Imaging periodic penetrable scattering objects is of interest for non-destructive testing of photonic devices. The problem is motivated from the decreasing size of periodic structures in photonic devices, together with an increasing demand in fast non-destructive testing. In this project, we consider the problem of imaging a periodic penetrable structure from measurements of scattered electromagnetic waves. Qualitative inverse scattering techniques are particularly attractive here since they do not use time consuming optimization techniques for reconstruction but rather directly transform measured data into a picture of the scattering object. We show that the Factorization method can be used as an algorithm for imaging of a special class of periodic dielectric structures known as diffraction gratings. Our sampling method computes a picture of the shape of the periodic structure from measured near-field data in a rapid way .

The MUSIC algorithm is a well-known imaging technique in signal processing for determining the location of emitters from sensors with arbitrary locations and directional characteristics in a noisy environment. Recent research in the inverse scattering literature has sought to apply this technique to determine not only the location, but the shape of extended scatterers. In a joint work with Tilo Arens and D. Russell Luke we relate the MUSIC algorithm to the Factorization method and show that MUSIC is actually applicable to inverse scattering problems without constraint on the object size. These results are also extended to scattering from cracks. With explicit constructions in hand, we are also able to provide error and stability estimates for practical implementations in noisy environments with limited data. In particular, we address the relation of the spectral properties of the continuous far field operator to those of the discrete version used implicitly in numerical examples appearing in the literature .

The RG-LSM 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 is 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 is for instance well suited for medical imaging techniques using microwaves (to detect tumors and malignancies characterized by strong variation in dielectric properties). 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 only 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. In a joint work with O. Ozdemir we first extended this approach to the case of rough interfaces . Motivated by microwave imaging experiments, we are currently investigating the cases where the inclusions are buried under thin rough layers for which the use of generalized interface conditions would be appropriate. A long time prospective of this work is to tackle the 3-D electromagnetic case.

We are interested in solving the inverse problem of determining a screen (or a crack) from multi-static measurements of electromagnetic (or acoustic) scattered field at a given frequency. An impedance boundary condition is assumed to be verified at both faces of the screen. We extended in a first step the use of the linear sampling method and the reciprocity-gap sampling method to retrieve the shape of the screen and we are currently analyzing the accuracy of these methods with respect to the impedances values as well as using this analysis to derive a priori estimates on the impedances values. This work is pursued in collaboration with F. Ben Hassen.

In collaboration with P. Monk and Q. Chen from the University of Delaware, we extended the use of sampling methods to inverse scattering problems with time dependent data. We considered in this first investigation the scalar problem and obstacles with Dirichlet boundary conditions. Motivated by ground penetrating radar experiments, we treated the case of near field scattered data generated by incident point sources with causal pulses. We first formulate the sampling algorithm using appropriate convolution in time. The causality assumption introduces additional difficulty is carrying out usual theoretical analysis of the method since one cannot rely on the use of Fourier transform. We provide a factorization of the sampling operator using retarded potentials which are then analyzed with the help of a Fourier-Laplace analysis. We also performed preliminary numerical simulations where the sampling equation is solved using truncated singular value decomposition. The obtained numerical results show good reconstructions and provide a satisfactory validation of our approach.

The so-called interior transmission problem plays an important role in the study of inverse scattering problems from (anisotropic) inhomogeneities. Solutions to this problem associated with singular sources can be used for instance to establish uniqueness for the imaging of anisotropic inclusions from muti-static data at a fixed frequency. It is also well known that the injectivity of the far field operator used in sampling methods is equivalent to the uniqueness of solutions to this problem. The frequencies for which this uniqueness fails are called transmission eigenvalues. We are currently developing approaches where these frequencies can be used in identifying (qualitative informations on) the medium properties. Our research on this topic is mainly done in the framework of the associate team ISIP http://www-direction.inria.fr/international/PHP/Networks/LiEA.php with the University of Delaware. Three contributions have been accomplished:

The main topic of the PhD thesis of A. Cossonnière is to extend some of the results obtained above (for the scalar problem) to the Maxwell's problem. In this perspective, theoretcial results related to solutions of the interior transmission problem for medium with cavities and existence of transmission eigenvalues have been obtained. During September-December 2009, A. Cossonnière visited the UDEL and studied the continuity of transmission eigenvalues with respect to the medium properties. Parallel to this work, G. Giorgi, who started in 2009 a PhD thesis co-directed by H. Haddar and M. Piana begun investigating (during a three months training at the DEFI team) a new procedure to improve the lower bound on medium index from observed transmission eigenvalue based on ideas inspired by the work of Cakoni-Gintides-Haddar mentioned above.

