Project proposal for INRIA Saclay Ile de France Shape Reconstruction - - PDF document

Project proposal for INRIA Saclay Ile de France Shape Reconstruction and Identification (D´ etermination de Formes et Identification) (DeFI)

http://www.cmap.polytechnique.fr/~defi 01/06/2008

Project coordinator

• Houssem Haddar (Directeur de Recherche (DR2), INRIA)

Members

• Gr´

egoire Allaire (Prof. de Math. Appl., Ecole polytechnique)

• Olivier Pantz (Prof. charg´

e de cours, Ecole Polytechnique) PostDoc

• Ozgur Ozdemir (2007-2008), Armin Lechleiter (2008-2009), Alex Kelly (2008-2009)

PhD students

• Ridha Mdimagh (2005), Yosra Boukari (2007), Anne Cossonni`

ere (2008) Associated members1

• Laurent Bourgeois (Enseignant-Chercheur, ENSTA)
• Antonin Chambolle (Directeur de Recherche, CNRS, CMAP)

PhD students co-supervised outside the group

• Berang`

ere Delourme (2007, POems, INRIA-Rocquencourt),

1Associated members are expected to spend 20% of their research activity within the DeFI team

An Overall Quick Presentation

The research activity of our team is dedicated to the design, analysis and implementa- tion of efficient numerical methods to solve inverse and/or shape and topological optimiza- tion problems in connection with acoustics, electromagnetism, elastodynamics, and waves in general. Sought practical applications include radar and sonar applications, bio-medical imaging techniques, non-destructive testing, structural design, composite materials, ... Roughly speaking, the model problem consists in determining information on, or opti- mizing the geometry (topology) and/or the physical properties of unknown targets from given constraints or measurements, for instance measurements of diffracted waves. In general this kind of problems is non linear. The inverse ones are also severely ill-posed and therefore require special attention from regularization point of view, and non trivial adaptations of classical optimization methods. Our scientific research interests are three-fold:

• Theoretical understanding and analysis of the forward and inverse mathematical

models, including in particular the development of simplified models for adequate asymptotic configurations.

• The design of efficient numerical optimization/inversion methods which are quick and

robust with respect to noise. Special attention will be paid to algorithms capable of treating large scale problems (e.g. 3-D problems) and/or suited for real-time imaging.

• Development of prototype softwares for precise applications or tutorial toolboxes.

Our team is born after the association of two groups: the inverse problem component of the POems project (INRIA) and the shape and topology optimization group OPTOPO at the CMAP (Ecole Polytechnique). As it will be developed in the presentation of our scientific issues and research themes, this association is a natural consequence of several scientific convergences, that we can quickly summarize in 1) the similarity and complementarity of developed tools: Asymptotic methods, Level Set methods, Sampling methods, 2) common problematics: the need of special regularization strategies, retrieving forms without a priori knowledge of its topology, the application to problems in connection with waves, and 3) the perspective of coupling methods to efficiently treat large scale problems and the desire

• f addressing challenging time-domain problems.

The DeFI group is hosted by the department of applied mathematics (CMAP) at Ecole Polytechnique. 1

2

Contents

1 Introduction: general presentation 5 2 Scientific issues through a model problem 8 2.1 Mathematical settings for the inverse problem . . . . . . . . . . . . . . . . 9 2.2 On general theoretical issues . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Approximate models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 The optimization point of view . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Special sampling methods for multi-static data . . . . . . . . . . . . . . . . 15 2.6 Some potential applications . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6.1 Imaging of urban infrastructures from multi-static data . . . . . . . 17 2.6.2 Bio-medical Applications . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6.3 Innovative methods in non destructive testing and parameter iden- tification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Description of research themes 20 3.1 Real-time algorithms for imaging at a fixed frequency in the non-linear regime 20 3.2 The conformal mapping method for EIT problems. . . . . . . . . . . . . . 21 3.3 Identifictaion/Invisibility of anisotropic media . . . . . . . . . . . . . . . . 22 3.4 Regularization and stability issues . . . . . . . . . . . . . . . . . . . . . . . 23 3.5 Shape and topological optimization methods . . . . . . . . . . . . . . . . . 24 3.6 Asymptotic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.7 Hybrid inversion methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.8 Sampling and optimization methods for time dependent problems . . . . . 27 4 Softwares 28 5 Results Dissemination 28 6 Collaborators 29 6.1 Academic (National and International) . . . . . . . . . . . . . . . . . . . . 29 6.2 Positioning within INRIA . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.3 Industrial grants and contacts . . . . . . . . . . . . . . . . . . . . . . . . . 32 3

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1 Introduction: general presentation

The research activity of our team will be dedicated to the design, analysis and imple- mentation of efficient numerical methods to solve inverse and/or shape and topological

• ptimization problems in connection with acoustics, electromagnetism, elastodynamics,

and waves in general. We are particularly interested in the development of fast methods that are suited for real-time imaging and/or large scale problems. These goals require to work on both the physical and the mathematical models, and indeed require a solid expertise in related numerical solvers. This introduction intends to give a general overview of our research interests and

• themes. We choose to present them through the specific example of inverse scattering

problems (from inhomogeneities), which will be central in most of foreseen developments. The practical problem would be to identify an inclusion from measurements of diffracted waves generated after sending some incident waves into the probed medium. Typical ap- plications include biomedical imaging where using micro-waves one would like to probe the presence of pathological cells, non destructive testing of (composite) materials where

• ne would like to identify the presence of faults from measurements of ultrasonic waves or

also imaging of urban infrastructures where using ground penetrating radars (GPR) one is interested in finding the location of buried facilities such as pipelines, waste deposits, . . . . This kind of applications requires in particular fast and reliable algorithms. By “imaging” we shall refer to the inverse problem where the concern is only the location and the shape of the inclusion, while in the “identification” problem one would also be interested in getting information on the inclusion physical parameters. Both problems (imaging and identification) are non linear and ill-posed (lack of stability with respect to measurements errors if some careful constrains are not added). Moreover, the unique determination of the geometry or the physical parameters is not guaranteed in general if sufficient measurements are not available. As an example, in the case of anisotropic inclusions, one can show that an appropriate set of data uniquely determine the geometry but not the material properties. These theoretical considerations are in gen- eral very difficult to address and often relies on ad-hoc techniques, especially in the case of inverse vectorial problems (e.g. the electromagnetic problem modeled by Maxwell’s equa- tions). More developed theories, based on the so-called Carleman estimates, are available for the scalar case. Even though not central to our research activity, our group will pursue some investigations in this direction through the study of the Cauchy problem associated with the Laplace or the Helmholtz operator in domains with singularities. We are also in- terested in applying these results to quantify the stability properties of impedance inverse boundary value problems or impedance tomography problems (see Section 3.4). These theoretical results (uniqueness, stability) are not only important in understanding the mathematical properties of the inverse problem, but are also useful for qualitative interpretation of numerical results (which information can be stably reconstructed) and the choice of appropriate regularization techniques. The latter point will also be discussed in 5

Section 3.4 of our research themes. Moreover, uniqueness proofs are in general constructive proofs, i.e. they implicitly contain a numerical algorithm to solve the inverse problem, hence their importance for practical applications. The sampling methods introduced below are one example of such algorithms. We shall be mainly concerned with numerical methods applied to the first type of inverse problems, where only the geometrical information is sought. In its most general setting, (using for instance the full 3-D Maxwell equations as a forward model and making no a priori assumptions on the targets) the inverse problem is very challenging, and it is commonly agreed that no available method would provide an universal satisfactory solution to it (regarding the balance cost-precision-stability). This is why in the majority

• f the practically employed algorithms, some simplifications of the underlying mathematical

model is used, according to the specific configuration of the imaging experiment . The most popular ones are geometrical optics (the Kirchhoff approximation) for high frequencies, and weak scattering (the Born approximation) for small contrasts or small obstacles. They actually give full satisfaction for a wide range of applications as attested by the large success of existing imaging devices (radar, sonar, echography, X-ray tomography, . . .).. Generally speaking, the used simplifications result into a linearization of the inverse problem and therefore are usually valid only if the latter is weakly non-linear. The devel-

