIBCs can be seen as approximate models for thin coatings, corrugated surfaces or highly absorbing media. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also consider the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact subset of a finite dimensional space.
keywords: uniqueness generalized impedance boundary conditions Inverse scattering problems in electromagnetism and acoustics stability.
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.
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
An inverse obstacle problem for the wave equation in a finite time domain
Laurent Bourgeois Dmitry Ponomarev Jérémi Dardé
Inverse Problems & Imaging 2019, 13(2): 377-400 doi: 10.3934/ipi.2019019

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.

keywords: Inverse obstacle problem quasi-reversibility level set method unique continuation wave equation lateral Cauchy data

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