Anthony Ventresque
email: aventresque@ntu.edu.sg

Sylvie Cazalens

Thomas Cerqueus

Philippe Lamarre

Gabriella Pasi
email: pasi@disco.unimib.it

Personalization through Query Explanation and Document Adaptation

Keywords: Query Explanation, Document Adaptation, Similarity and Propagation, Semantic Vector Space

come

INTRODUCTION

Personalization is nowadays an important issue for many data and information retrieval applications, aimed at enhancing the user experience and business profits. Coping with a huge and increasing amount of data accessible from the Web, retrieval systems need to display not only relevant information to a specific query but also information that match specific users needs, interests, preferences. And this is seen as an important marketing tool and a requirement for many electronic businesses.

Most of the personalization models are based on two important and complementary aspects: (i) implicit or explicit collection and consequent representation of user's behavior, Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, to post on servers or to redistribute to lists, requires a fee and/or special permission from the publisher, ACM. VLDB '10, September 13-17, 2010, Singapore Copyright 2010 VLDB Endowment, ACM 000-0-00000-000-0/00/00. preferences, interests; (ii) leveraging that knowledge during the retrieval process. This is mainly done by expanding queries [START_REF] Nie | Integrating logical operators in query expansion invector space model[END_REF], by re-ranking search results or by re-indexing documents [START_REF] Bordogna | Personalised indexing and retrieval of heterogeneous structured documents[END_REF]. Some of these tasks, like expanding queries, can be achieved either at the user's side or at the information providers side (i.e. server's side). This paper presents a personalized information retrieval approach which does not assume any user profiling by the information server. Our model favors (semi-)automatic query explanation at the user's side. Then the server reconsiders its document representations in the light of both the query and the explanation it received, thus enhancing the query evaluation process by specifically adapting it to the user. This approach can be useful when the server privacy policy commits it to not profiling the user.

In parallel to an ever increasing amount of information, these last ten years have witnessed a huge work concerning semantics, in particular with the definition of many domain ontologies (like in medecine, biology, almost any domain) and linked technologies. While a specific language, namely SPARQL, has been designed to query triple stores, other approaches consider using the concepts of the ontology to represent both data (let it be semi-structured or unstructured) and queries, thus considering search at a more semantic level. Our approach relies on the usage of a domain ontology, with queries and documents indexed by its set of concepts. We represent both queries and documents by semantic vectors [START_REF] Berry | Matrices, vector spaces, and information retrieval[END_REF]. Each concept is weighted according to its representativeness in the document (respectively the query). As in the classical vector space model, relevance is then modeled as a distance between the two vectors.

This paper does not focus on the indexing, which can be achieved (semi-)automatically by several indexers. Then, given a user query (user's side) and document vectors (server's side), the objectives of our approach are to define (i) a query explanation process and (ii) a light weight adaptation process which adapts the document representations to the query (but does not require re-indexing) and thus conceptually enhancing the query evaluation process. Both objectives are linked, as the adaptation process uses the query explanation. The proposed solution relies on several assumptions and choices.

First, each weighted concept of the query is explained separately. This seems more precise to us than a classical query expansion, which may lead to an unbalance of the original components of the query [START_REF] Nie | Integrating logical operators in query expansion invector space model[END_REF].

Second, the explanation of a given concept considers two notions : given the user's domain ontology, we assume that a similarity function on the set of concepts specifies to which extent a given concept is similar to another one. This is part of the user's modeling of the domain. However, to our view, similarity is not enough to express the importance of a linked concept in a query. Indeed, two different search contexts may require to give more or less importance (interest) to a same similar concept. We formalize this intuition by introducing the concept of propagation function. For each concept, it expresses the importance in the search of concepts with given similarity values. Both the similarity and importance values are computed automatically, but the user can keep control on the process. Given that importance propagation may vary depending on the search context, the user should have a toolbox with several propagation functions which she can choose or which are automatically proposed depending on the context (there may be a profiling module at the user's side which helps).

Finally, both the initial query and the concept explanations (which are vectors) are sent to the information provider, which has to evaluate the relevance of its documents. As explained before, our choice is to avoid re-indexing. Then, given the document representations (i.e. vectors), there are two options : (i) comparing each concept explanation with each document, and then aggregate the results to get a global relevance measure or (ii) computing an adapted document representation (without changing the initial one) which characterizes better each document with respect to the search needs expressed by the concept explanations. We have chosen the second option, in which, in the end, the relevance computation considers the initial query and the document adaptations.

Our contributions are: (i) a new formal approach to personalization which encompasses a query personalization process from users together with a light document adaptation; (ii) our solution does not require that the document provider maintains user profile; (iii) our solution is non-intrusive for existing systems, i.e. it can be plugged in systems without need of document reindexing, query reformulation nor special relevance function.

In the remaining of this paper, we first present a motivating example which shows that questions "how concepts are similar?" and "how much of them are interesting and to what extent?" are crucial for personalized retrieval; this is the core of our solution, and hence we present its architecture (Section 2). Then we formally define query personalization (Section 3) and document adaptation (Section 4). After a discussion on main assumptions of our work (namely ontological heterogeneity and similarity and propagation functions) in Section 6, we conclude (Section 7).

MOTIVATING EXAMPLE AND SYSTEM ARCHITECTURE

While selecting query terms that are fully compatible to documents' providers terms (index terms) is in itself a difficult problem, a same term could also have slightly different meanings to different users (term ambiguity).For instance let us assume three users Alice, Bob and Chikako, willing to adopt a dog. Their request, e.g. "I would like a dog", is very straightforward, and they can send it to an animal welfare organization nearby their living place. Unfortunately, there are a lot of sheltered dogs in these organizations, and scanning their data base could be tedious. Hence, it should be useful to specify which kind of dog each people intend to adopt. On the other hand, if they want a pedigree dog, they could be disappointed when using the keyword "dog". Indeed, dog breeders are canine specialists, and their animals shall not be deemed to be "dog", but "labrador", "akita", or whatever. In both cases, a more accurate description of the user's preferences, i.e. the intended objects should be useful to specify or to expand queries.

Let's Alice and Bob more likely consider as a dog prototype the labrador, while Chikako's dog prototype is represented by akitas (see Figure 1). Even if the concept dog has the same meaning in their mind, the descendant (more specific) concepts of dogs are not all similar, being some are more relevant than others. The consequence is that when querying the animal welfare organization data base, they don't look for the exactly same items. Then, a solution to improve the results expected from the evaluation of their queries should take into account the users' prototypal concepts and their similar concepts. This means to consider a central concept (the prototype) and to formalise a concept similarity function; the user will finally decide which items in the ranked list will be relevant to her/him.Thus, while labradors are the prototype "dogs" to Alice and Bob, it may happen that dalmatians and akita are still acceptable to Alice while not to Bob. Likewise, Chikako considers that dalmatians and labradors, even if they are less relevant than akita, are still ok. We call this combination of proximity to the central concept and interest values a propagation function. In Figure 1 the interest values that Alice, Bob and Chikako give to the concepts according to their similarity to dog. The propagation function aims at describing a dimension of a query, i.e. one of its main concept. As each user manages its own propagation (own similarity and own interest values), we call such a description a personalized dimension of the query. Once the query is personalized, our solution is to keep the query unchanged, but to transform the document representations according to the personalized dimensions. It brings us to adapted documents. Finally, the adapted documents are compared to the initial query. Architecture of our system is composed of five modules (Figure 2), over both user and document provider. Actually, the basic modules (white) are already provided by current systems. You can see two semantic indexing modules on both user's and document provider's side; these modules represent the queries and documents based on the representation model of the IR system (in our case: semantic vectors). Every system has also a relevance computation module (matching module), which ranks documents according to their relevance to the query (cosine). We add to this classical architecture three new modules (grey). On the user's side, the query personalization module explains the central concepts of the query, according to user's similarity and propagation functions. We see below in this paper how a user could obtain these functions (see Section 6.2). The descriptions are then given to the data adaptation module, which transform the document representation w.r.t. them.

QUERY PERSONALIZATION

In this section, after a description of the semantic vectors, we present a formalization of the propagation of users' interests, which constitutes the main process of query presonalization. We provide some inputs on similarity and propagation functions later in this paper (see Section 6.2).

Semantic Vectors

In the vector space model [START_REF] Berry | Matrices, vector spaces, and information retrieval[END_REF], both queries and documents are represented as vectors of keywords (terms). If there are n keywords, each query or document is represented by a vector in the n-dimensional space. Relevance of a document to a query can then be calculated by measuring the proximity of the two vectors. An approach based on semantic vectors [START_REF] Woods | Conceptual indexing: A better way to organize knowledge[END_REF] uses the same kind of multi-dimensional linear space except that it no longer considers as dimensions the keywords, but concepts belonging to a considered ontology: the content of each query (respectively document) is represented by a semantic vector according to each concept.

We consider a very general definition of ontology [START_REF] Gómez-Pérez | Ontological Engineering[END_REF]: it is a set of concepts together with a set of relations between those concepts. The only assumption we make is to be able to compute a similarity between concepts of an ontology, whatever the relations used. In the rest of the paper, we assume the existence of an ontology Ω, CΩ being its set of concepts. Then, a simple formal definition of a semantic vector can be the following:

Definition 1 (Semantic Vector). A semantic vec- tor -→
vΩ is an application defined on the set of concepts CΩ of the ontology:

∀c ∈ CΩ, -→ vΩ : c → [0, 1]
Reference to the ontology will be omitted whenever there is no ambiguity.