• pment of these simplified models and the improvement of their efficiency is still a very

active research area. With that perspective we are particularly interested in deriving and studying higher order asymptotic models associated with small geometrical parameters such as: small obstacles, thin coatings, wires, periodic media, . . . (see Section 3.6).The incorporation of higher order asymptotics usually introduce some non-linearity (and insta- bility) in the inverse problem. But this non-linearity is somehow “less stronger” than in the case of the exact model, and therefore would be easier to cope with, from the numerical point of view. A larger part of our research activity will be dedicated to algorithms that avoid the use of such approximations and that are efficient where classical approaches fail: i.e. when the non linearity of the inverse problem is sufficiently strong. This type of configuration is motivated by the applications mentioned above, and occurs as soon as the geometry of the unknown media generates non negligible multiple scattering effects (multiply-connected and closely spaces obstacles) or when the used frequency is in the so-called resonant region (wave- length comparable to the size of the sought medium). Our ideas to tackle this problem will be motivated and inspired by recent advances in shape and topological optimization methods and also the introduction of novel classes of imaging algorithms, so-called sampling methods. The sampling methods are fast imaging solvers adapted to muli-static data (multiple receiver-transmitter pairs) at a fixed frequency. Even if they do not use any linearization the forward model, they rely on computing the solutions to a set of linear problems of small size, that can be performed in a completely parallel procedure. Our team has already a solid expertise in these methods applied to electromagnetic 3-D problems. The success of 6

such approaches was their ability to provide a relatively quick algorithm for solving 3-D problems without any need for a priori knowledge on the physical parameters of the targets. These algorithms solve only the imaging problem, in the sense that only the geometrical information is provided. Despite the large efforts already spent in the development of this type of methods, either from the algorithmic point of view or the theoretical one, numerous questions are still open and will be partly exposed in Section 3.1 of our research themes. These attractive new algorithms also lack experimental validations, due to the need of multistatic data. We also would like to invest on this side by developing collaborations with experimental research groups (see Section 6). From the practical point of view, the most potential limitation of sampling methods would be the need of a large amount of data to achieve a reasonable accuracy. On the

• ther hand, optimization methods do not suffer from this constrain but they require good

initial guess to ensure convergence and reduce the number of iterations. Therefore it seems natural to try to combine the two class of methods in order to calibrate the balance between cost and precision (see Section 3.7). Among various shape optimization methods, the Level Set method seems to be partic- ularly suited for such a coupling. First, because it shares similar mechanism as sampling methods: the geometry is captured as a level set of an “indicator function” computed on a caretisian grid. Second, because the two methods do not require any a priori knowledge on the topology of the sought geometry. Beyond the choice of a particular method, the main question would be to define in which way the coupling can be achieved. Obvious strategies consist in using one method to pre-process (initialization) or post-process (find the level set) the other. But one can also think of more elaborate ones, where for instance a sampling method can be used to optimize the choice of the incident wave at each iteration step.The latter point is closely related to the design of so clalled “focusing incident waves” (which are for instance the basis of applications of the time-reversal principle). In the frequency regime, these incident waves can be constructed from the eigenvalue decomposition of the data operator used by sampling methods. The theoretical and numerical investigations

• f these aspects are still not completely understood for electromagnetic or elastodynamic

problems. Other topological optimization methods, like the homogenization method or the topo- logical gradient method, can also be used, each one has its pros-and-cons depending on considered problem (see Section 2.4). It is evident that the development of these methods is very suited to inverse problems and provide substantial advantage compared to classical shape optimization methods based on boundary variation. Their applications to electro- magnetic inverse problems has not been fully investigated (see Section 3.5). The efficiency

• f these optimization methods can also be increased in some asymptotic configurations.

For instance small amplitude homogenization method can be used as an efficient relaxation method for the inverse problem in the presence of small contrasts. On the other hand, the topological gradient method has shown to perform well in localizing small inclusions with 7

• nly one iteration (see Section 3.6).

For the identification problem, one would also like to obtain information of the physical properties of the targets. Of course optimization methods is a tool of choice for this type of

• problems. However, in some applications only qualitative information would be sufficient

and obtaining them in a cheaper way can be performed using asymptotic theories combined with sampling methods (see Section 3.3). A broader perspective of our research themes would be the extension of the above mentioned techniques to time-dependent problems. Taking into account time-dependent data is necessary for applications where some non-linearity is present in the forward model: design of electromagnetic absorbing coating (non-linear materials, meta-materials), study

• f crashworthiness in the case of structural design, . . ..

It is also important from the practical point of view since usually measurement devices can perform multi-frequency acquisition at a lower cost (see Section 3.8) The remaining of this document is organized as follows. We present in Section 2 the scientific issues related to our research activity through a model problem: the electromag- netic inverse scattering problem. We choose this model since 1) it was central in most of

• ur former research activity 2) it is representative of most of the physical models we are

interested in: acoustics, aeroacoustics, elastodynamics, and also the zero frequency limit of these problems (static case). Section 3 is dedicated to the research themes, where some key scientific research perspectives are presented. Most of these themes are dictated by some potential applications presented in Section 2.6. A quick summary of developed softwares and our plan for results dissemination is given in Sections 4-5. We end this document by indicating our positioning within INRIA, and within the international scientific community as well as our industrial contacts and grants.

2 Scientific issues through a model problem

The goal of this section is to present the scientific issues related to our research activity through a typical mathematical model: Maxwell’s equations in an inhomogeneous medium. In this model, the electric and magnetic fields denoted E and H respectively satisfy ε∂E ∂t + σE − curl H = J , (1) µ∂H ∂t + curl E = 0, (2) where the electric permittivity ε and conductivity σ are spatially dependent (i.e. different for propagative media and inclusions) and where we choose the magnetic permeability µ to be constant (which is usually the case for most applications). Sources of the field are described by the data vector function J . Of course these equations need to be comple- mented by initial and boundary data depending on the nature of the objects. Most inverse scattering algorithms we shall consider are posed in the frequency domain in which it is 8

assumed that E(x, t) = ℜ(ε−1/2 E(x) exp(−iωt)) and H(x, t) = ℜ(µ−1/2 H(x) exp(−iωt)) where ε0 and µ0 are the permittivity and permeability of free space (air) and ω is the temporal pulsation. Using these expressions in the above time dependent system, assum- ing a similar time dependence for J , and using the assumption that µ = µ0 gives the time-harmonic Maxwell system curl H + ik n E = −J (3) curl E − ik H = 0 (4) where the wave-number k = √ε0µ0ω and n = n(x) := ε(x)

ε0 + i σ(x) ε0ω denotes the medium

• index. This scaling gives n = 1 in the air. In general these parameters also depend on the

frequency (for instance, absorption usually increases with the frequency) but we shall only consider a range of frequencies for which this parameter can be assumed constant. For most of the applications the background medium has a piecewise constant index n = nb which is known a priori. Detectable target objects are then assumed to have n = nb. These targets may have anisotropic behaviour which is then modeled by letting n to be a 3 × 3 matrix. The time-harmonic Maxwell system (3-4) needs to be complemented with suitable boundary data and a radiation condition at infinity [?] to obtain a complete description

• f the field and we shall give some more details shortly. Note that in practice the Fourier

transform can be used to obtain time-harmonic fields from time-dependent fields and vice versa.

2.1 Mathematical settings for the inverse problem

Typically, the inverse imaging problem is stated as the problem of retrieving the support D of n = nb from the knowledge of measured diffracted waves, at a given frequency or for multiple frequencies. These diffracted waves result from sending some incident waves into the probed medium. To fix the ideas we shall consider the case where incident waves are due to electric dipoles located at x0 ∈ Σ ⊂ R3 with polarization p ∈ R3. These waves {Ei(·, x0, p), Hi(·, x0, p)} satisfy curl xcurl xEi − k2nbEi = p δ(x − x0) in R3 with appropriate radiation condition, and ik Hi = curl Ei. Due to the linearity with respect to p, the electric field can be expressed in terms of the 3 × 3 Green’s tensor G(x, x0) as Ei(x, x0, p) = G(x, x0)p. The total field {E(·, x0, p), H(·, x0, p)} is then a solution of (3-4) with J(x) = p δ(x − x0) and can be decomposed as {E(·, x0, p), H(·, x0, p)} = {Ei(·, x0, p), Hi(·, x0, p)} + {Es(·, x0, p), Hs(·, x0, p)} where {Es(·, x0, p), Hs(·, x0, p)} is the scattered field satisfying an appropriate radiation

• condition. In the case of dielectric scatterers the scattered field can be expressed in terms
• f volume potentials as

Es(x, x0, p) = k2

• R3

tG(x, y) (n(y) − nb(y))E(y, x0, p) dy

(5) 9

for x ∈ R3 \ D. This expression indicates in particular the non linear dependence of Es with respect to n−nb (since E depends on n−nb) and therefore indicates the non linearity

• f the application D → Es.

