HABILITATION A DIRIGER DES RECHERCHES

pr´esent´ee a` L’UNIVERSITE PARIS DAUPHINE

Sp´ecialit´e Math´ematiques appliqu´ees et applications des math´ematiques

Imagerie et Mod`eles Asymptotiques en Electromagn´etisme

Pr´esent´ee par Houssem HADDAR

Rapporteurs: Abderrahmane BENDALI Andreas KIRSCH Patrick JOLY (coordinateur)

Examinateurs: Gregoire ALLAIRE David COLTON Maria J. ESTEBAN Mohamed JAOUA Rainer KRESS Benoit PERTHAME

Contents

1 Asymptotic Models for Scattering Problems 9 1.1 Introduction ...... 9 1.2 Thin layer approximations for time dependent problems ...... 11 1.2.1 The model problem ...... 11 1.2.2 Scaled asymptotic expansion ...... 12 1.2.3 Stability and error analysis ...... 13 1.2.4 Expression of the GIBCs ...... 15 1.2.5 Numerical Efficiency ...... 17 1.3 Scattering from imperfectly conducting bodies ...... 19 1.3.1 The Helmholtz case ...... 20 1.3.2 The Maxwell case ...... 20 1.3.3 Numerical validation ...... 23 1.4 Perspectives and related developments ...... 25

2 Asymptotic Models and Ferromagnetic Materials 27 2.1 Introduction ...... 27 2.2 Study of the forward problem ...... 31 2.3 Approximate models for thin ferromagnetic coatings ...... 34 2.3.1 Expression and stability of GIBCs ...... 34 2.3.2 Theoretical validation ...... 36 2.3.3 Numerical scheme and validation ...... 36 2.4 Homogenized model for periodic ferromagnetic media ...... 36 2.4.1 Description of the mathematical model ...... 37 2.4.2 Description of the homogenized model ...... 38 2.4.3 Convergence results ...... 40 2.4.4 Applications and validation ...... 41 2.5 Perspectives ...... 41

3 Imaging techniques and inverse problems 43 3.1 Introduction ...... 43 3.2 The Linear Sampling Method ...... 44 3.2.1 A brief (historical) review ...... 44 3.2.2 A model problem: Maxwell’s equations for anisotropic media . . . . . 45 3.2.3 The linear sampling method ...... 48 3.2.4 Numerical validation ...... 50 3.2.5 Interior Transmission Problems ...... 52

3 4 CONTENTS

3.3 Inversion methods using the Reciprocity Gap principle ...... 54 3.3.1 Inverse planar screen problem ...... 55 3.3.2 A sampling method based on RG ...... 56 3.4 Iterative Methods ...... 59 3.4.1 A result in connection with the Newton Methods ...... 59 3.4.2 Electrostatic Imaging via Conformal Mapping ...... 59 3.5 Perspectives ...... 61 Introduction (in French)

