Uniqueness results for inverse Robin problems with bounded coefficient

In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain \Omega\subset\RR^n, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\Omega)$, r\textgreater{}n. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context

Identification of generalized impedance boundary conditions in inverse scattering problems

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 set. We also extend local stability results to the case of back-scattering data

The Morozov's principle applied to data assimilation problems

This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation

Generalized impedance boundary conditions with vanishing or sign-changing impedance

We consider a Laplace type problem with a generalized impedance boundary condition of the form $\partial_\nu u=-\partial_x(g\partial_xu)$ on a flat part $\Gamma$ of the boundary. Here $\nu$ is the outward unit normal vector to $\partial\Omega$, $g$ is the impedance parameter and $x$ is the coordinate along $\Gamma$. Such problems appear for example in the modelling of small perturbations of the boundary. In the literature, the cases $g=1$ or $g=-1$ have been investigated. In this work, we address situations where $\Gamma$ contains the origin and $g(x)=\mathbb{1}_{x>0}(x)x^\alpha$ or g(x)=-\mbox{sign}(x)|x|^\alpha with $\alpha\ge0$. In other words, we study cases where $g$ vanishes at the origin and changes its sign. The main message is that the well-posedness in the Fredholm sense of the corresponding problems depends on the value of $\alpha$. For $\alpha\in[0,1)$, we show that the associated operators are Fredholm of index zero while it is not the case when $\alpha=1$. The proof of the first results is based on the reformulation as 1D problems combined with the derivation of compact embedding results for the functional spaces involved in the analysis. The proof of the second results relies on the computation of singularities and the construction of Weyl's sequences. We also discuss the equivalence between the strong and weak formulations, which is not straightforward. Finally, we provide simple numerical experiments which seem to corroborate the theorems

The "exterior approach" applied to the inverse obstacle problem for the heat equation

International audienceIn this paper we consider the " exterior approach " to solve the inverse obstacle problem for the heat equation. This iterated approach is based on a quasi-reversibility method to compute the solution from the Cauchy data while a simple level set method is used to characterize the obstacle. We present several mixed formulations of quasi-reversibility that enable us to use some classical conforming finite elements. Among these, an iterated formulation that takes the noisy Cauchy data into account in a weak way is selected to serve in some numerical experiments and show the feasibility of our strategy of identification. 1. Introduction. This paper deals with the inverse obstacle problem for the heat equation, which can be described as follows. We consider a bounded domain D ⊂ R d , d ≥ 2, which contains an inclusion O. The temperature in the complementary domain Ω = D \ O satisfies the heat equation while the inclusion is characterized by a zero temperature. The inverse problem consists, from the knowledge of the lateral Cauchy data (that is both the temperature and the heat flux) on a subpart of the boundary ∂D during a certain interval of time (0, T) such that the temperature at time t = 0 is 0 in Ω, to identify the inclusion O. Such kind of inverse problem arises in thermal imaging, as briefly described in the introduction of [9]. The first attempts to solve such kind of problem numerically go back to the late 80's, as illustrated by [1], in which a least square method based on a shape derivative technique is used and numerical applications in 2D are presented. A shape derivative technique is also used in [11] in a 2D case as well, but the least square method is replaced by a Newton type method. Lastly, the shape derivative together with the least square method have recently been used in 3D cases [18]. The main feature of all these contributions is that they rely on the computation of forward problems in the domain Ω × (0, T): this computation obliges the authors to know one of the two lateral Cauchy data (either the temperature or the heat flux) on the whole boundary ∂D of D. In [20], the authors introduce the so-called " enclosure method " , which enables them to recover an approximation of the convex hull of the inclusion without computing any forward problem. Note however that the lateral Cauchy data has to be known on the whole boundary ∂D. The present paper concerns the " exterior approach " , which is an alternative method to solve the inverse obstacle problem. Like in [20], it does not need to compute the solution of the forward problem and in addition, it is applicable even if the lateral Cauchy data are known only on a subpart of ∂D, while no data are given on the complementary part. The " exterior approach " consists in defining a sequence of domains that converges in a certain sense to the inclusion we are looking for. More precisely, the nth step consists, 1. for a given inclusion O n , in approximating the temperature in Ω n × (0, T) (Ω n := D \ O n) with the help of a quasi-reversibility method, 2. for a given temperature in Ω n × (0, T), in computing an updated inclusion O n+1 with the help of a level set method. Such " exterior approach " has already been successfully used to solve inverse obstacle problems for the Laplace equation [5, 4, 15] and for the Stokes system [6]. It has also been used for the heat equation in the 1D case [2]: the problem in this simple case might be considered as a toy problem since the inclusion reduces to a point in some bounded interval. The objective of the present paper is to extend the " exterior approach " for the heat equation to any dimension of space, with numerical applications in the 2D case. Let us shed some light on the two steps o