The prototype of inverse imaging problems we shall be concerned with is the one pre- sented in Figure 1 and can be formulated as the problem of determining D from the measurements of Es(x, x0, p) × ν for all x0 ∈ Σ, p ∈ Tx0, and x ∈ Γ, where ν denotes the normal to the measurements surface location Γ and Tx0 denotes the tangent plane to Σ at x0. The identification problem related to this inverse problem consists in getting further informations on the material properties (n − nb).

Measurements Scatterers Ei(x; x0, p) = G(x; x0)p Γ x Es(x; x0, p) × ν ν Σ x0 D D D

Figure 1: Prototype of an imaging experience

2.2 On general theoretical issues

The theoretical questions related to the inverse problem are basically the following ones:

• Identifiability: Does the set of data uniquely determine the geometry D and/or the

physical parameters?

• Stability: How does the reconstruction behave with respect to measurement errors?

Let us indicate that in general, the inverse problem is unstable and even if unique determi- nation can be proved for the continuous setting, it is not guaranteed for the discrete one. The precise answer to these two questions greatly depends on the assumptions made on the geometry (regularity, bound, . . .) and on the set of data. For the identifiability question, most challenging problems correspond to configurations where the inverse problem is formally well posed (as many data as unknowns). For instance, in the case of back-scattering data (we measure Es(x0, x0, p) × ν for all x0 ∈ Σ, p ∈ Tx0)

• r in the case of a finite number of incident waves (we measure Es(x, x0, p) × ν for all

x ∈ Σ and for a finite number of x0) then the question of unique determination of D is largely open. Some results are known in particular configurations: if D is a perfect conductor (σ = ∞) and is sufficiently small with respect to the wavelength [?], or if it has a polyhedral boundary [?, ?], then unique determination can be proved with only several incident waves. The case of multi-static data is in general more accessible and unique determination of D and isotropic index n (under suitable regularity conditions) can be proved that case. Let 10

us also indicate that uniqueness proofs for these problems are in general constructive proofs, i.e. they implicitly contain a numerical algorithm to solve the inverse problem, hence their importance for practical applications. The sampling methods introduced below are one example of such algorithms [?]. The case of anisotropic inclusion introduces new difficulty to the inverse problem since

• ne can show that (under suitable assumptions on n − nb) D is uniquely determined from

multi-static data but not n. Some of recent works try to exploit this mathematical property in the design of cloaking materials [?, ?]. We are interested in this kind of problem by exploiting the use of so-called interior transmission problems to obtain estimates on the physical parameter of anisotropic inclusions (see Section 3.3). The stability question is indeed important as it would indicate what one can or cannot

• btain from a given set of data. A typical (optimal) result is a continuity of logarithmic

type with respect to admissible noise, assuming extra regularity on D or n. Let us quote that most of known results are restricted to the scalar case [?], and their generalizations to vectorial problems, like the Maxwell system, are open questions. Some Lipshitz stability can be restored if one reduces the inverse problem to a finite dimensional setting (finite parametrization of the unknowns). An interesting question then would be to measure how the stability constant blows up as the dimension goes to infinity [?]. This information can be used for instance in regularization strategies. In the case of frequency-dependent problems, one can also ask whether a quantification of the stability constant in terms of the frequency can be obtained? Some partial results has been obtained in [?] but an optimal answer to this question is still an open problem.

2.3 Approximate models

As indicated in the introduction existing imaging devices extensively use some simplifica- tions of the forward model that typically lead to a linearization of the inverse one. In the case of dielectrics, the most known simplification is the Born approximation,

• r weak scattering assumption, and consists into neglecting the scattered field inside the

inclusion which yields the following expression of the scattered field: Es(x, x0, p) ≃ k2

• R3

tG(x, y) (n(y) − nb(y))Ei(y, x0, p) dy.

(6) One clearly see that the inverse problem is now linear and to retrieve D or n one needs to invert the linear operator on the right hand side. However, since this operator is compact, the inversion is unstable and may induce poor results if a large amount of data is not

• available. A second major advantage of this approximation is that it leads to a quasi-

explicit reconstruction procedure for back-scattering data at different frequencies, which is the configuration commonly encountered in practical applications. This approximation is only valid if n − nb or |D| are sufficiently small. The quantifi- cation of how small they should be (in terms of the wavelength for instance) is generally determined by a rule of thumb. However, asymptotic studies may be used to quantify 11

the order of accuracy with respect to the small parameters. They also can be used to improve the accuracy by including higher order terms in the asymptotic expansions of the fields (as in so-called “distorted Born approximation”). This procedure usually breaks the linearity of the obtained inverse model and therefore may lead to a substantial increase of the inversion cost. Let us notice that in some other asymptotic configurations (thing coatings, highly os- cillating media, rough interfaces, high conductivity . . .), the use of asymptotic expansions does not lead in general to a linearization of the inverse problem. However, the use of asymptotic models in those cases may simplify the inverse problem and increase the effi- ciency of non-linear inversion methods (see Section 2.4 and Section 3.6).

2.4 The optimization point of view

When the inverse problem is strongly non-linear, a classical approach would be to formulate the problem as a minimization problem. The sought domain D is for instance characterized as being the global minimizer (under some constraints) of the functional J(D) = Es

mes − Es[D]L2(Γ×Σ)

where Es

mes denotes the set of measurements and Es[D] the values computed for a given ge-

• metry D: we here assume for simplicity that the value of n is known, which is for instance

the case when the target is made of a single dielectric with constant known material. This setting is indeed similar to the one encountered in shape and topology optimiza- tion problems (with different expression and meaning of the functional J(D)). A typical example is structural optimization where the shape of a structure is optimized in order to miminize its weight and maximize its strength. The functional J(D) can be in this case the co-called compliance (external forces work) or also the first eigenvalue of the struc-

• ture. Furthermore, for these problems not only the shape has to be optimized but also its

topology (number of holes or components). For the inverse problem the last issue is very important since in general the topology of the sought domain is not known a priori and therefore may be different from the initial guess. Several iterative schemes, like (Quasi-)Newton methods (see for instance [?], [?] and references therein), have been applied to the inverse problem, and as a general observation

• ne can state that if: 1) one has a good a priori knowledge on the physical parameters
• f the unknown targets, 2) one starts with a relatively good initial guess (including an

exact knowledge of the topology) and 3) one applies appropriate regularization techniques (taking into account the exponentially ill-posed behavior of the inverse problem), then this kind of optimization method would provide us with the best “stable” information that can be extracted from a given set of data. A single measurement is sometimes sufficient to

• btain reasonable reconstructions.

However, since this accuracy is obtained at the expenses of many strong a priori in- formation, the practicality of such procedure is very limited. Let also indicate that, from the theoretical point of view, no convergence analysis of the iterative scheme is available 12

for the fully non linear problem. This difficulty is due to the “strong” instability of the application Es[D] → D, which prevents the application of classical convergence theories from shape optimization area. For 3-D problems one has to also deal with more limitations on the algorithmic level. The first limitation is due to the cost of such methods, since they rely of the computation

• f solutions to the forward problem. Even with the constant improvements of forward

solvers for scattering problems, these computations are still very expensive. Most com- petive solvers are those based on integral equation methods but their use is restricted to (piecewise) homgeneous backgrounds. Let us also emphasize once more that some asymp- totic models can also be used to reduce the cost of the forward model and allow their use in an iterative procedure. The second limitation comes from the difficulty of efficiently handle topology changes during iterations, which in turn increases the risk of converging to a local minimum. The change of topology is not possible for classical optimization strategy based on boundary variation procedure, where the surface is monitored by carefully chosen basis functions. Many alternatives has been developed in the topology optimization area to overcome these difficulties, but no universal solution exists. The efficiency of a given approach depends

• n the constrains linked with a specific application. We hereafter indicate three classes of

methods that have (or would have) potential applications to the inverse problem.

• Level set method: It is a method for front propagation, due to Osher and Sethian [?].

Recently, it has been introduced in the field of topology optimization [?] where its ability to change topology has been remarkable. It is also a classical tool in image segmentations and in many inverse problems [?]. Combined to other ingredients (such as the topological gradient method) it is a very effective and cheap approach for solving shape reconstruction problems (see Figure 2).