Propagation of Interest

Conceptual similarity is a function centered on a concept: it gives a value to every concept according to its similarity to the central concept. We have suggested some propagation functions in [START_REF] Ventresque | Improving interoperability using query interpretation in semantic vector spaces[END_REF]. They are inspired by membership functions used in fuzzy logic, i.e. the most similar concepts are given the value 1, the most dissimilar are weighted with 0, and in between, concepts receive a value according to their similarity:

Pfc(x) = f l 1 ,l 2 (x) = 1 if x ≥ l1 1 l 1 -l 2 x + l 2 l 1 -l 2 if l1 > x > l2 0 if l2 ≥ x

Semantic Personalized Query

As we said in the introduction we do not expand (by propagation) in a single new vector the weights of the central concepts of the query. Moreover, each central concept c of a query -→ q is personalized in a separate vector. Thus a personalized dimension ----→ persDc is a semantic vector which records the propagation of one concept only. We use C-→ q = {c : c is a central concept}; a central concept is an important concept, e.g. any weighted concept, a concept weighted with a greater value than a threshold, etc.

Definition 4 (personalized dimension).

Let -→ q be a query vector and let c be a concept in C-→ q . A semantic vector

----→ persDc is a personalized dimension (persD), iff ∃Pfc∀c ∈ CΩ, ----→ persDc[c ] = Pfc(simc(c )) and ----→ persDc[c] = 1.
The mathematical expression ending the definition means that no matter how the personalized dimension is obtained the only restriction is that no concept can have a greater weight than c. And most importantly, the value of the central concept c in a ----→ persDc is always 1, and not the original value, because a personalized dimension is an explanation of a dimension of the query. The original query, and then the original values of the central concepts, are kept for the matching process. A personalized query is the set of personalized dimensions, one for each central concept of a query: persQ-→ q = { ----→ persDc : c ∈ C-→ q }. Figure 3 shows the personalization of a query -→ q with two weighted concepts c4 and c7.

DOCUMENT ADAPTATION

Once user has explained central concepts of his/her query -→ q , s/he sends the query personalized to the document provider, who adapts his/her documents to the persQ-→ q . Adaptation is not a reindexing of the documents, like with Bordogna and Pasi [START_REF] Bordogna | Personalised indexing and retrieval of heterogeneous structured documents[END_REF] for instance. It is a lightweight filter of documents through what the query explains as important in its personalized dimensions. If a concept ci is relevant for ----→ persDc j , then every document with this concept ci should give that information in its adaptation. In fact, for any ----→ persDc j , documents retain a unique value in the adaptation vector for concept cj, which is the best correlation between personalization value of ci ∈ ----→ persDc j and value

-→ d [ci]
. While all concepts involved in some personalized dimension are already captured by this process, their values are then null in the adaptation. Other concepts, not relevant for any personalized dimension, keep their original value, as they show some dimensions of the document not relevant for the personalized query. Indeed, the norm of the vector gets higher (and consequently, its relevance lower). For example, this is the case for concepts c1 and c9 in Figure 4.

Algorithm 1 details the computation of the personalization of document representation -→ d .This algorithm ensures that all the central concepts of the initial query vector are also weighted in the personalized document representation as far as it is related to them. With respect to the query, the personalization of the document representation is more accurate because it somewhat enforces the characterization of the document over each dimension of the query.

The example of Figure 4 illustrates the computation of a persR. Each persD of the personalized query is combined with the semantic vector of the document. Let us consider ----→ persDc 4 . In the document, the weight of c4 is null. However, the personalized dimension related to c4 weights other concepts. In particular, we have

----→ persDc 4 [ c2 ] = 0.3. As -→ d [ c2 ]= 1,
the resulting product is 0.3. This value improves -→ d [ c4 ] (which is null), so we keep it in the adaptation of the document representation. Hence, in the persR, we can express that the document is related to concept c4 of the query, even if it wasn't the case initially. Likewise, three concepts of the document (c6, c7 and c11) are important to ----→ persDc 7 , and the adaptation retains only one value for input : a semantic vector -→ d and a personalized query persQ-→ q on an ontology Ω output: a semantic vector persR

- → q - → d . begin forall c ∈ C -→ q do forall c : ----→ persDc[c ] = 0 do persR - → q - → d [c] ← max( -→ d [c ] × ----→ persDc[c ], persR - → q - → d [c]); forall c ∈ C-→ q do if ∃c ∈ C-→ q : ----→ persD c [c] = 0 then persR - → q - → d [c] ← 0 else persR - → q - → d [c] ← -→ d [c]; return -→ i d ; end ----→ persDc 7 [
c7] = 0.6. Note that while a classical expanded query would have given one single value from central concept c4 (but possibly on c2 instead of c4), expansion should have weighted 3 concepts from c7 (c6, c7 and c11). Our solution does not add as much noise as it could with classical expansion. Concepts c1 and c9 eventually keep their original value in persR of document d because they are not involved in any persD.

EXPERIMENTS

Our goal is to validate our approach in several steps: first step is cost analysis which enables to quantify the additional costs induced by the method. Second one is just to verify that given a query and different search contexts, the users get different results; this is can be viewed as a minimum requirement to get personnalized results. Finally, the method should be faced with a significant number of users who would estimate whether (or to which extent) they get personalized results. As we currently judge our number of users not significant enough, the paper focuses on the first two steps.

Complexity of our solution relies on the complexity of similarity and propagation functions, which together define query personalization, and document adaptation. There exist a lot of similarity measures (see Section 6.2) but they always consist on two nested loops. Let n be the number of concepts in the ontology, then similarity computation is in O(n 2 ). Propagation gives a weight to every concept; and there are as many propagation functions as there are central concepts. Assume m the number of central concepts, then we need O(m × n) to compute propagation. As m is generally very small compared to n, query personalization has a complexity of O(n 2 + n). However, these two steps can be computed before and cached; so there is not always needed to compute them whenever user queries the system. Adaptation mainly consists of a loop on every concept of the ontology for every document and persD. Then, for every central concept and every document, values are inserted at indices of central concepts. Finally, a loop is processed on concepts to put the values of concepts not involved in any persD. If d is the number of documents, we have an adaptation computation in O(d(2(m × n) + m × log(n) + n)). But this can be strongly reduced by using proper data model: big vectors (as many indices as there are concepts in the ontology) are useless while documents and queries are not expressed on all the concepts but a very small subset, etc. The worst case is not realistic and we assume a fast computation, by replacing at least n by n which is drastically smaller.

Our experiments use the Cranfield corpus and WordNet (considered as a "lightweight" ontology) to index the documents. We developped a prototype software called Mysins [START_REF] Ventresque | Mysins : make your semantic information system[END_REF] with a service oriented architecture. In Mysins search can be personalized by choosing the similarity and propagation functions. Later versions will have a more friendly interface, with automatic or semi-automatic personalized explanations (which can be obtained through profiling for example). The server side of Mysins runs the document adaptation module. We ran the 225 queries of the corpus. The number of retrieved documents is 50 (among the 1400 of the corpus). This later assumption seems reasonable as several user behavior analysis show that users generally consult the first result pages only. We use two different similarity measures: Wu and Palmer [START_REF] Wu | Verb semantics and lexical selection[END_REF] noted sim 1 , and a modified version of sim 1 (which permutes values of three closest groups of concept) as sim 2 . Likewise we use two different propagation functions: prop 1 = f0.95,0.9 and prop 2 = f0.8,0.6.

In order to compare two sets of retrieved documents with their relevance value, we consider two measures. First one (Jaccard coefficient) measures the similarity of the two sets of documents under consideration (without considering their relevance value). It is defined as the number of documents in the intersection divided by the number of documents in the union the two sets. Second one takes into account the order in the ranking of retrieved documents. We have chosen a modified version of Rank Distance (RD) [START_REF] Dinu | On the classification and aggregation of hierarchies with different constitutive elements[END_REF]: each document is given a value according to its position in the top-50, 50 for the 1st, 49 for the 2nd, etc. and 0 for the 51st onwards. Value of each document in first list is then compared to its value in second list. This measure gives of course more importance to permutations on top of the list of retrieved documents. You can see in Figure 5 (a), (b) and (c) the results for (sim 1 , prop 1 ), (sim 1 , prop 2 ), (sim 1 , prop 1 ), (sim 2 , prop 1 ) and (sim 1 , prop 1 ), (sim 2 , prop 2 ) respectively. Every dot corresponds to the comparison of answer lists of a query, using the given parameters. X-axis shows Jaccard measure and y-axis the home-made RD.

It is first worth noticing that dots are not close to (0,0), which means that there are differences between the two results sets. Most dots have Jaccard values between 0.05 and 0.4, while their RD values are between 0.1 and 0.6. Results sets are different (not the same collection of documents) and their ranking are even more different. Propagation eventually seems more important than similarity, because Figures 5 (a) and (c) have more scattered dots, with higher dissimilarity and/or disorder values in average. This section has proven that: (i) additional cost of our solution is limited and (ii) in different contexts the results sets are different and show a personalization of the retrieval. Future work intends to validate the approach with "real" users.

DISCUSSION AND RELATED WORK

In this section, we first position our assumptions and propositions w.r.t. related work. We then discuss how collection of similarity and propagation functions can be thought.

IR, Personalization and Ontologies

Context formalization for IR has focused a lot of attention in past few years [START_REF] Mylonas | Personalized informtion retrieval based on context and ontological knowledge[END_REF]. Many work address this problem through the construction of a contextual space, collecting information on past queries, users clicks, etc. While most of them use terms to characterize the context, Mylonas et alii [START_REF] Mylonas | Personalized informtion retrieval based on context and ontological knowledge[END_REF] propose to use an ontology. Their work is very interesting and can be compared to ours. But it does not use semantic vectors and our solution is more lightweight.

Query expansion has been seen promising to enhance smallsize queries in order to help IR engines [START_REF] Chirita | Personalized query expansion for the web[END_REF]. But while query expansion is a worthwhile contribution to IR, offering more relevant results, it often adds noise in the retrieval. So IR systems need to know when to use it [START_REF] Teeman | To personalize or not to personalize: Modeling queries with variations in user intent[END_REF]. Our solution do not use a query expansion, but a description of central concepts of the query through a propagation function on the concepts of the ontology. We have proven in [START_REF] Ventresque | Improving interoperability using query interpretation in semantic vector spaces[END_REF] that our solution performs better than expansion in general case. And it is specifically more resistant to the use of many concepts.