Despite Newton-like methods are among the classical techniques for solving non-linear inverse problems, their convergence analysis is still incomplete. In a joint project with Andreas Rieder, we develop a general convergence analysis for an entire class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. The methods under consideration consists of two components: the outer Newton iteration (stopped by a discrepancy principle) and an inner regularization scheme which provides the update of the iteration. In this paper we give a novel and unified convergence analysis which is not restricted to a specific inner regularization scheme. Indeed, our analysis applies to a variety of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients .

It is well admitted that optimization methods offer in general a good accuracy but are penalized by the cost of solving the direct problem and by requiring a large number of iterations due to the ill-posedness of the inverse problem. However, profiting from good initial guess provided by sampling methods these method would become viable. Among optimization 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 only 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.

Dimitri Nicolas started his PhD on September 2009 on this topic under the supervision of G. Allaire. Preliminary 2-d numerical experiments have been conducted by initializing a geometric optimization algorithm with the shape provided by the linear sampling method. The obtained results validate the efficiency of this coupling in the case of simply connected obstacles. More complex configurations are under investigations.

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 electrostatic imaging. In the fashion of a decomposition method, the reconstruction of the boundary shape ${\Gamma }_0$ of a perfectly conducting or a nonconducting inclusion within a doubly connected conducting medium $D\subset {ℝ}^2$ from over-determined Cauchy data on the accessible exterior boundary ${\Gamma }_1$ is separated into a nonlinear well-posed problem and a linear ill-posed problem. The approach is based on a conformal map $\Psi :B\to D$ that takes an annulus $B$ bounded by two concentric circles onto $D$. In the first step, in terms of the given Cauchy data on ${\Gamma }_1$, by successive approximations one has to solve a nonlocal and nonlinear ordinary differential equation for the boundary values ${\Psi |}_{C_1}$ of this mapping on the exterior boundary circle of $B$. Then in the second step a Cauchy problem for the holomorphic function $\Psi$ in $B$ has to be solved via a regularized Laurent expansion to obtain the unknown boundary ${\Gamma }_0=\Psi \left(C_0\right)$ as the image of the interior boundary circle $C_0$.

In a joint work with R. Kress we proposed an extension of this approach to two-dimensional inverse electrical impedance tomography with piecewise constant conductivities. A main ingredient of our method is the incorporation of the transmission condition on the unknown interior boundary via a nonlocal boundary condition in terms of an integral equation. We present the foundations of the method, a local convergence result and exhibit the feasibility of the method via numerical examples .

With Alex Kelly, presently a post-doc at CMAP, we are studying a two phase optimal design problem where the state equation is a wave equation. As usual this type of problem is ill-posed, namely it does not admit a solution. Establishing its relaxed formulation is a difficult task, so we simplify the problem by making an assumption on the small amplitude of the contrast. We then perform a second-order asymptotic expansion of the original problem with respect to this small aspect ratio. It is still not a well-posed problem but its relaxation is much more simple, using the notion of $H$-measures, which is easier to manipulate than homogenization theory. This yields a satisfying existence theory as well as an efficient numerical method for computing the optimal designs. We are currently writing a paper on the topic.