• Topological gradient method: In the context of structural design it was introduced

in [?], [?], but other versions of the method have been independently developed for inverse problems [?]. Roughly speaking the topological gradient method amounts to decide whether or not it is favorable (for decreasing the objective function) to nucleate a small hole in a given shape. As a matter of fact, creating a hole changes the topology and is thus one way of escaping local minima (due to topological constraint). The coupling of topological and shape gradients in the level set framework has been a great improvement (at least in 2-d) with respect to previous methods and is much less prone to finding local, non global, optimal shapes [?].

• Homogenization method. It has been widely used in shape and topology optimization

[?], [?]. However its use for inverse problems has not been so much investigated (see nevertheless [?]). In the context of inverse problems, the homogenization method can be seen as a relaxation approach for ill-posed problems. However, it is still important to include some a priori knowledge of the results in order to improve the efficiency

• f such relaxation approaches. One such a priori knowledge is the small contrast or

13

amplitude assumption which is at the root of the small amplitude homogenization method described in Section 3.6. Figure 2: Example of 3-D shape and topological optimization results using the Level Set method (the results correspond to robust compliance minimization with different levels

• n uncertainties on the transverse loading). These results are due to Allaire-De Gournay-

Jouve (see http://www.cmap.polytechnique.fr/~ optopo) We end this section by mentioning another important class of iterative methods specif- ically designed for (scattering) inverse problems and known as decomposition methods. The goal of these methods is to accelerate the convergence of an iterative procedure by reducing the cost of an iteration step. For various developed methods, (like the dual space method introduced by Colton-Monk [?] and also the closely connected iterative method by Kirsch-Kress [?], point source and related methods by Potthast [?, ?], . . .), a common characteristic is to avoid the use of a forward solver for the direct problem by using a suitable approximation (of this problem). We refer to them as decomposition methods since they split the inversion procedure into an ill-posedness part (approximate model) and a non-linear one (matching the data). Such methods are usually more attractive than a general optimization scheme in the sense that they take into account the specificity of the scattering problem, and therefore would be more robust. However, like iterative methods, they require a good initial guess and a priori knowledge on the unknown target (material properties, boundary conditions, topology, . . .). 14

2.5 Special sampling methods for multi-static data

Muti-static data basically correspond to the configuration shown in Figure 1, i.e. data is collected at a an extended grid of measurement points (Γ) and are generated by sending multiple incident waves (which are here point sources at Σ). So-called sampling methods are known to provide a simple algorithm with spectacular efficiency and relatively good precision for this type of data. These methods has been developed after the work of Colton-Kirsch in 1996 [?]) where a simple method (later denoted by the Linear Sampling Method (LSM)) has been intro- duced to solve the following basic inverse scattering problem: determine the shape of an

• bstacle from the knowledge of the acoustical scattering amplitude (or far field) at a fixed
• frequency. Beyond its confusing simplicity, the main originality of this method was to

break, in principle, with classical non linear optimization techniques. As mentioned above these methods are iterative and require a quick forward solver and a priori knowledges on the physical properties of the inclusion. The LSM does not need any of these requirements and is not iterative. This is why it is a relatively cheap algorithm. On the other hand, large amount of data is needed in general to achieve reasonable accuracy. Number of papers have been dedicated to the development of this method. Two key works for the scalar problem are [?] and [?] (see [?] and the monograph by Cakoni-Colton [?] for a review). The extension to the 3-D Maxwell case has been initiated in [?] for perfect scatterers and continued in a series of papers reviewed in [?]. The principle of this algorithm is to first construct a data (linear sampling) operator F, which in our case is defined as acting on tangential functions g on Σ by Fg(x) =

• Σ

Es(x, x0, g(x0)) ds(x0) x ∈ Γ then to solve for any point z in the probed region the linear (ill-posed) equation Fgz = G(·, z)q (7) where q ∈ R3 \ {0} is an arbitrary polarization. The imaging function z → 1/gz then provides an approximation of the indicator function of the region n = nb, in the sense that it is almost zero outside this region. We hereafter give some numerical examples obtained with LSM for the electromagnetic inverse scattering problem. The ones chosen here may not reflect real experiments but were the first representative of the potentialities of the method in reconstructing very complex geometries with simple tools. These results are summarized in Figures 3-6 and correspond to a perfect scatterer with full aperture far-field data (measurements and sources are in the far-field region). They were obtained in collaboration with F. Collino and M’B. Fares from Cerfacs (see http://www-rocq.inria.fr/~ haddar for extended results). 15

Figure 3: Exact geometry of the perfect scatterer. Figure 4: Reconstructed geometry with a wave number k = 28 and using 252 uniformly distributed incident direc- tions. Figure 5: Reconstructed geometry with a wave number k = 56 and using 252 uniformly distributed incident direc- tions. Figure 6: Reconstructed geometry with a wave number k = 84 and using 492 uniformly distributed incident direc- tions. Since the introduction of the linear sampling method in 1996, a variety of related inversion schemes have been proposed that share the same spirit as the linear sampling

• method. In particular, we mention the factorization method of Kirsch [?, ?], that turned
• ut to be a generalization of the well-known MUSIC algorithm [?, ?, ?], and the probe

methods of Potthast and his co-workers [?]. These variants differ depending on the definition of the operator F, and/or the right hand side of equation (7) and/or the choice of the imaging function. However, they all share the same principle: For each sampling point z, solve a linear equation of small size independent of the forward model and independent of other sampling points. This parallel procedure, combined with the size of the linear system lead to very quick algorithms. The accuracy of such procedure however greatly depends on the number of measurements, and 16

numerical experiments showed that a resolution of λ/4 where λ is the wavelength, can be attained for the linear sampling method. Recently, Colton and Haddar [?] have proposed a new sampling method based on the use

• f Cauchy data that offer interesting advantages for applications to the imaging of buried
• bjects. This new formulation is based on the reciprocity gap principle that was classically

used in order to obtain quasi-explicit reconstruction formulas for imaging of simplified structures such as planar cracks or point sources [?, ?, ?, ?]. This algorithm can be seen as a generalization of the linear sampling method principle and offers a flexible mathematical framework, especially in choosing the parametrization of the approximating solutions. The method has been generalized to the 3-D Maxwell problem in [?] with numerical examples using synthetic data attesting the viability of this algorithm in imaging buried scatterers. Other related methods, like the convex scattering support of Kusiak-Sylvester [?, ?], or the enclosure method by Ikehata [?] share similarities with the linear sampling methods but are dedicated to single measurements.

2.6 Some potential applications

We shall here present typical foreseen applications in three different domains. These appli- cations are not exhaustive examples but give a flavor of the kind of problems that can be tackled using the above presented methods, and motivate our research themes presented in Section 3. 2.6.1 Imaging of urban infrastructures from multi-static data Conventional radar imaging techniques (ISAR, or GPR [?, ?]) use backscattering data to image targets. The commonly used inversion algorithms are mainly based on the use of weak scattering approximations such as the Born or Kirchhoff approximation leading to very simple linear models, but at the expense of ignoring multiple scattering and polar- ization effects. The success of such an approach is evident in the wide use of synthetic aperture radar techniques. However, the use of backscattering data makes 3-D imaging a very challenging prob- lem (it is not even well understood theoretically) and as pointed out by Brett Borden in the context of airborne radar [?]: “In recent years it has become quite apparent that the problems associated with radar target identification efforts will not vanish with the devel-

• pment of more sensitive radar receivers or increased signal-tonoise levels. In addition it

has (slowly) been realized that greater amounts of data - or even additional “kinds” of radar data, such as added polarization or greatly extended bandwidth - will all suffer from the same basic limitations affiliated with incorrect model assumptions. Moreover, in the face of these problems it is important to ask how (and if) the complications associated with radar based automatic target recognition can be surmounted.” This comment also applies to the more complex GPR problem. Our research themes will incorporate the development, analysis and testing of several novel methods, such as sampling methods, level set methods or topological gradient meth- 17

• ds, for ground penetrating radar applications, such as imaging of urban infrastructures,

using multistatic data. This effort will be conducted in close collaboration with the Univer- sity of Delaware that has contacts with Witten Technologies Inc (WTI), a company based in Boston and the winner of numerous awards for the development of innovative technol-

• gy for the detection of underground objects (c.f. www.wittentech.com): Figure 7 shows

a subsurface imaging GPR array system - called the CART Imaging System - combining multiple standard GPR antenna in an array configuration. We recall that sampling approaches are particularly attractive since 1) even though they avoid the need for the weak scattering approximations they are still linear methods that are easily implementable and 2) they require no a priori assumptions on the material

• r topological properties of the objects being imaged. This second point is of particular

importance since the objects lying beneath the surface of an urban environment range from abandoned facilities, rock formation, unmarked burial sites to corroded chemical waste deposits and a priori assumptions on the material or topological properties of such

• bjects would be totally unrealistic.