Cette introduction est un r´esum´e succint des diff´erentes ´etapes de mon activit´e de recherche. Elle est destin´ee a` donner une vue d’ensemble d´econnect´ee de l’ordonnancement th´ematique du reste de ce m´emoire. C’est aussi la seule partie r´edig´ee en fran¸cais... Le choix de l’anglais pour la partie technique, qui n’est pas une vraie commodit´e pour moi, a ´et´e dict´e par la pr´esence de plusieurs membres non francophones dans le jury. Mes recherches ont d´ebut´e avec mes travaux de th`ese, sous la direction de P. Joly, sur les mod`eles asymptotiques de revˆetements ferromagn´etiques de faible ´epaisseur. Cette th´ematique ´etait soutenue par Dassault-Aviation et la DRET (jusqu’`a la disparition de celle- ci). Au del`a de l’´etude d’un mod`ele non lin´eaire assez particulier, les ´equations de Maxwell- Landau-Lifshitz [A10], ces travaux avaient aussi pour objectif de proposer des m´ethodes de d´erivation de conditions aux limites stables en temps qui permettent de reproduire avec suff- isamment de pr´ecision la diffraction des ondes ´electromagn´etiques par un revˆetement de faible ´epaisseur. Ces travaux ont fait l’objet de plusieurs publications, ou` dans les deux premi`eres [A13, A14] seul le probl`eme 1D ´etait a` l’´etude, permettant de se focaliser sur la sp´ecificit´e non lin´eaire des ´equations du ferromagn´etisme et ´egalement la d´ependance temporelle des solutions. Ces travaux ont ´et´e ensuite g´en´eralis´es au cas 3D [A15], ou` les difficult´es tech- niques li´ees a` la prise en compte des caract´eristiques g´eom´etriques de l’obstacle diffractant ont ´et´e trait´ees. Je me suis ´egalement int´eress´e aux cas des revˆetements poss´edant des car- act´eristiques p´eriodiquement et rapidement variables le long de la surface. Des techniques d’homog´en´eisation bas´ees sur les outils de convergence a` double ´echelle ont ´et´e employ´ees pour la d´erivation et la justification du mod`ele limite [A16]. Le mod`ele obtenu est non stan- dard en homog´en´eisation du fait de la non lin´earit´e des ´equations. Seul le cas stratifi´e a ´et´e compl`etement ´etudi´e. La justification math´ematique du mod`ele dans le cas d’une p´eriodicit´e quelconque pose des difficult´es techniques encore non r´esolues. Ma th`ese a ´et´e suivie par un stage post doctoral a` l’universit´e de Delaware ou` j’ai choisi de m’initier a` un th`eme tr`es diff´erent du ferromagn´etisme : il s’agit de l’imagerie ´electromagn´etique ou acoustique appliqu´ee a` l’identification d’objets par des ondes a` fr´equence fixe. Apr`es avoir ´etudi´e pendant ma th`ese un moyen de camoufler un objet par des revˆetements sophistiqu´es (ferromagn´etiques), je me suis retrouv´e a` d´evelopper une m´ethode de visualisa- tion qui tente de faire abstraction de sa nature physique ! Il s’agit de la Linear Sampling Method. Le principe de cette m´ethode, qui a ´et´e initialement introduite par Colton-Kirsch en 1996 pour l’acoustique, est de faire correspondre a` chaque point du domaine sond´e un crit`ere permettant de savoir s’il est a` l’int´erieur de l’objet ou pas. Ce crit`ere est obtenu par la r´esolution d’un syst`eme lin´eaire construit a` partir des mesures d’ondes diffract´ees et d’une fonction test bien choisie. Mes travaux sur cette m´ethode, pendant le post-doc et apr`es mon entr´ee a` l’INRIA, ont essentiellement port´e sur son extension a` l’´electromagn´etisme [A7, A24]

5 6 CONTENTS et aux milieux anisotropes [A11, A2, P4], en collaboration avec F. Cakoni, D. Colton et P. Monk de l’Universit´e de Delaware pour les aspects th´eoriques et avec F. Collino et M’B. Fares du Cerfacs sur les aspects num´eriques [A5]. C’est avec ces deux derniers que je me suis associ´e pour r´ealiser des tests num´eriques d’inversion a` ´echelle r´eelle. Ces travaux ont fait, entre autres, l’objet d’un article de synth`ese [A8]. Apr`es mon entr´ee a` l’INRIA, outre la continuation des mes recherches initi´ees durant mon post-doc, j’ai pu diversifier mes activit´es a` l’int´erieur des deux th´ematiques, probl`emes inverses et mod`eles asymptotiques, notamment a` travers de nouvelles collaborations.