Assuming a total agreement on ontologies on both sides is not realistic: an ontology is a conceptualization of knowledge upon the world, and we can hardly imagine a unique model of the world for every users. Alignment of ontologies, i.e. mappings between parts of the ontologies [START_REF] Euzenat | Ontology matching[END_REF], are mostly used to address these problems. However these alignments are often incomplete: either because the process is time or resource consuming, or because users do not want to share all their conceptualizations, or because it is not always possible. While query or document indexing could be done on unshared parts of ontologies, it is useless. Indeed, every unshared element could not be understand, and hence no document would be relevant (cosine works only on common parts of queries and documents vectors). However, these unshared parts are meaningful for users and document providers and they are worthy of being used. We propose in [START_REF] Ventresque | Improving interoperability using query interpretation in semantic vector spaces[END_REF] an interpretation process which let users and providers free to use their own ontologies during the information retrieval process. We are still working on an extension of the system described in this paper to heterogeneous context.

Similarity and Propagation

Similarity functions have been studied for a very long time [START_REF] Resnik | Using information content to evaluate semantic similarity in a taxonomy[END_REF][START_REF] Tversky | Features of similarity[END_REF], etc. There exist a lot of different similarity functions, depending on the application and some desired properties. While most of them are context-independant, some takes it into account [START_REF] Kashyap | Semantic and schematic similarities between database objects: a context-based approach[END_REF][START_REF] Keßler | The effect of context on semantic similarity measurement[END_REF]. Even if the problem of finding a personalized similarity function is not exactly addressed in these studies, we assume it could be done quite easily. For instance, we could imagine to collect the relative use frequency of sister concepts (e.g. labrador and akita) to give them different similarity values. We do not address either the issue of propagation function personalization. It is a topic in itself and we focus here on the general process of personalization. However, we can imagine to first use a basic propagation, like we used in the worked mentioned before; then the system could collect feeedback from the user and change this basic propagation, according or not to some context. We would like to focus later on this issue.

CONCLUSION

Personalization of answering, content filtering, recommendation systems, etc. have been a topic of immense interest in recent times. While some solutions use a collection of user's behavior at providers' side or may substantially modify the retrieval system, our solution does not require that the information server maintains any user profile and is non-intrusive for retrieval systems. Moreover, we focus on a description of the query in order to watch documents in the light of its needs, and do not invent a new query formulation paradigm, or a reindexing of documents. Once users provide similarity and propagation functions, our system is lightweight and can be integrated in most information systems, assuming the system use semantic vector representations for queries and documents; then our solution can be used with documents, comments in blog, etc.

Figure 1 :

 1 Figure 1: Alice, Bob and Chikako's similarity ranking of concepts and the propagation of their interest, both w.r.t. concept dog.

Figure 2 :

 2 Figure 2: System architecture. All new grey modules are included in a classical semantic retrieval system.

Definition 2 (

 2 Similarity Function). Let c be a concept of CΩ. simc: CΩ → [0, 1], is a similarity function iff simc(c) = 1 and 0 ≤ simc(cj) ≤ 1 for all cj = c in CΩ. Given a similarity function and a central concept c, we define a propagation function as a function which describes the importance of every concept according to c. We assume this function is monotonically decreasing. Definition 3 (Propagation function). Let c be a concept of CΩ; and let simc be a similarity function. A function Pfc : [0, 1] → [0, 1] simc(c ) → Pfc(simc(c )) is a propagation function from c iff Pfc(simc(c)) = 1, and ∀c k , c l ∈ CΩ simc(c k ) ≤ simc(c l ) ⇒ Pfc(simc(c k )) ≤ Pfc(simc(c l ))

Figure 3 :

 3 Figure 3: A personalized query composed of 2 personalized dimensions.

Algorithm 1 :

 1 Adaptation of document representation wrt. a query.

Figure 4 :

 4 Figure 4: Obtaining the adapted document representation.

Figure 5 :

 5 Figure 5: Comparisons of three pairs of parameters: same similarity and different propagation functions (a), different similarity and same propagation functions (b) and different similarity and different propagation functions, using Jaccard (x-axis) and home-made RD (y-axis).


Identification de conditions d'impédances généralisées: quelques aspects numériques

Résumé : 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 λ et µ à identifier. Nous commençons par établir une estimation originale de stabilité des fonctions λ et µ 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 λ et µ, 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.

Mots-clés : Diffraction inverse, Condition d'impédance généralisée, Estimation de stabilité, Méthode de gradient

Introduction

We are interested in this work by the identification of boundary coefficients in so-called Generalized Impedance Boundary Conditions (GIBC) on a given obstacle from measurements of the scattered field far from the obstacle associated with one or several incident plane waves at a given frequency. More specifically we shall consider boundary conditions of the type ∂u ∂ν + div ∂D (µ∇ ∂D u) + λu = 0 on ∂D where µ and λ are complex valued functions, div ∂D and ∇ ∂D are respectively the surface divergence and the surface gradient on ∂D and ν denotes the outward unit normal on ∂D. In the case µ = 0 this condition is the classical impedance boundary condition (also known as the Leontovitch boundary condition) used for instance to model imperfectly conducting obstacles. The wider class of GIBCs is commonly used to model thin coatings or gratings as well as more accurate models for imperfectly conducting obstacles (see [START_REF] Bendali | The effect of thin coating on the scattering of time-harmonic wave for the helmholtz equation[END_REF][START_REF] Duruflé | Higher order generalized impedance boundary conditions in electromagnetic scattering problems[END_REF][START_REF] Haddar | Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings[END_REF][START_REF] Haddar | Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: The scalar case[END_REF]).

Addressing this problem is motivated by applications in non destructive testing, identification problems or modeling related to stealth technology or antennas. The use of GIBCs has at least two advantages for the inverse problem as compared to the use of an exact model. First, the identification problem becomes less unstable. Second, since solving the forward problem with GIBC is less time consuming, using such model in iterative non-linear methods is more advantageous.

The classical case µ = 0 has been addressed in the literature by several authors, from the mathematical point of view in [START_REF] Sincich | Stable determination of the surface impedance of an obstacle by far field measurements[END_REF][START_REF] Labreuche | Stability of the recovery of surface impedances in inverse scattering[END_REF] and from the numerical point of view in [START_REF] Cakoni | The determination of the surface conductivity of a partially coated dielectric[END_REF][START_REF] Cakoni | The determination of boundary coefficients from far field measurements[END_REF]. The problem of recovering both the shape of the obstacle and the impedance coefficient is also considered in [START_REF] Liu | Reconstruction of the shape and surface impedance from acoustic scattering data for an arbitrary cylinder[END_REF][START_REF] Serranho | A hybrid method for inverse scattering for shape and impedance[END_REF][START_REF] Sini | Reconstruction of shapes and impedance functions using few far-fields measurements[END_REF][START_REF] Sini | Obstacle and boundary determination from scattering data[END_REF]. The case of GIBC has only been recently addressed in [START_REF] Bourgeois | Identification of generalized boundary conditions in inverse scattering problems[END_REF] where uniqueness and local stability results have been reported.

The present work complements these first investigations in two directions. The first one is on the theoretical level. We prove stability of the reconstruction of the impedances on a perturbed geometry relying on two properties: continuity of the far field with respect to the obstacle, uniformly with respect to the impedance coefficients, stability for the inverse coefficient problem for a known obstacle. This kind of stability result would be useful for instance when the geometry has been itself reconstructed from measurements using some qualitative methods (e.g. sampling methods [START_REF] Cakoni | Qualitative Methods in Inverse Scattering Theory[END_REF][START_REF] Kirsch | The factorization method for inverse problems[END_REF]) and therefore is known only approximately. This result may be also useful in understanding the convergence of iterative methods to reconstruct both the obstacle and the coefficients where the updates for the geometry and the physical parameters are made alternatively. Let us also mention that the proof of our stability result can be straightforwardly extended to other identification problems that enjoy the two properties indicated above.

In a second part, we investigate a numerical optimization method to identify the boundary coefficients. We propose a reconstruction procedure based on a steepest descent method with H 1 (∂D) regularization of the gradient. The accuracy and stability of the inversion scheme is tested through various numerical experiments in a 2D setting. Special attention is given to the case of non regular coefficients and inexact knowledge of the boundary ∂D.

The outline of our article is the following. In section 2 we introduce and study the forward and inverse problems. Section 3 is dedicated to the derivation of a stability result with respect to the geometry. The numerical part is the subject of section 4.

The forward and inverse problem 2.1 The forward scattering problem

= u s + u i such that    ∆u + k 2 u = 0 in Ω div ∂D (µ∇ ∂D u) + ∂u ∂ν + λu = 0 on ∂D (1)
and u s satisfies the Sömmerfeld radiation condition lim

R→∞ |x|=R ∂u s ∂r -iku s 2 ds = 0
where k is the wave number,

u i = e ik d•
x is an incident plane wave where d belongs to the unit sphere of R d denoted S d-1 ,

u s ∈ V := {v ∈ D ′ (Ω), ϕv ∈ H 1 (Ω) ∀ ϕ ∈ D(R d ) and v |∂D ∈ H 1 (∂D)} is the scattered field,

INRIA

div ∂D and ∇ ∂D are respectively the surface divergence and the surface gradient on ∂D, ν is the outward unit normal on ∂D.