In most shape optimization problems, the optimal solution does not belong to the set of genuine shapes but is a composite structure. The homogenization method consists in relaxing the original problem thereby extending the set of admissible structures to composite shapes. From the numerical viewpoint, an important asset of the homogenization method with respect to traditional geometrical optimization is that the computed optimal shape is quite independent from the initial guess (even if only a partial relaxation is performed). Nevertheless, the optimal shape being a composite, a post-treatment is needed in order to produce an almost optimal non-composite (i.e. workable) shape. The classical approach consists in penalizing the intermediate densities of material, but the obtained result deeply depends on the underlying mesh used and the details level is not controllable. We proposed (in a joint work with K. Trabelsi) a new post-treatment method for the compliance minimization problem of an elastic structure. The main idea is to approximate the optimal composite shape with a locally periodic composite and to build a sequence of genuine shapes converging toward this composite structure. This method allows us to balance the level of details of the final shape and its optimality. Nevertheless, it was restricted to particular optimal shapes, depending on the topological structure of the lattice describing the arrangement of the holes of the composite. We lifted this restriction in order to extend our method to any optimal composite structure for the compliance minimization problem. We intend to extend this approach to the minimization of other cost functions and are currently working on the multiload case.

With F. Jouve et N. Van Goethem we worked on the numerical implementation of the Francfort-Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith energy release rate. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation, including kinking and branching. Several numerical examples in $2d$ and $3d$ are discussed in and a full article will soon appear.

In a joint work with M. Palombaro and J. Rauch, we studied the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period $\epsilon$. We consider initial data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schrödinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schrödinger equation , .

Jointly with L. Friz, we extended these previous results in the case of a locally periodic media. In such a case, on top of homogenization appears another effect, called localization (similar to the so-called Anderson localization for the Schrödinger equation in quantum mechanics). We consider initial data that are localized Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave at a given frequency $\xi$ and of a smooth envelope function whose support is concentrated at a point $x$ with length scale $\sqrt{\epsilon }$. We assume that $(\xi ,x)$ is a stationary point in the phase space of the Hamiltonian $\lambda (\xi ,x)$, i.e., of the corresponding Bloch eigenvalue. Upon rescaling at size $\sqrt{\epsilon }$ we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian $\lambda (\xi ,x)$ is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger equation is made of a countable sequence of finite multiplicity eigenvalues with exponentially decaying eigenfunctions .

In a first work, in collaboration with S. Chun and J. Hesthaven from Brown University, we established transmission conditions modelling thin anisotropic media in time dependent electromagnetic diffraction problems. The derived interface conditions turn out to be well suited for Discontinuous Galerkin methods since the latter implicitly support discontinuities between elements. The interface conditions only results into a modification of the numerical flux used in DG methods. These conditions has been successfully tested in the 1-D case up the fourth order where stabilization in time has been applied to the fourth order condition. It is also worth noticing that the expression of these conditions in the anisotropic case cannot be simply deduced from the isotropic one by just replacing constant coefficients with their matrix equivalent. We extended the 1-D case to the 2-D and 3-D ones, where stable conditions are designed for curved geometries up tor order 3 and for flat ones up to order 4. These conditions are numerically validated in the 2-D case .

Jointly with B. Delourme and P. Joly we are investigating the extension of this work to the cases where the thin interface has (periodic) rapid variations along tangential coordinates. Motivated by non destructive testing experiments of tires, we considered the case of cylindrical geometries and time harmonic waves. We already obtained a full asymptotic description of the solution in terms of the thickness in the scalar case using so called matched asymptotic expansions. This asymptotic expansion is then used to derive generalized interface conditions and establish error estimates for obtained approximate models. The case of 3-D Maxwell's equations is under study.

We studied so-called Generalized Impedance Boundary Conditions (gibc) in the context of time-harmonic rough surface and rough layer scattering. In such problems one considers scattering objects like an unbounded hypersurface or an inhomogeneous infinitely extended layer. For a variety of interface and boundary conditions including the GBICs, we showed existence and uniqueness of solution for scattering of acoustic or TE/TM polarized electromagnetic waves from such structures. This result is achieved by Rellich identities yielding explicit a-priori bounds on the solution - those also allow to transfer the asymptotic analysis of GBICs for bounded obstacles to the rough surface setting . Currently under investigation is whether these results can be extended to the full electromagnetic rough layer scattering problem.