Mes recherches sur les mod`eles asymptotiques se sont poursuivies en collaboration avec P. Joly et H.M. Nguyen par la d´erivation de conditions aux limites g´en´eralis´ees mod´elisant la diffraction par des obstacles fortement absorbants [A17, AS18]. Leur validation num´erique a fait l’objet d’une r´ecente note en collaboration avec P. Joly et M. Durufl´e [A9]. Par ailleurs, j’ai pu profiter de mon passage a` l’universit´e de Brown pour initier avec Y. Hesthaven une coop´eration avec le projet ONDES sur des sujets d’int´erˆets communs, dont la d´erivation de conditions d’imp´edances g´en´eralis´ees associ´ees a` des revˆetements ou interfaces de faibles ´epaisseurs. Ceci a aboutit au projet INRIA-NSF “Collaborative Effort on Approximate Boundary Conditions for Computational Wave Problem”. Un travail en cours s’int´eresse a` l’utilisation de m´ethodes de Galerkin Discontinu pour la discr´etisation des mod`eles approch´ees d’interfaces de faibles ´epaisseurs en ´electromagn´etisme. C’est aussi en collaboration avec J. Hesthaven que nous projetons d’´editer en 2006 un volume sp´ecial du journal JCAM d´edi´e a` la conf´erence WAVES’05 qui s’est tenue a` l’Universit´e de Brown. Tr`es r´ecemment, et dans le cadre du travail de th`ese de X. Claeys, que je co-encadre avec P. Joly, je m’int´eresse a` la d´erivation de mod`eles filaires (pour des probl`emes de diffraction en ´electromag´etisme) adapt´es a` des m´ethodes volumiques d’ordre ´elev´e. Nous nous int´eressons plus particuli`erement a` l’application de la m´ethode des d´eveloppement asymptotiques rac- cord´ees pour l’obtention de ces mod`eles.

Mon investissement sur la th´ematique des probl`emes inverses s’est intensifi´e, a` commencer par l’organisation de l’´ecole des ondes sur le th`eme des probl`emes directs et inverses en diffraction (en 2003). Cette ´ecole a permis de d´ebuter une collaboration avec R. Kress sur les m´ethodes it´eratives de type Gauss-Newton appliqu´ees a` la d´etection d’objets revˆetus de couches absorbantes [A20]. Nous explorons depuis peu l’utilisation des applications conformes comme outil d’imagerie ´electrostatique en 2D. Le premier travail concernait des inclusions parfaitement conductrices ou insulatrices [A21]. Le cas interm´ediaire est en cours d’´etude. J’ai ´egalement d´evelopp´e une collaboration tr`es ´etroite avec le laboratoire LAMSIN (en Tunisie) et plus particuli`erement avec A. Ben-Abda. Elle a pour objectif d’´etendre l’application du principe de “Reciprocity Gap” initialement con¸cu pour la d´etection de fissures planes en ´electrostatique au cas des ondes. La conjugaison de ce concept avec la LSM m’a permis d’´etendre celle-ci aux probl`emes d’imageries autres que ceux rencontr´es en diffraction, no- tamment celui de la d´etection d’objets ou d´efauts enfouis. La th`ese de F. Delbary que je co-encadre avec A. Ben Abda s’int´eresse a` son application au cas des fissures [A1]. J’ai ´egalement appliqu´e ce nouveau concept a` la d´etection des mines antipersonnelles en collab- oration avec D. Colton [A6] en consid´erant d’abord une mod´elisation 2D simplifi´ee. Le cas de l’´electromagn´etisme 3D fait l’objet d’un article r´ecemment soumis [AS3] en collaboration avec F. Cakoni et M’B Fares. Cette derni`ere th´ematique en est a` ses d´ebuts d’exploration et plusieurs questions th´eoriques et num´eriques sont encore en suspens. CONTENTS 7