The application of sampling algorithms to the imaging of a network of buried pipelines may suffer from loss of resolution due to multiple scattering effects. The combination with shape optimization methods like the Level Set method and gradient topological method would provide an interesting solution to fix this problem. 2.6.2 Bio-medical Applications Among emerging medical imaging techniques we are particularly interested in those using low to moderate frequency regimes. These include Microwave Tomography [?, ?, ?, ?]), Electrical Impedance Tomography [?, ?] and also the closely related Optical Tomography technique [?, ?]. They all have the advantage of being potentially safe and relatively cheap modalities and can also be used in complementarity with well established techniques such as X-ray computed tomography or Magnetic Resonance Imaging. With these modalities tissues are differentiated and, consequentially can be imaged, based on differences in dielectric properties (some recent studies have proved that dielectric properties of biological tissues can be a strong indicator of the tissues functional and pathological conditions, for instance, tissue blood content, ischemia, infarction, hypoxia, malignancies, edema and others). The main challenge for these functionalities is to built a 3-D imaging algorithm capable of treating multi-static measurements to provide real-time images with highest (reasonably) expected resolutions and in a sufficiently robust way. Another important biomedical application is brain imaging. We are for instance inter- ested in the use of EEG and MEG techniques as complementary tools to MRI [?, ?, ?]. They are applied for instance to localize epileptic centers or active zones (functional imag- ing). Here the problem is different and consists into performing passive imaging: the epileptic centers act as electrical sources and imaging is performed from measurements of induced currents. Incorporating the structure of the skull is primordial in improving the resolution of the imaging procedure. Doing this in a reasonably quick manner is still an active research area, and the use of asymptotic models would offer a promising solution to 18

Figure 7: The WTI CART GPR system. Top left: A graphic showing how separate transmitters and receivers are arranged in the antennae box. Top right: The source wavelet and its frequency spectrum indicating the operating frequency for the experiments. Bottom left: The antennas. Bottom right: The CART system towed by an automobile. 19

fix this issue. 2.6.3 Innovative methods in non destructive testing and parameter identifi- cation One challenging problem in this vast area is the identification and imaging of defaults in anisotropic media. For instance this problem is of great importance in aeronautic con- structions due to the growing use of composite materials. It also arises in applications linked with the evaluation of wood quality, like locating knots in timber in order to op- timize timber-cutting in sawmills, or evaluating wood integrity before cutting trees. The anisotropy of the propagative media renders the analysis of diffracted waves more complex since one cannot only relies on the use of backscattered waves. Another difficulty comes from the fact that the micro-structure of the media is generally not well known a priori. Our concern will be focused on the determination of qualitative information on the size of defaults and their physical properties rather than a complete imaging which for anisotropic media is in general impossible [?]. For instance, in the case of homogeneous background, one can link the size of the inclusion and the index of refraction to the first eigenvalue of so-called interior transmission problem [?]. These eigenvalues can be deter- mined form the measured data and a rough localization of the default. Our goal is to extend this kind of idea to the cases where both the propagative media and the inclu- sion are anisotropic. The generalization to the case of cracks or screens has also to be investigated. In the context of nuclear waste management many studies are conducted on the pos- sibility of storing waste in a deep geological clay layer [?, ?]. To assess the reliability of such a storage without leakage it is necessary to have a precise knowledge of the porous media parameters (porosity, tortuosity, permeability, etc.). The large range of space and time scales involved in this process requires a high degree of precision as well as tight bounds on the uncertainties. Many physical experiments are conducted in situ which are designed for providing data for parameters identification. For example, the determination

• f the damaged zone (caused by excavation) around the repository area is of paramount

importance since microcracks yield drastic changes in the permeability. Level set methods are a tool of choice for characterizing this damaged zone.

3 Description of research themes

3.1 Real-time algorithms for imaging at a fixed frequency in the non-linear regime

Participants: H. Haddar, A. Lechleiter and O. Ozdemir Using low/moderate frequencies for imaging purposes is a new trend in many appli-

• cations. It has been dictated by various constrains depending on the application itself.

For instance, in medical imaging, using very low frequency excitations (like in EIT) or 20

microwaves would result in a much more safer technologies than those based on high fre- quencies (X-ray, Laser). For mine detection, probing wet soils (conductive media) requires the use of relatively low frequencies so that the incident wave can penetrate the probed

• medium. At these regimes, the imaging resolution is low and suffers from the severe insta-

bility and non-linearity of the underlying inverse problem, which prevent known classical imaging algorithms (tomography) to perform well. We believe that sampling methods, like those developed in our group, provide the appropriate and relevant remedy; their de- velopment is therefore an important issue for that perspective. Our research guidelines in the future will be 1) adapt this kind of algorithms to realistic experiment settings 2) promote their efficiency by improving their speed (coupling with multipole techniques, clever factorization) and testing them against real data. We shall benefit, for the latter point, from developed cooperation with engineering groups having experimental facilities (Electromagnetic research group at ITU). As far as the first point is concerned we are particularly interested in the adaptation

• f the RG-LSM method to urban infrastructure imaging problems [?, ?]. We recall that

this algorithm has been introduced by Colton-Haddar in 2005 as a reformulation of the linear sampling method in the cases where measurements consist of Cauchy data at a given surface, by using the concept of reciprocity gap. The main advantage of this algorithm was to avoid the need of computing the background Green tensor (as required by classical sampling methods) as well as the Dirichlet-to-Neumann map for the probed medium (as required by sampling methods for impedance tomography problems). This method would be for instance well adapted to medical imaging techniques using microwaves. However, in many other practical applications, like imaging of embedded facilities in the soil or mine detection, the required data at the interface cannot be easily obtained and one has

• nly access to measurements of the scattered wave in the air. In order to overcome this

limitation we proposed to couple the RG-LSM algorithm with a continuation method that would provide the Cauchy data from the scattered field. We showed that the obtained scheme has the same convergence properties as RG-LSM with exact data and remains competitive with respect to classical approaches. Preliminary numerical results in a 2-D configuration confirmed these conclusions and also gave further insight on the sampling resolution: Due to the ill-posedness of the first step, only the propagative part of the wave is well reconstructed, which may results in poor approximations of the field. However, the second step (RG-LSM) seems not being affected by this error and therefore is the reconstruction of the target. We are currently extending this approach to the case of rough interfaces and would like to apply these techniques to the 3-D electromagnetic case.

3.2 The conformal mapping method for EIT problems.

Participants: Y. Boukari and H. Haddar In a series of recent papers [?, ?, ?] Akduman, Haddar and Kress have developed a new simple and fast numerical scheme for solving two-dimensional inverse boundary value problems for the Laplace equation that model non-destructive testing and evaluation via 21

electrostatic imaging. In the fashion of a decomposition method, the reconstruction of the boundary shape Γ0 of a perfectly conducting or a nonconducting inclusion within a doubly connected conducting medium D ⊂ R2 from over-determined Cauchy data on the accessible exterior boundary Γ1 is separated into a nonlinear well-posed problem and a linear ill-posed problem. The approach is based on a conformal map Ψ : B → D that takes an annulus B bounded by two concentric circles onto D. In the first step, in terms

• f the given Cauchy data on Γ1, by successive approximations one has to solve a nonlocal

and nonlinear ordinary differential equation for the boundary values Ψ|C1 of this mapping

• n the exterior boundary circle of B. Then in the second step a Cauchy problem for the

holomorphic function Ψ in B has to be solved via a regularized Laurent expansion to obtain the unknown boundary Γ0 = Ψ(C0) as the image of the interior boundary circle C0. As a first step toward solving the impedance tomography problem we considered the case where the solution satisfies an impedance boundary condition on Γ0 (which can be seen as an approximation for a transmission problem between two conducting media). In that case the algorithm does not completely decompose the inverse problem into a well-posed nonlinear ordinary differential equation and an ill-posed Cauchy problem. Consequently its analysis and implementation is more involved and require different tools than those used for the Dirichlet or Neumann case. The first investigations by Haddar and Kress showed that when the impedance is relatively small the algorithm becomes unstable. We also noticed that the instability is also linked to the size of the interior boundary. To overcome these difficulties, we proposed to use the Dirichlet to Neumann operator associated with the conjugate harmonics and proved how this leads to a stabilization (and convergence) of the algorithm for small impedances. However, our analysis is still not completely satisfactory as it does not indicate whether the two algorithms (for large and small impedances) are complementary for all configurations or not. This work in ongoing and we plan to treat in the very near future the case of Electrical Impedance Tomography (EIT) problem, with realistic electrode models. This work will be conducted in a collaboration with R. Kress and the newly hired Post-Doc, F. Delbary at the university of Goettingen.