In collaboration with B. Aslanyurek, who was visiting our group for 9 months in 2009, we derived Generalized Impedance Boundary Conditions that model thin dielectric coatings with variable width. We treated the 2-D electromagnetic problem for both TM and TE polarizations. The expressions of the gibcs are derived up to the third order (with respect to the coating width). The order of convergence is numerically validated through various numerical examples. A particular attention is given to the cases where the inner boundary has corner singularities .

We are interested here in the identification of a medium impedance from the knowledge of far measurements of a scattered wave at a given frequency. Assuming that the unknown medium occupies a domain $D$, the medium impedance is understood as a “local” operator that links the Cauchy data of the field $u$ on the medium boundary $\Gamma :=\partial D$. More precisely we consider the cases where a boundary condition of the form: $\partial u/\partial \nu +Zu=0$ on $\Gamma$ is satisfied, where $Z$ is a boundary operator and $\nu$ denotes the outward normal field on $\Gamma$.

The exact impedance operator $Z$ corresponds to the so-called Dirichlet-to-Neumann (DtN) map, i.e. ${f\mapsto -\partial u/\partial \nu |}_{\Gamma }$ where $u$ solves the Hemholtz equation inside $D$ and satisfies $u=f$ on $\Gamma$. Consequently determining this map is “equivalent” to identify the physical properties inside $D$, which is in general a severely ill-posed problem that requires more than a finite number of measurements.

We are interested here in situations where the operator $Z$ is an approximation of the exact DtN map. In general these approximations correspond to asymptotic models associated with configurations that involve a small parameter. These cases include small amplitude roughness, thin coatings, periodic gratings, highly absorbing media, $...$

The simplest form is the case where $Z$ is a scalar function, which corresponds in general to the lowest order (non trivial) approximations, for instance in the case of very rough surfaces of highly absorbing media (the Leontovich condition). However, for higher order approximations or in other cases the operator $Z$ may involve boundary differential operators. For instance when the medium contains a perfect conductor coated with a thin layer of width $\delta$ then for TM polarization, the approximate boundary conditions of order 1 corresponds to $Z=1/\delta$ while for the TE polarization it corresponds to $Z=\delta ({\partial }_{ss}+k^{2}n)$ where $s$ denotes the curvilinear abscissa, $k$ the wave number and $n$ is the mean value of the thin coating index with respect to the normal coordinate. Higher order approximations would include curvature terms or even higher order derivatives. This type of conditions will be referred to as Generalized Impedance Boundary Conditions gibc. One easily sees, from the given example, how the identification of the impedance would provide information on some effective properties of the medium (for instance, the thickness of the coating and the normal mean value of its index). Determining these effective properties would be less demanding in terms of measurements than solving the inverse problem with the exact DtN map (the unknown parameters have one dimension less) and we also expect that the inherent ill-posedness to be less severe.

In a first work with L. Bourgeois and motivated by the example above we addressed the question of unique identification and stability of the reconstruction of $Z=\mu {\Delta }_{\Gamma }+\lambda$ from the knowledge of one scattered wave. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also considered the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact set. We also extend local stability results to the case of back-scattering data .

The general goal of the PhD thesis of Nicolas Chaulet (started in October 2009) is to extend this work to more complex expressions of the impedance operator and validate theoretical results through numerical experiments. Some 2-D numerical results are obtained in this direction as well as stability results with respect to error on the boundary location. We also would like to investigate the problem where the boundary is also unknown and analyze whether a gibc induces cloaking.

The Lippmann-Schwinger integral equation describes scattering electromagnetic waves from penetrable objects. If the modeling of the inhomogeneous medium involves space dependent coefficients in the highest order terms of the underlying partial differential equation, then the corresponding integral operators typically fail to be compact. In a joint work with Andreas Kirsch we investigate such cases and study the arising integral equations in weighted spaces of square integrable functions. The two examples we treat are acoustic scattering from a medium with a space dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann-Schwinger type. Therefore we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in $L^2$ affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in $L^2$ does not suffer from our $L^2$ setting, compared to schemes in higher order Sobolev spaces .