For a precise definition of the surface operators we refer to Chapter 5 of [START_REF] Henrot | Variation et optimisation de formes[END_REF]. For v ∈ H 1 (∂D) the surface gradient

∇ ∂D v lies in L 2 t (∂D) := {V ∈ (L 2 (∂D)) d , V • ν = 0}. Moreover, div ∂D (µ∇ ∂D u) is defined in H -1 (∂D) for µ ∈ L ∞ (∂D) by div ∂D (µ∇ ∂D u), v H -1 (∂D),H 1 (∂D) := - ∂D µ∇ ∂D u • ∇ ∂D vds ∀v ∈ H 1 (∂D) .
In order to prove well-posedness of the solution of problem (1) and then to numerically solve the problem, we introduce the so-called Dirichlet-to-Neumann map so that we can can give an equivalent formulation of (1) in a bounded

domain Ω R = Ω ∩ B R where B R is the ball of radius R such that D ⊂ B R . The Dirichlet-to-Neumann map, S R : H 1/2 (∂B R ) → H -1/2 (∂B R ) is defined for g ∈ H 1/2 (∂B R ) by S R g := ∂u e /∂r| ∂BR
where u e is the radiating solution of the Helmholtz equation outside B R and u e = g on ∂B R . We have the following properties (see [START_REF] Nédélec | Acoustic and Electromagnetic Equations[END_REF] p.97)

ℜe S R g, g ≤ 0 and ℑm S R g, g ≥ 0 ∀g ∈ H 1/2 (∂B R ), where •, • is the duality product between H -1/2 (∂B R ) and H 1/2 (∂B R ). Solving (1) is equivalent to find u in V R := {v ∈ H 1 (Ω R ); v |∂D ∈ H 1 (∂D)} such that: P(λ, µ, ∂D)            ∆u + k 2 u = 0 inside Ω R div ∂D (µ∇ ∂D u) + ∂u ∂ν + λu = 0 on ∂D ∂u ∂r -S R (u) = ∂u i ∂r -S R (u i ) on ∂B R (2)
We assume

Assumption (H1) ∂D is Lipschitz continuous, (λ, µ) ∈ (L ∞ (∂D)) 2 are such that ℑm(λ) ≥ 0 , ℑm(µ) ≤ 0 a.e. in ∂D
and there exists c > 0 such that ℜe(µ) ≥ c a.e. in ∂D.

In the following, K will be a compact subset of (L ∞ (∂D)) 2 such that there exists a constant c K > 0 such that if (λ, µ) ∈ K then assumption (H1) holds with the constant c = c K .

Theorem 2.1. If assumption (H1) is satisfied then problem P(λ, µ, ∂D) has a unique solution u in V R .

Proof. In order to prove this result, we use the variational formulation of (2) given by : find

u ∈ V R such that for all v ∈ V R a(u, v) = l(v) , with a(u, v) := ΩR (∇u • ∇v -k 2 uv)dx + ∂D (µ∇ ∂D u • ∇ ∂D v -λuv)ds -S R u, v l(v) := ∂BR ∂u i ∂r -S R (u i ) vds Remark that V R equipped with the scalar product (•, •) VR := (•, •) H 1 (ΩR) + (•, •) H 1 (∂D)
is a Hilbert space. Thanks to the Riesz representation theorem we define two operators J R and

K R from V R into itself that satisfy for all u, v ∈ V R (J R u, v) VR = (u, v) H 1 (ΩR) + ∂D (µ∇ ∂D u • ∇ ∂D v)ds -S R u, v , (K R u, v) VR = -(1 + k 2 )(u, v) L 2 (ΩR) - ∂D λuvds and F R an element of V R uniquely determined by (F R , v) VR = l(v) ∀v ∈ V R .
The variational formulation is equivalent to find u ∈ V R such that

(J R + K R )u = F R in V R .
As soon as ℜe(µ(x)) ≥ c > 0 for almost every x ∈ ∂D the operator J R is an isomorphism of V R . Moreover the Rellich theorem tells us that H 1 (Ω R ) and H 1 (∂D) are compactly embedded into L 2 (Ω R ) and L 2 (∂D) respectively. From that we deduce that K R is a compact operator on V R and therefore our problem is of Fredholm type. As a consequence, it is sufficient to prove uniqueness to have existence. Let u be satisfying

a(u, v) = 0 ∀v ∈ V R
then taking v = u and the imaginary part of this equation we have

∂D (ℑm(µ)|∇ ∂D u| 2 -ℑm(λ)|u| 2 )ds -ℑm S R u, u = 0.

Assumption (H1) implies that

ℑm S R u, u = 0 which implies that u| ∂BR = 0 (see [START_REF] Nédélec | Acoustic and Electromagnetic Equations[END_REF] p. 103). Therefore ∂u ∂r | ∂BR = S R u = 0 thanks to the boundary condition on ∂B R . To conclude, u is solution to a Cauchy problem inside Ω R with data equal to zero on ∂B R , hence u = 0 on Ω R and P(λ, µ, ∂D) has a unique solution in V R .

INRIA

For λ and µ satisfying (H1) let us define the operator A λ,µ by

(A λ,µ u, v) VR = ΩR (∇u•∇v-k 2 uv)dx+ ∂D (µ∇ ∂D u•∇ ∂D v -λuv)ds-S R u, v . The map (λ, µ) -→ A λ,µ ∈ L(V R , V R )
where L(V R , V R ) is the set of bounded operators on V R , is continuous and for every (λ, µ) that satisfy (H1) A λ,µ is invertible on V R . We establish in the following an uniform bound for its inverse A -1 λ,µ with respect to the impedance coefficients.

Proposition 2.2. There exists a constant C K > 0 such that for every (λ, µ)

∈ K |||A -1 λ,µ ||| ≤ C K where ||| • ||| stands for operators norm.
Proof. Consider the map

A : (λ, µ) ∈ K -→ A -1 λ,µ ∈ L(V R , V R ).
We shall prove that this map is continuous and since K is compact, we obtain the desired uniform estimate. Take (λ, µ) ∈ K and (h,

l) ∈ (L ∞ (∂D)) 2 such that (λ + h, µ + l) ∈ K and for F ∈ V R define u and u h,l in V R by A λ,µ u = F A λ+h,µ+l u h,l = F.
We have

A -1 λ+h,µ+l F -A -1 λ,µ F VR = u h,l -u VR = A -1 λ+h,µ+l (A λ,µ -A λ+h,µ+l )u VR but A λ,µ depends continuously on (λ, µ) lim (h,l)→0 |||A λ,µ -A λ+h,µ+l ||| = 0 (3)
and using the results on Neumann series of chapter 10 in [START_REF] Kress | Linear integral equations[END_REF] there exists a constant C(λ, µ) which depends on λ and µ such that for every (h, l) sufficiently small

|||A -1 λ+h,µ+l ||| ≤ C(λ, µ). (4)
(3) and (4) together imply that lim

(h,l)→0 |||A -1 λ+h,µ+l -A -1 λ,µ ||| = 0
in other words, the map A is continuous on the compact set K.

From now on we shall denote A := A λ,µ . Under the sufficient conditions (H1) on the impedance coefficients λ and µ that ensure existence and uniqueness for the forward problem we can study the inverse coefficients problem and this is the aim of the next section.

Formulation of the inverse problem

We recall the following asymptotic behaviour for the scattered field (see [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]):

u s (x) ∼ e ikr r (d-1)/2 u ∞ (x) + O 1 r r -→ +∞
uniformly for all the directions x = x/r ∈ S d-1 . The far-field u ∞ ∈ L 2 (S d-1 ) is uniquely determined by this expression and we have an integral representation on the boundary Γ of some regular open domain that contains D :

u ∞ (x) = Γ u s (y) ∂Φ ∞ (y, x) ∂ν(y) - ∂u s (y) ∂ν Φ ∞ (y, x) ds(y) ∀x ∈ S d-1 , (5)
where Φ ∞ is the far-field associated with the Green function of the Helmholtz equation and is defined by

in R 2 Φ ∞ : (y, x) ∈ R 2 × S 1 -→ e iπ/4 √ 8πk e -iky•x in R 3 Φ ∞ : (y, x) ∈ R 3 × S 2 -→ 1 4π e -iky•x ,
and the second integral has to be understood as a duality product between H -1/2 (Γ) and H 1/2 (Γ). We can also define the far-field map

T : (λ, µ, ∂D) → u ∞
where u ∞ is the far-field associated with the scattered field u s = uu i and u is the unique solution of problem P(λ, µ, ∂D). The inverse coefficients problem is the following : given an obstacle D an incident direction d ∈ S d-1 , its associated far-field pattern u ∞ for all x ∈ S d-1 , reconstruct the corresponding impedance coefficients λ and µ. In other words the inverse problem amounts to invert the map T with respect to the coefficients λ and µ for a given ∂D. The first natural question related to this inverse problem is injectivity of T and stability properties of the inverse map. These questions have been addressed in [START_REF] Bourgeois | Identification of generalized boundary conditions in inverse scattering problems[END_REF] where for instance results on local stability in compact sets have been reported. Our subsequent analysis on the stability with respect to perturbed obstacles of the reconstruction of λ and µ will depend on stability for the inverse map of T . We therefore shall assume the following

Assumption (H2)
there exists a compact K ⊂ (L ∞ (∂D)) 2 such that for (λ, µ) ∈ K there exists a constant C(λ, µ, K) which depends on λ, µ and K such that for every ( λ, µ) ∈ K we have

λ -λ L ∞ (∂D) + µ -µ L ∞ (∂D) ≤ C(λ, µ, K) T (λ, µ, ∂D) -T ( λ, µ, ∂D) L 2 (S d ) .

INRIA

We refer to [START_REF] Bourgeois | Identification of generalized boundary conditions in inverse scattering problems[END_REF] for examples of compacts K for which (H2) holds. We notice for instance that uniqueness with single incident wave fails in general except if one assumes that parts of λ and µ are known a priori. Moreover we may need to add some restriction for the incident direction or for the geometry of the obstacle (see [START_REF] Bourgeois | Identification of generalized boundary conditions in inverse scattering problems[END_REF] for more details). There are also global stability results available in the literature (that is C(λ, µ, K) does not depend on λ) for classical impedance boundary conditions (i.e. µ = 0), see [START_REF] Labreuche | Stability of the recovery of surface impedances in inverse scattering[END_REF][START_REF] Sincich | Stable determination of the surface impedance of an obstacle by far field measurements[END_REF].

Remark 2.3. u ∞ is an analytical function on S d-1 (see [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]) hence assuming that we know the far-field everywhere on S d-1 is equivalent to assume that we know it on a non-empty part of S d-1 .