3.3 Identifictaion/Invisibility of anisotropic media

Participants: A. Cossoni` ere, H. Haddar and A. Lechleiter The inverse problems for anisotropic media have gained a large interest in recent years due to the importance of such problems in many practical applications : imaging of ur- ban infrastructures (deposit monitoring), non destructive testing of composite materials and more recently in stealth technology and antennas (exploiting cloaking properties of anisotropic effective media). This is a challenging (and interesting) problem due to non uniqueness issues associated with anisotropy, which induces an extra difficulty to inverse problem: non-linear, unstable and non uniquely solvable! We shall continue, in collabo- ration with university of Delaware, our investigations on this area by exploiting the link between the so-called “interior transmission eigenvalues” and some effective properties of the anisotropy (those that are uniquely determined: support and matrix invariants) and 22

its use in the identification problem. In a first work, in collaboration with F. Cakoni we studied the interior transmission problem in the case of anisotropic Maxwell’s equations and with possibly coated parts (that is modeled with an impedance boundary condition). The results extend previous work by Haddar [?] where only the case of anisotropies with norm greater than one is studied, and also show that the set of the eigenvalues for this transmission problem is at most discrete [?]. Showing the existence of these eigenvalues is still an open problem due to the fact that the operator involved is not self-adjoint. Some recent progress has been made by showing existence of eigenvalues when the contrast is sufficiently large. The analysis also showed that some lower bounds can be obtained on these eigenvalues in terms of the inhomogeneity shape and the index norm. This was the starting point

• f the idea of exploiting the eigenvalues to get information on the index of the sought

inhomogeneity. More precisely, using the sampling operator with fixed sampling point and by varying the frequency one can localize the presence of eigenvalues when the norm

• f the solution becomes large. This procedure has been successfully tested in the case
• f isotropic circles where the eigenvalues can be computed in terms of some equation

involving the Bessel functions. The first eigenvalue can be used to get estimate on the index norm of the inclusion. The procedure has been validated in the scalar case (TM and TE modes) and for the case of a full aperture [?]. We would like first to extend these numerical results to the case of limited apertures and 3-D problems. We are also studing the sensitivity of these eigenvlaues with respect to perturbations of the anisotropic medium. These investigations are motivated by non-destructive evaluation of structures (presence

• f holes) or identification problems in cluttered media (obstacles under trees, caves, . . .).

3.4 Regularization and stability issues

Participants: L. Bourgeois, A. Chambolle and H. Haddar The optimal choice of regularization techniques for LSM has not yet been fully investi-

• gated. In most existing work, Tikhonov type regularization has been used. But since one is

interested in computing the characteristic function of a domain, this regularization would not be optimal. The use of other techniques similar to the method of total variation would be more appropriate. However, an adaptation has to be found since the discontinuity is not present in the solution of the ill-posed linear integral equation but in the indicator function constructed from solutions associated with different choices of the right hand side. This also leads to interesting new issues in “edge-preserving” image reconstruction

• r shape/pattern detection in image analysis. A bunch of new numerical techniques for

variational methods in imaging have been developed in the past ten years, including fixed point/dual methods, combinatorial analysis techniques (e.g., max flow/min cut algorithms, and variants) [?, ?], and these new approaches are of great help in quite a few reconstruc- tion tasks. These new tools in regularization of ill-posed problems all share in common the ability of representing well discontinuous objects and preserve edges in images. The computational cost of these approaches is not as great as it used to be: we therefore can 23

expect some improvement of standard techniques. Of course, there is still some work to make this effective, especially in 3-D problems. In connection with regularization issue we are also interested in the theoretical quan- tification of the inverse problem stability. The general electromagnetic imaging problem is a largely open and a hard issue. More accessible problems are those associated with the static scalar case. Preliminary results have been obtained by L. Bourgeois for the Cauchy problem (for domains with corners) and applied to quantify the convergence rate of the quasi-reversibility method. The extension of these results to the Helmholtz problem will be investigated. For the latter problem, it would be desirable to have explicit dependen- cies between the stability and the used frequency. This would be very interesting from practical perspectives since it can applied to relate expected spatial resolution to the used wave-length.

3.5 Shape and topological optimization methods

Participants: G. Allaire and O. Pantz The typical optimization problem in structural design would be to find the strongest structure having the minimal weight for a given exterior load. Classical optimization meth-

• ds are based on boundary variations by computing the sensitivity of the structure with

respect to the parameters that define the geometry. This classical approach, called geo- metrical approach, has the main drawback of not allowing a change in the shape topology (number of connected components or holes) during iterations, and therefore preserve the

• ne of the initial guess. However, so-called topological optimization methods are those that

allow such a change in the topology. These methods have of course a decisive advantage since quite often one has no a priori knowledge on the optimal shape topology. Our main recent contributions in the framework of topological optimization methods are first the extension of the Level Set Method (introduced by Osher and Sethian for front propagation) to solve optimization problems in structural design. Indeed one of the main attractive part of this method is to allow a change in the topology with simple implementation even for complicated objective functions [?], [?]. We also have proposed a special post-treatment of the so-called homogenization method. This method is based on relaxing the minimization problem by extending the minimization set to composite shapes (containing many tiny holes). In other words, the minimization sequences ”converge” toward a generalized (or composite) shape and not toward a genuine shape (i.e. an open set). Hence, a post-treatment is needed in order to find a genuine shape whose cost is close to the optimal one. The most common method used for that purpose consists in adding a penalization of intermediate material density to the cost-function and to resume the optimization process starting with the composite optimal shape. Unfor- tunately, this method does not allow to control neither the topological and geometrical complexity of the final shape, nor the additional cost. We have proposed a new method, which consists in building a sequence Ωn of genuine shape that ”converges” toward the

• ptimal shape. As n increases, the shape Ωn presents an increasing number of details

24

whereas its cost converges toward the optimal one [?, ?]. The extension of these methods to the case of imaging problems would be the first step toward the perspective of coupling them with the non-iterative methods mentioned above. Let also mention that for applications where only the amplitude is measured (no in- formation on the phase), which is typically the case of high frequencies, and where the use of the Kirchhoff approximation is not permitted (typically when multiple scattering effect is important), the use of iterative non-linear methods cannot be avoided. Some re- cent works on solving this inverse problem using boundary variation methods, showed that iterative methods provide relatively good results for simply connected geometries. The use

• f topological optimization methods would then be promising for this type of applications.