On simultaneous identification of a scatterer and its generalized impedance boundary condition

We consider the inverse scattering problem consisting in the identification of both an obstacle and two functional coefficients of a generalized boundary condition prescribed on its boundary, from far--fields due to several plane waves. After proving a uniqueness result for such inverse problem, we define and compute appropriate derivative of the far--field with respect to an obstacle with non constant impedances. A steepest descent method is then applied to retrieve both the obstacle and the functional impedances from the measured far--fields. The feasability of the method is demonstrated with the help of some 2D numerical experiments

Identification of generalized impedance boundary conditions: some numerical issues

We are interested in the identification of a Generalized Impedance Boundary Condition from the far fields created by one or several incident plane waves at a fixed frequency. We focus on the particular case where this boundary condition is expressed as a second order surface operator: the inverse problem then amounts to retrieve the two functions $\lambda$ and $\mu$ that define this boundary operator. We first derive a new type of stability estimate for the identification of $\lambda$ and $\mu$ from the far field when inexact knowledge of the boundary is assumed. We then introduce an optimization method to identify $\lambda$ and $\mu$, using in particular a $H^1$-type regularization of the gradient. We lastly show some numerical results in two dimensions, including a study of the impact of some various parameters, and by assuming either an exact knowledge of the shape of the obstacle or an approximate one.Ce travail concerne l'identification d'une condition aux limites d'impédance généralisée (GIBC) sur le bord d'un objet diffractant à partir du champ lointain créé par une ou plusieurs ondes planes, dans le cas particulier où cette condition est caractérisée par un opérateur d'ordre 2 sur le bord, défini par deux fonctions $\lambda$ et $\mu$ à identifier. Nous commençons par établir une estimation originale de stabilité des fonctions $\lambda$ et $\mu$ cherchées vis à vis du champ lointain, en présence d'une erreur commise sur la forme de l'obstacle. Nous introduisons ensuite une méthode d'optimisation pour identifier $\lambda$ et $\mu$, une régularisation de type $H^1$ du gradient étant utilisée. Nous montrons enfin des résultats numériques de reconstruction en deux dimensions incluant une étude de sensibilité par rapport aux différents paramètres, en supposant une connaissance exacte ou approchée de la forme de l'obstacle

Imaging an acoustic waveguide from surface data in the time domain

International audienceThis paper deals with an inverse scattering problem in an acoustic waveguide. The data consist of time domain signals given by sources and receivers located on the boundary of the waveguide. After transforming the data to the frequency domain, the obstacle is then recovered by using a modal formulation of the Linear Sampling Method. The impact of many parameters are analyzed, such as the numbers of sources/receivers and the distance between them, the time shape of the incident wave and the number and the values of the frequencies that are used. Some numerical experiments illustrate such analysis

A remark on Lipschitz stability for inverse problems

An abstract Lipschitz stability estimate is proved for a class of inverse problems. It is then applied to the inverse Robin problem for the Laplace equation and to the inverse medium problem for the Helmholtz equation