3 A stability estimate for the impedances in the presence of a perturbed obstacle

The forward and inverse problems on a perturbed obstacle

The question we will try to answer in this section concerns the stability of the reconstruction of the impedances with respect to the geometry ∂D : assume that we reconstruct the impedance coefficients on a "perturbed" obstacle, do we have some stability estimates on this reconstruction?

Remark 3.1. Here "perturbed" means that we have an approximate knowledge of the obstacle (see Figure 2) that can be obtained after (or during) a numerical reconstruction of the obstacle for example. More precisely, we may need the stability result we present hereafter to ensure convergence of hybrid reconstruction methods of both the impedance coefficients and the obstacle as presented in [START_REF] Cakoni | The determination of the surface conductivity of a partially coated dielectric[END_REF], [START_REF] Sini | Reconstruction of shapes and impedance functions using few far-fields measurements[END_REF] or [START_REF] Serranho | A hybrid method for inverse scattering for shape and impedance[END_REF] for the case of a classical impedance condition.

First we shall define the "perturbed" problem and then present a continuity result with respect to the obstacle uniformly with respect to the impedance coefficients. The stability result is then deduced thanks to the stability estimate for the inverse coefficients problem with a known obstacle given in assumption (H2).

Figure 2: Illustration of the "perturbed" geometry.

We assume that D is of class C 1 and we consider problem (2) with a "perturbed" geometry D ε ⊂ B R of D. We say that D ε is a perturbation of D as soon as one can find a function ε ∈ (C 1 (R d )) d sufficiently small so that Id + ε is a C 1 -diffeomorphism and

∂D ε = {x + ε(x) ; x ∈ ∂D} .
Recall that whenever [START_REF] Henrot | Variation et optimisation de formes[END_REF]).

ε (W 1,∞ (R d )) d < 1 the map f ε := Id + ε where Id stands for the identity of R d , is a C 1 -diffeomorphism of R d (see chap 5. of
The inverse problem. In this situation of perturbed knowledge of the obstacle, the inverse problem consists in finding the better approximations λ ε and µ ε of the exact impedance coefficients for example minimizing the distance between T (λ, µ, ∂D) and the far-field T (λ ε , µ ε , ∂D ε ). Following this idea, let us assume that for some δ ≥ 0

T (λ ε , µ ε , ∂D ε ) -T (λ, µ, ∂D) L 2 (S d-1 ) ≤ δ .
From this information and from the distance between ∂D and ∂D ε , can we have an estimate on the boundary coefficients? In other words, do we have

λ ε • f ε -λ L ∞ (∂D) + µ ε • f ε -µ L ∞ (∂D) ≤ g(δ, ε) (6)
for some function g such that g(δ, ε) → 0 as δ → 0 and ε → 0? In order to prove such a result, we first need a continuity property of T with respect to ∂D.

Continuity of the far-field with respect to the obstacle

In the following, for a given ε 0 > 0 we assume that Assumption (H3)

D is an open bounded domain of R d of class C 1 , ε ∈ (C 1 (R d )) d with ε < 1,
where

• stands for the (W 1,∞ (R d )) d norm • = • L ∞ (R d )) d + ∇ • (L ∞ (R d )) d×d .
To prove the continuity of the far-field with respect to the obstacle, we first establish this result for the scattered field. Define

λ ε = λ • f -1 ε and µ ε = µ • f -1 ε two elements of L ∞ (∂D ε ).
To evaluate the distance between the solution u of P(λ, µ, ∂D) and the solution u ε of P( λ ε , µ ε , ∂D ε ) we first transport u ε on the fixed domain Ω R by using a diffeomorphism between

Ω ε R := Ω R \ D ε and Ω R . Let R 1 > 0 be such that ∀ε satisfying (H3) D ε ⊂ B R1 and R 2 > R 1 sufficiently large (see remark 3.2) such that there exists a C ∞ (R d ) cut-off function ψ such that ψ is compactly supported in B R2 , ψ = 1 on B R1 and ψ W 1,∞ (R d ) < 1. Then Id + ψε is a C 1 -diffeomorphism satisfying (Id + ψε)(Ω R ) = Ω ε R .
We shall still denote by f ε this diffeomorphism. From now on we take R := R 2 in the formulation (2). Define u ε := u ε •f ε where u ε is the solution of P( λ ε , µ ε , ∂D ε ), then the following estimate holds. Theorem 3.3. Let (λ, µ) be in K. Suppose that (H3) is satisfied, there exists two constants ε 0 > 0 and C K which depend only on K such that for every ε ≤ ε 0 we have:

INRIA

u ε -u H 1 (ΩR) ≤ C K ε . (7)
We denote by O(y) a C ∞ (R) function such that

|O(y)| ≤ C|y| ∀y ∈ R (8)
for C > 0 independent of y. In the proof of the Theorem we will need the following technical Lemma whose proof is postponed to the end of the Theorem's proof.

Lemma 3.4. There exists a constant C > 0 such that for every ε satisfying (H3) and for every u, v in H 1 (∂D) and µ ∈ L ∞ (∂D) we have

∂Dε (µ • f -1 ε )∇ ∂Dε (u • f -1 ε ) •∇ ∂Dε (v • f -1 ε )ds ε - ∂D µ∇ ∂D u • ∇ ∂D vds ≤ C u H 1 (∂D) v H 1 (∂D) µ L ∞ (∂D) ε .
Proof of Theorem 3.3. The weak formulation of

P( λ ε , µ ε , ∂D ε ) is : find u ε ∈ V ε R := {v ∈ H 1 (Ω ε R ); v |∂Dε ∈ H 1 (∂D ε )} such that for all v ε ∈ V ε R we have : a ε (u ε , v ε ) = l ε (v ε ) RR n 7449 with a ε (u ε , v ε ) = Ω ε R (∇u ε • ∇v ε -k 2 u ε v ε )dx ε + ∂Dε ( µ ε ∇ ∂Dε u ε • ∇ ∂Dε v ε -λ ε u ε v ε )ds ε -S R u ε , v ε l ε (v ε ) = ∂BR ∂u i ∂r -S R (u i ) v ε ds We define a new bilinear form on V R a ε (u, v) := a ε (u • f -1 ε , v • f -1 ε ) ∀u, v ∈ V R . As f ε (∂B R ) = ∂B R , we have l(v) = l ε (v • f -1 ε ) and u ε is solution of a ε ( u ε , v) = l(v) ∀v ∈ V R .
In addition for every

v ∈ V R S R v = S R (v • f -1 ε )
and as a consequence for every u, v

∈ V R we have S R u, v = S R (u • f -1 ε ), v • f -1 ε
. Using the change of variables formula for integrals (see chap. 5 of [START_REF] Henrot | Variation et optimisation de formes[END_REF]), we have :

a ε (u, v) = ΩR (∇u • P ε • ∇v -k 2 u v)J ε dx + ∂Dε µ ε [∇ ∂Dε (u • f -1 ε )] • [∇ ∂Dε (v • f -1 ε )]ds ε - ∂D λu vJ ν ε ds -S R u, v
where

J ε = |det(∇f ε )|
, where det stands for the determinant of a matrix,

P ε is the matrix (∇f ε ) -1 (∇f ε ) -T , J ν ε = J ε (∇f ε ) -T ν
(∇f ε (x)) -1 = (Id + ∇ε(x)) -1 uniformly for x ∈ ∂D (∇f ε ) -1 (x) = ∞ n=0 (-1) n (∇ε(x)) n = (1 + O( ε ))Id.
As a consequence P ε (x) expands uniformly for x ∈ ∂D as

P ε (x) = Id(1 + O( ε ))
and we also have

J ε (x) = 1 + O( ε ) and J ν ε (x) = 1 + O( ε ) INRIA since det(Id+∇ε(x)) = 1+div(ε)+O( ε 2
). Using all these results and Lemma 3.4 we are able to prove that the difference between the bilinear forms a ε and a is small. For u and v two functions of V R we have

ΩR (∇u • P ε • ∇vJ ε -k 2 u vJ ε -∇u • ∇v + k 2 u v)dx ≤ C u VR v VR ε ∂D {λu vJ ν ε -λu v}ds ≤ C λ L ∞ (∂D) u VR v VR ε ∂Dε (µ • f -1 ε )∇ ∂Dε (u • f -1 ε ) •∇ ∂Dε (v • f -1 ε )ds ε - ∂D µ∇ ∂D u • ∇ ∂D vds ≤ C u VR v VR) µ L ∞ (∂D) ε .
where the constant C does not depend on ε. One obtains for the bilinear forms a ε and a:

| a ε (u, v) -a(u, v)| ≤ C K u VR v VR ε (9)
where C K does not depend on λ, µ and ε. Thanks to the Riesz representation theorem we uniquely define

A ε from V R into itself by (A ε v, w) VR = a ε (v, w) ∀ v, w ∈ V R .
We recall the definition of the operator

A of V R (Av, w) VR = a(v, w) ∀v, w ∈ V R and of the element F of V R (F, w) VR = l(w) ∀w ∈ V R .
Thanks to inequality (9), we have

|||A ε -A||| ≤ C K ε .
To have information on the scattered field we should have information on the inverse of the operators and we will use once again the results on Neumann series of [START_REF] Kress | Linear integral equations[END_REF]. Actually, as soon as

|||A -1 (A ε -A)||| ≤ 1/2 which is true when ε 0 ≤ 1/(2C 2 K ) (see Proposition 2.
2), the inverse operator of A ε satisfies

|||A -1 ε ||| ≤ |||A -1 ||| 1 -|||A -1 (A ε -A)||| ≤ 2C K .
From the identity

u ε -u = A -1 ε (A -A ε )u we deduce u ε -u VR = A -1 ε ((A -A ε )u) VR ≤ |||A -1 ε ||| (A ε -A)u VR ≤ 2C 2 K ε u VR ≤ 2C 3 K ε F VR
where we again used Proposition 2.2 for the last inequality. This provides the desired estimate.