3.6 Asymptotic methods

Participants: G. Allaire, Y. Boukari, H. Haddar and R. Mdimagh The use of asymptotic models, if applicable, generally leads to a simplification of the forward model and offer better alternatives to solve the inverse problem. We shall be mostly concerned by configurations where the small parameter in the forward problem is linked with geometry or physical parameters. We hereafter present three typical examples where the simplified models would provide efficient ways to answer some imaging and identification problems. The first case correspond to configurations where the size of the sought obstacle is small (compared to the wavelength!). Asymptotic developments (representation formu- las, matched asymptotics, . . . [?, ?, ?]) show that these obstacles asymptotically behave like point sources whose characteristics depend on the physical nature of the obstacles, the distance between obstacles and the nature of the propagative medium (homogeneous, periodic, . . .). The expression of the asymptotic expansion can be used to design appro- priate versions of the sampling methods (like MUSIC algorithms [?]) to localize the small

• inclusions. In the framework of the PhD thesis of R. Mdimagh, we extended these results

to the case of non homogeneous backgrounds and showed how an RG-LSM like algorithm is capable of determining both the location and the effective material properties of the inclusions. A second family of asymptotic models that would be of interest for the inverse prob- lem is represented by so-called “generalized impedance boundary conditions” [?]. These boundary conditions are used to model the presence of a coating or a corrugated part of the boundary. We shall be interested in methods to evaluate these impedances from the measurements of diffracted electromagnetic waves then use these impedances to get in- formation on the coating (thickness, material properties, presence of corroded parts, . . .). The case of generalized impedance boundary conditions is different form usual impedance boundary conditions, since the impedance operator is no longer a constant. It involves tangential derivatives as well as curvature terms. So far the derivation of these boundary conditions has been done for coatings with fixed width and slowly variable material prop- erties [?, ?, ?]. Our goal is to also derive these conditions for non constant coatings and 25

possibly fast variation along the boundary terms. This work will be conducted in collab-

• ration with the POems group at INRIA Rocquencourt, and more precisely in paralle to

the PhD work of B. Delourme (who is doing her PhD under the supervision of H. Haddar and P. Joly). From the inverse problem perspectives, going to the higher order conditions may not be useful since the accuracy of the reconstruction is deteriorated by the instability of the inverse procedure. A stability study would provide us with the correct balance between the

• rder of the approximate boundary condition and the stable information one can obtain

for the inverse problem. These apsects will be studied in the framework of the PhD thesis

• f Y. Boukari.

We shall also be interested in the development of so-called “small amplitude homoge- nization method”. The standard homogenization method is not always applicable since it requires many informations on optimal composite materials which may be not yet available for some problems. In order to circumvent this difficulty it is possible to make a simpli- fying assumption that the material phases have similar properties, in other words that their properties have a low contrast or amplitude. In this case, it is possible to compute explicitely a relaxation of the problem using the notion of H-measures instead of the full homogenization theory [?, ?]. This approach has been successfully implemented for two- phase optimal design and would be of interest in simulation of crack evolution [?] (work in collaboration with F. Jouve and N. Van Goethem).

3.7 Hybrid inversion methods

Participants: G. Allaire, L. Bourgeois, H. Haddar and O. Pantz It is widely agreed within the scientific community that none of the available methods give full satisfaction in solving the inverse problem of imaging or parameter identification in its general setting. However, coupling different techniques has the potential of offering better ways to handle the problem. Among other possibilities we hereafter give some typical examples of such coupling. It is well known that optimization methods offer in general a better accuracy but are penalized by the cost of solving the direct problem and requiring a large number of iterations due to the ill-posedness of the inverse problem. However, profiting for good initial guess provided by sampling methods these method would become viable. Among

• ptimization methods, the Level Set method seems to be well suited for such coupling

since it is based on capturing the support of the inclusion through an indicator function computed on a cartesian grid of probed media. Beyond the choice of an optimization method, our goal would be to develop coupling strategies that uses sampling methods not

• nly as an initialization step but also as a method to optimize the choice of the incident

(focusing) wave that serves in computing the increment step. The “notion” of focusing wave has its origin in the well-known time reversal principle and proving this property can be shown by the Factorization method for normal far-field operators. It would be of great interest to investigate the validity of this property for more general situations (absorption, 26

anisotropy). Our research activity on this subject is still at its prospective stage and we hope being able to hire a PhD student in the near future to work on these issues. Methods from shape and topology optimization can also be used as post processing

• f sampling procedures by optimizing the criteria used by these methods in order to ap-

proximate the indicator function of the sought object. This is particularly relevant for the RGLSM type of methods since one has a large flexibility in the choice of the parametriza- tion of the unknowns. Also an optimization procedure can be used to complement the geometrical information by evaluating some of the unknown physical properties. In a different context, linked with non destructive evaluation of structures, the coupling between the Level Set Method and the Quasi Reversibility Method offers an interesting solution to the detection of anomalies (like plasticity regions, cavities, · · · ) within elas- tic structures. The Quasi Reversibility Method has been shown successful in solving the Cauchy problem for the Laplace equation, and the analysis of this method (stability, con- vergence) is now well understood. It can be used to iteratively reconstruct the field inside a given structure from the Cauchy data at the outer boundary. The Level Set method can be used to update the admissible domain during iterations. This research activity is conducted by L. Bourgeois and his PhD student J. Darde.

3.8 Sampling and optimization methods for time dependent prob- lems

Participants: G. Allaire, H. Haddar A. Lechleiter and O. Pantz This research theme is one of the long term perspectives of our group. In the case of inverse problems, we have developed imaging algorithm that are efficient in handling the inverse problem in the harmonic regime (fixed frequency) at the resonance region. Most

• f imaging facilities offer the possibility of changing the frequency of incident waves at

a low cost by using wide-band pulses. It is therefore important, from practical point of view, to extend our methods so that it efficiently handle multi-frequency data. This is a first step toward another important practical problem: the imaging from multi-frequency back-scattering data (at resonance regimes) which correspond to radar and sonar most common configurations. Other practical problems involving non linearity on the forward model (non-linear materials) would also require working with time dependent data. Our first investigations on this area have started with Qiang Chen (PhD student from the university of Delaware) by extending the sampling method to time domain problems. The 2-D preliminary results are very promising. In the case of probe methods, the first attempt in [?] to use multi-frequency signals has shown that some link between the regularization parameter and the frequency has to be incorporated in order to optimize the results. It would be interesting to extend these results to the case of linear sampling methods. Studying the connection between these methods and time-reversal methods would also be very interesting. In the context of structural design most of shape and topology optimization algorithms have been restricted to a steady-state setting. However, many problems, such as damping 27

• f waves or shocks, crash-worthiness or frequency response, require the extension of these

techniques to the time-dependent case. In particular, the adjoint technique, which requires a backward time integration, may have a prohibitive CPU cost. Therefore, various alter- native strategies must be explored as back and forth-nudging, time domain decomposition for the optimization process, etc.

4 Softwares

We hereafter describe the currently developed softwares by the team members that will constitute the basis of our foreseen software developments linked with our research themes.

• Samplings-2D: This software is written in Fortran 90 and is related to forward and

inverse problems for the Helmholtz equation in 2-D. It includes three independent

• components. The first one solves to scattering problem using integral equation ap-

proach and supports piecewise-constant dielectrics and obstacles with impedance boundary conditions. The second one contains various samplings methods to solve the inverse scattering problem (LSM, RGLSM(s), Factorization, MuSiC) for near- field or far-field setting. The third component is a set of post processing functional- ities to visualize the results. The software is available at http://www.cmap.polytechnique.fr/~haddar

• Structural Optimization with FreeFem++: This is a toolbox that contains effi-

cient implementations of shape optimization methods in 2-D using the free finite element software FreeFem++. It supports boundary variation methods, homogeniza- tion method, and Level Set method. A web page of this toolbox is available at http://www.cmap.polytechnique.fr/~allaire/freefem_en.html

• RGLSM-3D & LSM-3D: These Fortran 90 codes are dedicated to the solution of the 3-

D electromagnetic inverse scattering problem using respectively, RGLSM and LSM. There are parallel versions of these codes that are coupled to the CESC code (solver for electromagnetic scattering problems using integral equation methods) developed at CERFACS. They also support imaging for doubly layered medium. Let us also mention that our team would also greatly benefit from forward and inverse solvers developed by some of our partners, such that the code MontJoie developed by M. Durufle in the POems group (INRIA Rocquencourt) dedicated to various 3-D scattering problems and the code optopo developed by F. Jouve at the CMAP (Ecole Polytechnique) dedicated to 2-D and 3-D shape and topology optimization methods.

5 Results Dissemination

The project members will of course heavily invest on disseminating the team results to the scientific community in the most adapted way: publications and workshops organizations 28

are the basics for a good distribution and promotion, as well as the organization of specific conference sessions and meetings. More precisely, the DeFI team will:

• Write high level publications (journal papers, conference papers) - this will concern all

the scientific aspects (theory, algorithms, applications). We quote that several books (on shape and topological optimizations methods) have already been published by

• G. Allaire [?, ?, ?].
• Organize conference special sessions (such as minisymposia), schools and workshops

(such as the one organized in Sestri-Levante, Italy, 6-8 May 2008)

• Develop an open-source software toolbox in order to capitalize the solutions to con-

crete toy problems and make them available through the web site.

• Maintain a web site, where DeFI activities will be presented clearly:

See http://www.cmap.polytechnique.fr/~defi.

• Transfer the knowledge acquired by DeFI team to industrial partners, through grants,

PostDocs and PhD students.