Waves in inhomogeneous media with variable refractive index can be described by the Lippmann-Schwinger integral equation. We investigate spectral methods for these integral equations that go back to an idea of Vainikko. In our analysis, we are especially interested in media with non-smooth physical characteristics and analyze the convergence order of adapted numerical schemes for this problem. We are also interested in special ways of discretizing the corresponding integral equations for multiple distant scattering objects. In the future, we aim to apply such spectral techniques to electromagnetic scattering problems in optics.

Scattering of electromagnetic waves from the surface of ground are often modelled by a time-harmonic scattering problem involving unbounded scattering objects. We are interested in theoretical and numerical studies of this type of problems via variational formulations.

In a first work, A. Lechleiter and S. Ritterbusch considered the scalar problem in dimension two and three. The refractive index describing physical properties of the medium can be real or (partially) complex valued and is allowed to jump across interfaces. However, the index needs to satisfy a non-trapping condition, which requires, roughly speaking, monotonicity in the direction normal to the layer. In the half space above and below the rough layer a radiation condition is set up using the angular spectrum representation. Due to the unbounded setting, integral formulas similar to Rellich's identity are derived to obtain a priori bounds for a variational solution of the rough layer scattering problem. This a-priori bound is the basis for formulated existence result. Regularity theory and bounds on its frequency dependence are also provided .

We are also investigating extensions of such approach to the more complicated 3-D electromagnetic problem. In that perspective we considered the scattering of time-harmonic electromagnetic waves from a metallic plate coated with a dielectric layer. This problem occurs for instance when monochromatic light propagates through photonic assemblies mounted on a plate. We first established a variational framework using the DtN map for Maxwell equation in half space. As opposed to the scalar case, the real part of this operator does not have a fix sign, which induces difficulties is establishing existence of solutions. The latter is done using an appropriate limiting absorption principle combined with a priori estimates derived from Rellich type identities. Our analysis only apply to small perturbation of stratified parallel layers. We are now interested in cases where the perfect conductor has a rough surface and also in widening the range of admissible material configurations.

We have developed a new contact algorithm for bodies undergoing finite deformations. Only the kinematic aspect of the contact problem has been investigated, that is the numerical treatment of the non-intersection constraint. In consequence, mechanical aspects like friction, adhesion or wear have not been considered and we restricted our analysis to the simplest frictionless case. On the other hand, our method allowed us to treat contacts and self-contacts, thin or non-thin structures in a single setting. This work has lead to the publications of two papers. One focus on the simulation of aortic valves, where complex self-contacts between the valves could occur . A c++ code has been developed to treat those contacts and has been coupled with a fluid structure code by the REO team of the INRIA. The other is less specialized to a particular application and give a presentation of the algorithm in a more general setting . It also contains several applications in a two dimensional case (dynamic of balloons, contacts and self-contacts between linear and non-linear elastic bodies). The codes where developed under Freefem++ and C++.

Blood is essentially composed of red blood cells, white blood cells and platelets suspended in a fluid (blood plasma). If it can be considered as a homogeneous fluid when circulating in vessels of large diameter, this approximation is no longer valid when it reaches vessels with diameter of an order of magnitude comparable to that of the cells it carries. In this case, the influence of the cells on the flow can no longer be homogenized. Therefore, the mechanical behavior of red blood cells (which account for $99\%$ of the cells presenting in the blood), their interaction with the surrounding fluid or between themselves (by contact) must be taken into account. Numerical tool plays thus an essential role: it enables to validate the advanced physical models, to access to data difficult to obtain experimentally and to determine the dependence of the flow behavior on the parameters of the model. In , we proposed a numerical method which allows to take into account these three essential aspects (mechanical behavior of red blood cells, fluid/structures interactions and structures/structures contact interactions). Our study is limited to the two-dimensional case which, although simplistic, allows us to reproduce a quite large range of experimental observations as shown in the numerical simulations obtained. We intend to extend our analysis to the three dimensional case, which is a lot more difficult to tackle. In particular, both flexural and membrane effects are present in the 3d setting (whereas only flexural effects are relevant in the 2D case). Moreover, the eventual management of the meshes of the RBC and of their interaction with the fluid is also challenging.