∂D ∩ V = {ϕ(ξ) ; ξ ∈ U }
where V is a neighbourhood of x 0 and ϕ(0) = x 0 and such that

e i := ∂ϕ ∂ξ i (0) , for i = 1, d -1
form a basis of the tangential plane to ∂D at x 0 . We can also use this parametrization to describe a surfacic neighbourhood of x 0,ε := f ε (x 0 ), similarly there exists a neighbourhood V ε of x 0,ε such that

∂D ε ∩ V ε = {ϕ ε (ξ) ; ξ ∈ U }
where ϕ ε := f ε •ϕ. We define the tangential vectors of ∂D ε at point x 0,ε = ϕ ε (0) by

e ε,i := ∂ϕ ε ∂ξ i (0) for i = 1, d -1
and thanks to the chain rule

e ε,i = ∇f ε (x 0 )e i . (10)
Remark that as ∇f ε (x 0 ) is an invertible matrix ( ε < 1), the family e ε,i is a basis of the tangent plane to ∂D ε at x ε,0 . Finally, we define the covariant basis of the cotangent planes of ∂D at point x 0 and of ∂D ε at point x 0,ε by e i • e j = δ i j and e i ε • e ε,j = δ i j for i, j = 1, d -1.

Using this definition and (10) we have the relation

e i ε = (∇f ε ) -T e i i = 1, d -1.
Finally in the covariant basis, the tangential gradient ∇ ∂D for w ∈ H 1 (∂D) at point x 0 is

∇ ∂D w(x 0 ) = d-1 i=1 ∂ w ∂ξ i (0)e i
where w := w • ϕ. Similarly, at point x 0,ε we have for

w ε ∈ H 1 (∂D ε ) ∇ ∂Dε w ε (x 0,ε ) = d-1 i=1 ∂ w ε ∂ξ i (0)e i ε
where

w ε := w ε • ϕ ε . As a consequence, for w ∈ H 1 (∂D) ∇ ∂Dε (w • f -1 ε )(x 0,ε ) = d-1 i=1 ∂ w ∂ξ i (0)e i ε = d-1 i=1 ∂ w ∂ξ i (0)(∇f ε (x 0 )) -T e i = (∇f ε (x 0 )) -T ∇ ∂D w(x 0 ) INRIA because w • f -1 ε • ϕ ε = w.
By this formula, we just proved that for every

x ε = f ε (x), x ∈ ∂D we have ∇ ∂Dε (w • f -1 ε )(x ε ) = (∇f ε (x)) -T ∇ ∂D w(x) (11)
for every w ∈ H 1 (∂D). For u, v and µ, change of variables in the boundary integral

(x = f -1 ε (x ε )) gives ∂Dε (µ • f -1 ε )∇ ∂Dε (u • f -1 ε ) • ∇ ∂Dε (v • f -1 ε )dx ε = ∂D µ(x)[∇ ∂Dε (u • f -1 ε )(f ε (x))] • [∇ ∂Dε (v • f -1 ε )(f ε (x))]J ν ε dx
and thanks to the relation ( 11)

∂Dε (µ • f -1 ε )∇ ∂Dε (u • f -1 ε ) • ∇ ∂Dε (v • f ε ) -1 dx ε = ∂D µ(x)[∇ ∂D u(x)](∇f ε (x)) -1 (∇f ε (x)) -T [∇ ∂D v(x)]J ν ε dx.
Finally recalling that

(∇f ε (x)) -1 (∇f ε (x)) -T = P ε (x) = (1 + O( ε ))Id and J ν ε = 1 + O( ε )
we may write

∂Dε (µ • f -1 ε )∇ ∂Dε (u • f -1 ε ) • ∇ ∂Dε (v • f -1 ε )dx ε = (1 + O( ε )) ∂D µ∇ ∂D u • ∇ ∂D vdx
which is the desired result.

Corollary 3.5. Let (λ, µ) be in K. Suppose that (H3) is satisfied, there exists two constants ε 0 > 0 and C K which depend only on K such that for every ε ≤ ε 0 we have:

T (λ • f -1 ε , µ • f -1 ε , ∂D ε ) -T (λ, µ, ∂D) L 2 (S d-1 ) ≤ C K ε .
Proof. Let u ∞ be the far-field that corresponds to the obstacle D and u ∞ ε the one that corresponds to the obstacle D ε . We now write the integral representation formula for the far-field on ∂B R and as

u ε | ∂BR = u ε | ∂BR we obtain u ∞ ε (x) = ∂BR u ε (y) ∂ Φ ∞ (y, x) ∂ν(y) - ∂ u ε (y) ∂ν Φ ∞ (y, x) ds(y) .
We know that the exterior DtN map defined in section 2 is continuous from H 1/2 (∂B R ) to H -1/2 (∂B R ) and as a consequence

∂ u ε ∂ν H -1/2 (∂BR) ≤ C u ε H 1/2 (∂BR) , finally u ∞ ε (x) -u ∞ (x) L 2 (S d-1 ) ≤ C u ε -u H 1/2 (∂BR)
. The trace is continuous from H 1 (Ω R ) into H 1/2 (∂B R ) so we combine this last inequality with [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF] to obtain the continuity result

T (λ • f -1 ε , µ • f -1 ε , ∂D ε ) -T (λ, µ, ∂D) L 2 (S d-1 ) ≤ C K ε .
This ends the first step of the stability result. In the next section we use this continuity result and the stability assumption of section 2.2 to prove a stability estimate with respect to the obstacle.

Conclusion

In this section we assume that we satisfy (H2) for a given K and (H3).

Theorem 3.6. There exists a constant ε 0 which depends only on K such that for every (λ, µ) ∈ K there exists a constant C(λ, µ, K) such that for every ε ≤ ε 0 and for every

(λ ε • f ε , µ ε • f ε ) ∈ K that satisfy T (λ ε , µ ε , ∂D ε ) -T (λ, µ, ∂D) L 2 (S d-1 ) ≤ δ for δ > 0 we have λ ε • f ε -λ L ∞ (∂D) + µ ε • f ε -µ L ∞ (∂D) ≤ C(λ, µ, K)(δ + ε ).
Proof. The idea of the proof is to split the uniform continuity with respect to the obstacle and the stability with respect to the coefficients and to use assumption (H2) . The aim is to prove that

T (λ ε • f ε , µ ε • f ε , ∂D) -T (λ, µ, ∂D) L 2 (S d-1 ) ≤ g(δ, ε )
for a suitable g. We have

T (λ ε • f ε , µ ε • f ε , ∂D) -T (λ, µ, ∂D) L 2 (S d-1 ) ≤ T (λ ε , µ ε , ∂D ε ) -T (λ ε • f ε , µ ε • f ε , ∂D) L 2 (S d-1 ) + T (λ, µ, ∂D) -T (λ ε , µ ε , ∂D ε ) L 2 (S d-1 )
but the hypothesis of the Theorem tells us that

T (λ ε , µ ε , ∂D ε ) -T (λ, µ, ∂D) L 2 (S d-1 ) ≤ δ
and as we have continuity with respect to the obstacle (see Corollary 3.5) we have

T (λ ε , µ ε , ∂D ε ) -T (λ ε • f ε , µ ε • f ε , ∂D) L 2 (S d-1 ) ≤ C K ε because we took λ ε • f ε and µ ε • f ε in a given compact set K. Finally we have T (λ ε • f ε , µ ε • f ε , ∂D) -T (λ, µ, ∂D) L 2 (S d-1 ) ≤ C K (δ + ε )
and the stability assumption in (H2) tells us that

λ ε • f ε -λ L ∞ (∂D) + µ ε • f ε -µ L ∞ (∂D) ≤ C(λ, µ, K) T (λ ε • f ε , µ ε • f ε , ∂D) -T (λ, µ, ∂D) L 2 (S d-1 )
which concludes the proof.

INRIA

Remark 3.7. The local nature of this estimate only depends on the local stability result for the impedances. In the case of a classic impedance boundary condition (µ = 0) we could use the global stability results of Sincich in [START_REF] Sincich | Stable determination of the surface impedance of an obstacle by far field measurements[END_REF] or of Labreuche in [START_REF] Labreuche | Stability of the recovery of surface impedances in inverse scattering[END_REF] and obtain a constant C(λ, µ, K) that only depends on K.

Numerical algorithm and experiments

This section is dedicated to the effective reconstruction of the impedance functional coefficients λ and µ from the observed far-field u ∞ obs ∈ L 2 (S d-1 ) associated to a given incident direction and a given obstacle. In this view we shall minimize the cost function

F (λ, µ) = 1 2 T (λ, µ) -u ∞ obs 2 L 2 (S d-1 ) (12)
with respect to λ and µ using a steepest descent method.

Fréchet derivative of the cost function

First of all, to minimize the cost function F , we have to compute its Fréchet derivative and hence to calculate the Fréchet derivative of the far-field map T .

Lemma 4.1. The far-field map T is Fréchet differentiable and for (λ, µ) ∈ (L ∞ (∂D)) 2 that satisfy (H1) its Fréchet derivative dT λ,µ :

L ∞ (∂D) 2 → L 2 (S d-1 ) maps (h, l) to v ∞ h,l such that v ∞ h,l (x) := p(., x), div ∂D (l∇ ∂D u) + hu H 1 (∂D),H -1 (∂D) , ∀x ∈ S d-1 ,
where u is the solution of (1) and p(., x) is the solution of problem [START_REF] Allaire | Conception optimale de strctures[END_REF] in which u i is replaced by Φ ∞ (., x).