6 Collaborators

6.1 Academic (National and International)

University of Delaware (USA) The collaboration with D. Colton, F. Cakoni and P. Monk from the university of Delaware has been very intense and fruitful since 2002 (7 publications are in connec- tion with this collaboration). The main topic of this collaboration is the development

• f non standard qualitative-methods for inverse scattering problems. It is worth men-

tioning that subsequent to our common work on the RG-LSM method, the Air-Force (USA) has engaged in collaboration with the Delaware team, a multi-million project to investigate the use of reciprocity gap functional method in the detection of under- ground cavities. Mutual scientific short time visits have been established on a regular base and the inria associated team ISIP created in 2008 (see http://www.cmapx.polytechnique.fr/~defi/equipe-associee-ISIP.html) will certainly reinforce this collaboration. The DeFI team has hosted the PhD stu- dent Qiang Chen from the Delaware team for three month during 2008. He has started in collaboration with H. Haddar a project on time-domain Linear sampling method. University of Goettingen (Germany) The well established and ongoing collaboration with Rainer Kress from the university

• f Goettingen focuses on iterative methods of Newton type or decomposition methods

to solve imaging problems. Our recent work is dedicated to the use of conformal 29

mapping in electrical impedance tomography. A Post-Doc has been hired by the Goettingen group to work on this common project. Ecole Nationale des Ing´ enieurs de Tunis (Tunisia) The collaboration with the LAMSIN in Tunisia has also been well developed and is related to inverse problems. In collaboration with A. Ben-Abda, H. Haddar super- vises the PhD work of R. Mdimagh on inverse source problems (with applications to the localization epileptic centers) and the PhD work of Y. Boukari on inverse crack problems (with application to non destructive evaluation of structures). H. Haddar is also an active member of the associated team ENEE (ENc´ ephalographie- Epid´ emiologie-Electronique) between INRIA and the LAMSIN. Since 2008, a Stic project DGRST(Tunisie)/INRIA M´ ethodes innovantes en imagerie et en contrˆ

• le

non destructif des structures between the DeFI team and the LAMSIN has been cre- ated and serves with the ENEE team as a financial support for our common PhD students. Brown University (USA) The research activity in collaboration with Jan Hesthaven from Brown university is concerned with the development of efficient generalized impedance boundary condi- tions adapted to higher order numerical schemes (like discontinuous Galerkin meth-

• ds) dedicated to the computation of time dependent solutions to scattering prob-
• lems. We actively participated to the organization of the Waves 2005 conference at

Brown university. H. Haddar and J. Hesthaven edited a special issue of the Journal

• f Computational and Applied Mathematics dedicated to this event.

CERFACS (France) The collaboration with CERFACS (with F. Collino and M’B Fares) focuses on large scale 3-D numerical experiments that simulate inverse electromagnetic scattering problems. Two published articles acknowledge this ongoing collaboration, where currently we explore the case of anisotropic media. The CINES computer resources have been used to simulate synthetic data and perform inversion experiments. We started in 2008 a new project on reduced point-sources models for scattering and inverse problems at high frequencies. Electrical and Electronics Engineering Faculty (Turkey) Active collaboration has been established between the electromagnetic group of I. Akduman and the DeFI team after hiring O. Ozdemir (Post-Doc 2007/2008). One

• f the main goals of this collaboration is the use of generalized impedance bound-

ary conditions for buried objects imaging and for coatings non destructive testing (a common project has been submitted to the Turkish National Science Foundation in 2008). The Turkish group is also developing experimental facilities for electromag- netic inverse problems that would provide us with experimental data to validate our methods. 30

6.2 Positioning within INRIA

Project POEMS (Rocquencourt) Project POems is specialized in the study and simulation of wave propagation phe-

• nomena. The main activity of the project is dedicated to the design of efficient nu-

merical methods to solve the direct problem. The project interests related to inverse problems have started with the recruitment of H. Haddar in 2002 then reinforced after the arrival of L. Bougeois in 2004. Most of the latter activity has now moved to the DeFI team. Of course strong interactions will continue between the two groups since for instance, one needs the forward solver to treat the inverse problem (either by generating synthetic data or by using the solver in an iterative procedure). Project ESTIME (Rocquencourt) Project ESTIME research interests are parameter estimates and modeling in hetero- geneous media. The latter theme is now the center of the research activity of the project: building numerical models of various flow in porous media to model contami- nant transport for environment studies, or oil displacement in petroleum engineering. The interaction on the part dedicated to parameter estimates would be of interest for the DeFI project. Project ANUBIS (Bordeaux) We have contacts with ANUBIS project through the associated team ENEE (ENc´ ephalographie-Epid´ emiologie-Electronique) between INRIA and ENIT, conducted by J. Henry (ANUBIS). We collaborate on methods for EEG problems, which are very specific applications of the imaging algorithm developed within our team. Project OPALE (Sophia Antipolis) The activity of project OPALE on shape optimization techniques is complementary to the DeFI team. However, the used techniques and applications are different. We focus on imaging problems related to scattering phenomena and shape and topol-

• gy optimization methods that avoid iterative methods using boundary variation

procedures. Project ODYSSEE (Sophia Antipolis) The project ODYSSEE has long expertise in biomedical imaging, mainly dedicated to post-processing of existing and well established imaging techniques (for instance IRM and fIRM). The goal of project DeFI with respect to these applications is rather the developments of new imaging modalities (micro-wave tomography), and therefore we are in a more prospective side. Project PARIETAL (Saclay) 31

Sought bio-medical imaging applications would be of interest for the PARIETAL group (member of the NeuroSpin plateform of CEA). As mentioned above, our ex- pertise for this type of applications is still at first stages, and indeed we would greatly benefit from the more applied nature of the NeuroSpin activity. The techniques we would like to explore and the tools involved are different from MRI or fMRI but can be complementary as mentioned in Section 2.6.2. Project Magic3D (Pau) The activity of this project is related to seismic applications, and is mainly dedicated to the development of efficient numerical methods to solve large scale elastodynamic problems in time-domain. Even though the physical phenomena and the applications involved are different from those we are interested in, the two projects belong to the broader mathematical and numerical community interested in direct and inverse problems in connection with wave propagation phenomena. Project APICS (Sophia Antipolis) The main interest of project APICS is the development of methods for modeling, identification and control of dynamical systems. Recently, they applied their tech- niques to some simplified geometrical inverse problem for the electrostatic problem. These techniques are based on harmonic analysis and therefore are hardly extendable to inverse problems for the wave equation.

6.3 Industrial grants and contacts

• DGA: As part of its strategic plan, DGA provided financial support for two years

(2007-2009) Post-Doc positions within the DeFI team to develop sampling methods for imaging of buried objects. The first year Post-Doc position has been occupied by O. Ozdemir from ITU university and the second year is currently occupied by A. Lecheiter from the university of Karlsruhe.

• EADS Foundation: a joint project DEFI-CERFACS has obtained 100k euros for

the development of reduced models associated with scattering problems at moderate

• frequencies. These models are expected to provide reliable signature of the scatterers

that serves the identification problem. The money is used to provide financial support for the PhD thesis of Anne Cossoni`

• ere. This work is conducted in collaboration with

the MBDA division of EADS.

• EADS (CCR): Characterization and optimization of composite materials for heat

propagation and lightning strike mitigation. The goal is to find accurate homogenized models (including boundary layer effects and periodicity defaults) of fiber-matrix composites and optimize their thermal properties as well as their lightning strike

• protection. A PhD thesis on this subject is under preparation.

32

• Chaire EADS/X/INRIA: Mod´

elisation Math´ ematique et Simulation Num´

• erique. This

chair has been created in September 2008 (for 4 years) and G. Allaire is the coordi- nator of this chair. See the web page: http://www.cmap.polytechnique.fr/mmnschair/index.html

• Witten Technologies Inc: WTI is a company based in Boston and the winner of

numerous awards for the development of innovative technology for the detection

• f underground objects (c.f. www.wittentech.com). The company is interested in

the application of sampling methods to the imaging of urban infrastructure using multistatic GPR data. The contact with company has been established through the University of Delaware team.

• CEA (DM2S, Saclay): Determination by inverse problem techniques of porosity and

permeability of underground nuclear waste repository. The goal is to find macroscopic porosities and permeabilites of porous media from in-situ diffusion experiments. This project is sponsored by the GDR MOMAS lead by G. allaire. 33