Proof. Following the proof of Proposition 6 in [START_REF] Bourgeois | Identification of generalized boundary conditions in inverse scattering problems[END_REF], we obtain that T is differentiable and that dT λ,µ (h, l) = v ∞ h,l where v ∞ h,l is the far-field associated with v s h,l

solution of              ∆v s h,l + k 2 v s h,l = 0 in Ω div ∂D (µ∇ ∂D v s h,l ) + ∂v s h,l ∂ν + λv s h,l = -div ∂D (l∇ ∂D u) -hu on ∂D lim R→∞ ∂BR |∂v s h,l /∂r -ikv s h,l | 2 ds = 0 . (13)
From ( 5), we have for all

x ∈ S d-1 v ∞ h,l (x) = ∂D v s h,l ∂Φ ∞ (., x) ∂ν - ∂v s h,l ∂ν Φ ∞ (., x) ds. ( 14
)
Since on ∂D

∂v s h,l ∂ν + div ∂D (µ∇ ∂D v s h,l ) + λv s h,l = -div ∂D (l∇ ∂D u) -hu
replacing Dv s h,l in ( 14) and integrating by part we obtain

v ∞ h,l (x) = v s h,l , div ∂D (µ∇ ∂D Φ ∞ (., x)) + ∂Φ ∞ (., x) ∂ν + λΦ ∞ (., x) H 1 (∂D),H -1 (∂D) + Φ ∞ (., x), div ∂D (l∇ ∂D u) + hu H 1 (∂D),H -1 (∂D) . ( 15
)
We introduce p(•, x), the adjoint state solution of [START_REF] Allaire | Conception optimale de strctures[END_REF]

with u i = Φ ∞ (•, x). p s (•, x) := p(•, x) -Φ ∞ (•, x) satisfies on ∂D div ∂D (µ∇ ∂D p s ) + ∂p s ∂ν + λp s = -div ∂D (µ∇ ∂D Φ ∞ ) - ∂Φ ∞ ∂ν -λΦ ∞ .
As v s h,l and p s are radiating solutions of a scattering problem the following identity holds:

∂D ∂p s ∂ν v s h,l -p s ∂v s h,l
∂ν ds = 0 , finally using the boundary condition for p s and then for v s h,l , equation ( 15) becomes

v ∞ h,l (x) = -p s (•, x), div ∂D (µ∇ ∂D v s h,l + ∂v s h,l ∂ν(y) + λv s h,l ) H 1 (∂D),H -1 (∂D) + Φ ∞ (., x), div ∂D (l∇ ∂D u) + hu H 1 (∂D),H -1 (∂D) = p(., x), div ∂D (l∇ ∂D u) + hu H 1 (∂D),H -1 (∂D)
which concludes the proof.

Theorem 4.2. The function F is differentiable and for (λ, µ) ∈ (L ∞ (∂D)) 2 that satisfy (H1) its Fréchet derivative satisfies

∀(h, l) ∈ (L ∞ (∂D)) 2 dF (λ, µ)•(h, l) = ℜe G, div ∂D (l∇ ∂D u) + hu H 1 (∂D),H -1 (∂D)
where u is the solution of the problem P(λ, µ, ∂D), G = G i + G s is the solution of P(λ, µ, ∂D) with u i replaced by

G i (y) := S d-1 Φ ∞ (y, x)(T (λ, µ) -u ∞ obs )dx.
Proof. By composition of derivatives and using the Fubini theorem we have for

(h, l) ∈ L ∞ (∂D) 2 dF (λ, µ) • (h, l) = ℜe (T (λ, µ) -u ∞ obs , dT (λ, µ) • (h, l)) L 2 (S d-1 ) = ℜe S d-1 (T (λ, µ) -u ∞ obs )(x) p(y, x), A(u)(y) H 1 (∂D),H -1 (∂D) dx = ℜe G, A(u) H 1 (∂D),H -1 (∂D)
with A(u)(y) = div ∂D (l(y)∇ ∂D u(y)) + h(y)u(y).

INRIA

Numerical algorithm

To minimize the cost function [START_REF] Kirsch | The factorization method for inverse problems[END_REF] we use a steepest descent method and we calculate the gradient of F with the help of Lemma 4.1. We solve the direct problems using a finite elements method (implemented with FreeFem++ [21]) with a truncated DtN map on the boundary of a circle of sufficiently large radius R, in order to bound the computational domain. We will consider different geometries for the obstacle D and we look for the imaginary part of a function λ with ℑm(λ) ≥ 0 and the real part of a function µ with ℜe(µ)(x) ≥ c > 0 for almost every x ∈ ∂D assuming that ℜe(λ) and ℑm(µ) are known in order to satisfy hypothesis presented in [START_REF] Bourgeois | Identification of generalized boundary conditions in inverse scattering problems[END_REF] for which uniqueness and stability hold. Moreover, for sake of simplicity we will choose these known parts of the impedances equal to zero. Let us give initial values λ init and µ init in the same finite element space as the one used to solve the forward problem. We update these values at each time step n as follows

λ n+1 = λ n -iδλ n
where the descent direction δλ n is taken proportional to dF (λ n , µ n ). Since the number of parameters for λ n is in general high, a regularization of dF (λ n , µ n ) is needed. We choose to use a H 1 (∂D)-regularization (see [START_REF] Allaire | Conception optimale de strctures[END_REF] for a similar regularization procedure) by taking δλ n ∈ H 1 (∂D) solution to

η 1 (∇ ∂D (δλ n ), ∇ ∂D ϕ) L 2 (∂D) + (δλ n , ϕ) L 2 (∂D) = α 1 dF (λ n , µ n ) • (iϕ, 0) (16)
for each ϕ in the finite element space and with η 1 the regularization parameter to choose and α 1 is the descent coefficient for λ. We used iϕ in ( 16) in order to compute the gradient with respect to the imaginary part of λ. We apply a similar procedure for µ, µ n+1 = µ nδµ n where δµ n solves

η 2 (∇ ∂D (δµ n ), ∇ ∂D ϕ) L 2 (∂D) + (δµ n , ϕ) L 2 (∂D) = α 2 dF (λ n , µ n ) • (0, ϕ).
We take two different regularization parameters for λ and µ since we observed that the algorithm has different sensitivities with respect to each coefficient.

From the practical point of view we choose large η i at the first steps to quickly approximate the searched λ and µ then we decrease them during the algorithm in order to increase the precision of the reconstruction. In all the computations (except for constant λ and µ), the parameters α 1 and α 2 are chosen in such a way that the cost function decreases at each step. A more precise description of the algorithm is given in the following where lambda and mu respectively denote λ n and µ n at time step n.

If newF > F -if n is even alphamu=alphamu/2; -if n is odd alphalambda=alphalambda/2;
If newF < F lambda = lambdanew; mu=munew;

-F=Fnew;

compute gradmu and gradlambda (evaluation of the adjoint state)

-alphamu = 2*alphamu (if n is even), alphalambda = 2*alphalambda; (if n is odd).
Stopping criterion We stop the algorithm if the descent coefficients α µ and α λ are too small or if the number of iterations is larger than 100 (in every cases there was not a concrete improvement of the reconstruction after 80 iterations).

Remark that we update alternatively λ and µ because the cost function is much more sensitive to λ than to µ and as a consequence, if we update both at each time step, we would have a poor reconstruction of µ.

Numerical experiments

In this section we will show some numerical reconstructions using synthetic data generated with the code FreeFem++ in two dimensions to highlight the importance of each parameter of the problem. The aim is to better understand the behaviour of the iterative algorithm we presented in the previous section. First of all, we will see that the use of a single incident wave is not satisfactory and we will quickly turn to the use of several incident waves. Then all the simulations will be done with several incident waves and with limited aperture data. Remark that all the theoretical results still hold in this particular case (see remark 2.3). In most of the simulations the obstacle will be an ellipse (see Figure 4) but we will show the influence of the convexity of the obstacle on the quality of the reconstruction. In most of experiments, the diameter of the obstacle will be more or less equal to the wavelength (2π/k). Moreover, since we consider a modelling of physical properties we rescale the equation on the boundary of the obstacle ∂D in order to deal with dimensionless coefficients λ and µ. The equation on ∂D becomes In all cases (except when we specify it), we simply reconstruct µ taking λ = 0 because the reconstruction of λ has been investigated for a long time (see [START_REF] Cakoni | The determination of the surface conductivity of a partially coated dielectric[END_REF] or more recently [START_REF] Cakoni | The determination of boundary coefficients from far field measurements[END_REF]). Finally, we will only consider star-shaped obstacles ∂D = {r = X(θ), θ ∈ [0, 2π[} and so we can define the impedance functions as functions of the angle θ. In the following we will represent λ and µ with the help of such a parametrization.

div ∂D µ k ∇ ∂D u + ∂u ∂ν + kλu = 0. INRIA

A single incident wave with full aperture

In this section we consider the exact framework of the theory developed at the beginning, namely we enlighten the obstacle with a single incident plane wave and we measure the far-field in all directions. As we could expect, the reconstruction is quite good in the enlightened area but rather poor far away from such area (see Figure 4.3.1).

Several incident waves and limited aperture

In this section we suppose that we measure several far-fields corresponding to several incident directions. We hence reproduce an experimental device (see Figure 6) that would rotate around the obstacle D. To be more specific, we denote u s (•, d) the scattered field associated with the incident plane wave of Figure 6: Several limited observations. direction d. Considering we have N incident directions d j and N areas of observation S j ⊂ S 1 such that S i = S 1 and S i ∩ S j = ∅ for i = j, we construct a new cost function

F (λ, µ) = 1 2 N j=1 T (λ, µ, d j ) -u ∞ obs (•, d j ) 2 L 2 (Sj ) .
To minimize F we use the same technique as before but now the Herglotz incident wave G i is associated with the incident direction d j and is given by

G i (y, d j ) = Sj Φ ∞ (y, x)(T (λ, µ) -u ∞ obs )dx.
More precisely, for the next experiments we send N = 10 incident waves that are uniformly distributed on the unit circle and the aperture of observation is of π/5 so that the global angle of observation is 2π. First of all let us show that this experimental setting provides more accurate results than the previous one.

Testing the accuracy of the algorithm. On Figure 4.3.2 we can see two reconstructions for two ranges of µ, µ ∼ 1 on the left and µ ∼ 10 on the right, we use exact data to verify that the algorithm converges and gives good results. Obviously, the results are much more accurate than in the case of a single incident wave. In order to evaluate the convergence of the algorithm, we introduce the following relative cost function:

Error := N i=1 T (λ, µ, d i ) -u ∞ obs (•, d i ) 2 L 2 (Si) N i=1 u ∞ obs (•, d i ) 2 L 2 (Si)
.

The obtained relative error is really small (< 1%) and is probably due to numerical errors on the far-field computation. From now on we will add some noise on the data to avoid "inverse crime". Precisely we handle some noisy data In the next experiments we study the impact of the level of noise (1% and 5%) on the quality of the reconstruction. Error(σ) will denote the final error with amplitude of noise σ.

u ∞ σ (•, d i ) such that u ∞ σ (•, d i ) -u ∞ obs (•, d i ) L 2 (Si) u ∞ obs (•, d i ) L 2 (Si) = σ .
Influence of the wavelength and the regularization parameter. We are now interested in the influence of the wavelength on the accuracy of the results, the first two graphics (Figure 4.3.2) show how the algorithm behaves with respect to the wavelength. We can see that if we decrease the wavelength (Figure 8(b)) the reconstructed impedance is very irregular, that's why on Figure 8(c) we add some regularization to flatten the solution, and then improve the reconstruction. Note that if the wavelength is too small, all the tested regularization strategies did not allow us to correctly capture the solution (see Figure 8(c) with 5% noise).

The case of non-smooth functional coefficients. We are able to identify a non-smooth coefficient µ since µ is expressed as a linear combination of functions of the finite element space. We present our results on Figure 4.3.2 for piecewise constant functions µ. To have a good reconstruction of a piecewise constant function, we need a small wavelength. However, we have just seen before that too small wavelength generates instability due to the noise that contaminates data. That's why we use a two steps procedure. First we use a large wavelength equal to 0.7 (k = 9, on the left) to quickly find a good approximation of the coefficient. Secondly, to improve the result, we use a three times smaller wavelength (k = 24 on the right). Hence, we combine the advantage of a large wavelength (low numerical cost) and the advantage of a small wavelength (good precision on the reconstruction of the discontinuity). We clearly improve the result compared with the reconstruction using a wave number equal to 9.

a) X(θ) = (0.3 cos(θ); 0.2 sin(θ)) (b) X(θ) = (0.3 cos(θ -π) + 0.1 cos(2(θ - π)); 0.2 sin(θ)) (c) X(θ) = (0.3 cos(θ -π) + 0.15 cos(2(θ - π)); 0.2 sin(θ)) (d) X(θ) = (0.3 cos(θ -π) + 0.2 cos(2(θ - π)); 0.2 sin(θ))
Figure 10: "Kite"-shaped obstacle.

In this perspective, we consider four different geometries represented on Figure 10, and on Figure 4.3.2 we present the reconstruction of a smooth µ for those four different geometries. We can clearly conclude that the non-convexity of the obstacle creates local minima for the cost function. More precisely we can see that the reconstruction is poor in the non-convex area of the obstacle.

INRIA

Simultaneous search for λ and µ. We now study the simultaneous reconstruction of λ and µ on the ellipse (see Figure 4). of functional impedance coefficients λ and µ using N = 10 incident waves with aperture equal to π/5. The reconstruction is quite good for 1% of noise and remains acceptable for 5% of noise. On Figure 4.3.2 we tried to improve these results by observing everywhere but we still poorly reconstruct µ with a high level of noise. Here, we should note that the reconstruction of the coefficient µ is a much harder task than the reconstruction of λ.

Stability with respect to a perturbed geometry. To illustrate the stability result with respect to the obstacle stated in Theorem 3.6 we construct numerically u ∞ obs (•, d i ) = T (λ 0 , µ 0 , ∂D, d i ) for a given obstacle D (left hand side of Figure 14) and we minimize the "perturbed" cost function

F ε (λ, = 1 2 N i=0 T (λ, µ, ∂D ε , d i ) -u ∞ obs (•, d i ) 2 L 2 (Si)
for a perturbation D ε (right hand side of Figure 14). The perturbation of the obstacle is denoted γ and defined by

γ := ε 0 diam(D)
.

Note that the perturbed obstacle is the convex hull of the non-convex obstacle D. In the following experiments we want to evaluate the impact of γ on the reconstruction of the coefficients. To satisfy the assumptions of Theorem 3.6 we have to check that we can find some λ ε and µ ε such that

F ε (λ ε , µ ε ) ≤ δ
for a small δ. If we take the same uniformly distributed incident directions with N = 10 and k = 9 as before (the wavelength is more or less equal to the diameter of D), we have

F ε (λ 0 , µ 0 ) N i=0 u ∞ obs (•, d i ) 2 L 2 (Si) = 8% if γ = 1% 27% if γ = 3% .
These levels of perturbation on the cost function are too high to hope a good reconstruction of the coefficients. It is reasonable to consider we do not enlight the obstacle in the direction of the non-convexity (since we have poor knowledge

INRIA

of such area). Let us suppose for example that we still have 10 incident waves but now the incident directions belong to [-π/2, π/2]. We have the following relative errors with the actual impedances

F ε (λ 0 , µ 0 ) N i=0 u ∞ obs (•, d i ) 2 L 2 (Si) = 3% if γ = 1% 9% if γ = 3% .
In this case we hope a good reconstruction of the impedance coefficients at least in the directions of incidence. On Figure 4.3.2 we can see that this is the case for γ = 1% even if we put σ = 5% of noise on the measurements. If we have γ = 3% the reconstruction remains quite good in the non-perturbed area and acceptable in the perturbed area.

Figure 1 :

 1 Figure 1: Configuration of the problem. Let D be an open bounded domain of R d , d = 2 or 3 with a Lipschitz continuous boundary ∂D, Ω = R d \ D and (λ, µ) ∈ L ∞ (∂D) 2 the impedance coefficients. The scattering problem with generalized impedance boundary conditions (GIBC) consists in finding u = u s + u i such that  

Figure 3 :

 3 Figure 3: Example of definition of the cut-off function ψ.

  where for any invertible matrix B, B -T is the transposition of the inverse of B. Thanks to Neumann series, we have the following development for

Proof of Lemma 3 . 4 .

 34 Let us consider three functions u ∈ H 1 (∂D), v ∈ H 1 (∂D) and µ ∈ L ∞ (∂D) and let x 0 ∈ ∂D . There exists a function ϕ of class C 1 and an open set U ⊂ R d-1 such that

  First step (if n > 0): update λ and µ if n is even : newmu = mu -alphamu*gradmu; newlambda = lambda; if n is odd : RR n 7449 newlambda = lambda -alphalambda*gradlambda; newmu = mu; Second step: computation of the cost function Fnew:=F(newlambda,newmu) solve the scattering problem for newlambda and newmu determine the associated far-field evaluate the new cost function newF Third step: validation

Figure 4 :

 4 Figure 4: Geometry of the obstacle, X(θ) = (0.3 cos(θ); 0.2 sin(θ)).

Figure 5 :

 5 Figure 5: Reconstruction of ℜe(µ 0 ) = 0.5(1 + cos 2 (θ)), λ = 0, µ init = 0.7, wave number k = 9, the incident angle is 0 on the left and π/2 on the right.

INRIA

Figure 7 :

 7 Figure 7: Wave number k = 9, regularization parameter η = 0, λ = 0, reconstruction of ℜe(µ 0 ) = 0.5(1 + cos 2 (θ)), µ init = 0.7 on the left and ℜe(µ 0 ) = 5(1 + cos 2 (θ)), µ init = 7 on the right.

  Error (1%) = 2.1% , Error (5%) = 10.2%

Figure 8 :

 8 Figure 8: Reconstruction of ℜe(µ 0 ) = 0.5(1 + cos 2 (θ)), µ init = 0.7, λ = 0, with no regularization procedure, wave number k = 2 (top left) and k = 24 (top right); with a regularization procedure and k = 24 (bottom).

  Error (1%) = 2.3% , Error (5%) = 11%

Figure 9 :

 9 Figure 9: Reconstruction of ℜe(µ 0 ) = 0.5 + 0.5χ [-π/2,π/2] , µ init = 0.7, λ = 0, the wave number is k = 9 on the left and k = 24 on the right.

(

  Error(1%) = 9.1% , Error(5%) = 16.2%

Figure 11 :

 11 Figure 11: Reconstruction of ℜe(µ 0 ) = 0.5(1 + cos 2 (θ)), µ init = 0.7, λ = 0, for the four obstacles of Figure 10.

Figure 12 :

 12 Figure 12: Wave number k = 9, limited aperture data with 10 incident waves, Error (1%) = 2.7% , Error (5%) = 9.8% reconstruction of ℑm(λ 0 ) = 0.5(1 + sin 2 (θ)), λ init = 0.7i (on the left) and ℜe(µ 0 ) = 0.5(1 + cos 2 (θ)), µ init = 0.7 (on the right).

Figure 13 :

 13 Figure 13: Wave number k = 9, full aperture data with 10 incident waves, reconstruction of ℑm(λ 0 ) = 0.5(1 + sin 2 (θ)), λ init = 0.7i (on the left) and ℜe(µ 0 ) = 0.5(1 + cos 2 (θ)), µ init = 0.7 (on the right).

Figure 14 :

 14 Figure 14: Exact(left side) and perturbed(right side) geometries.

  Error (1%) = 8% , Error (5%) = 10.8%

Figure 15 :

 15 Figure15: Perturbed obstacle (see Figure14), wave number k = 9, µ init = 0.7, λ = 0, γ = 1% on the left and 3% on the right .

Table 1

 1

	represents the

Table 1 :

 1 Simultaneous reconstruction of λ and µ constant with the wave number k = 9; σ = 1% of noise. simultaneous reconstruction of constant λ and µ and we observe a good reconstruction in the constant case. On Figure 4.3.2 we show the reconstruction

RR n 7449

Acknowledgement

The grant of Nicolas Chaulet is supported by a DGA grant.