Je voudrais terminer ce rapide “tour d’horizon” par citer quelques travaux plus ponctuels, qui ne sont pas couverts par le reste de cette synth`ese. Toujours en relation avec les ondes, mais sur un th`eme diff´erent, j’ai collabor´e avec D. Matignon sur l’´etude du mod`ele de Webster-Lokshin pour la mod´elisation d’instruments mu- sicaux a` vent. Ce mod`ele a la particularit´e de faire intervenir des d´eriv´ees fractionnaires comme termes dissipatifs. Nous avons ´etudi´e ce mod`ele sur le plan th´eorique et num´erique via l’utilisation de repr´esentations diffusives d’op´erateurs pseudo-diff´erentiels diffusifs [P12, P23]. Lors du passage de S. Kusiak en stage post doctoral a` l’ENSTA, nous avons collabor´e ensemble sur le d´eveloppement d’une m´ethode d’inversion qu’il a introduite dans sa th`ese, et qui partage certaines ressemblances avec la LSM. Elle est moins gourmande en donn´ees mais ne permet de retrouver que l’enveloppe convexe de l’objet recherch´e. Nous avons propos´e une extension de la m´ethode aux cas de donn´ees de type back-scattering [A22]. Par ailleurs, j’ai aussi particip´e a` l’arc Headexp de l’INRIA, ou` l’objectif ´etait de mod´eliser la propagation d’ondes ´elecromagn´etiques dans le cerveau. Mon investissement a concern´e l’encadrement de stage de DEA de S. V´erit´e pour l’impl´ementation d’une famille sp´eciale d’´el´ements finis th´etrah´edriques d’ordre 2 qui autorise la condensation de masse. Il a eu aussi pour objectif de proposer des mod`eles alternatifs pour la prise en compte des couches ext´erieures de faible ´epaisseur. 8 CONTENTS Chapter 1

Asymptotic Models for Linear Scattering Problems

Contents

1.1 Introduction ...... 9 1.2 Thin layer approximations for time dependent problems . . . . . 11 1.2.1 The model problem ...... 11 1.2.2 Scaled asymptotic expansion ...... 12 1.2.3 Stability and error analysis ...... 13 1.2.4 Expression of the GIBCs ...... 15 1.2.5 Numerical Efficiency ...... 17 1.3 Scattering from imperfectly conducting bodies ...... 19 1.3.1 The Helmholtz case ...... 20 1.3.2 The Maxwell case ...... 20 1.3.3 Numerical validation ...... 23 1.4 Perspectives and related developments ...... 25

1.1 Introduction

This part of my research is motivated by the design of efficient numerical treatment of elec- tromagnetic diffraction problems where a small parameter is present in the physical model. The smallness is measured with respect to the wavelength of harmonic regimes or the small- est one in the frequency content of the source term in time dependent problems. As later explained, this small parameter can be due to either the presence of fine geometrical details or the existence of materials with high contrasts. The classical and general solution to treat these problems is purely numerical and relies on the use of local mesh refinement to accurately take into account the effect of the small parameters. However this solution has several drawbacks: 1) it increases the size of the discrete problem, especially in 3-D situations and becomes prohibitive in extreme cases, 2) it requires the use of adapted local time stepping for time dependent problem in order to control the numerical dispersion (and therefore the precision).

9 10 CHAPTER 1. ASYMPTOTIC MODELS FOR SCATTERING PROBLEMS

Our alternative solution aims at avoiding this mesh refinement by modifying the equations (and the geometry) of the physical model in the continuous setting. We are able to construct, in some special cases, a hierarchy of simplified approximate models whose accuracy are func- tions of the small parameter. The typical example that falls within these considerations is the case of thin coatings. In order to avoid the use of mesh step proportional to the width of the coating, the idea is to replace the equations satisfied by the solution inside the thin layer by a boundary condition on the outer boundary of the scatterer, commonly denoted by Generalized Impedance Boundary Condition (GIBC). The development of this technique has gained in recent decades a sub- stantial attention in both engineering and mathematical community. Most of the work done in this area was concerned with harmonic regimes. To derive the approximate models, a vari- ety of techniques have been explored. Without pretending being exhaustive, we can mention the use of analytical expressions of the solutions in simple geometries (half space or sphere) [71, 111, 75, 39], or using Taylor expansions of the solutions [64], and more recently, using micro-local expansion of the exact impedance operator for high frequency regimes [91, 35]. The method that we adopted in our analysis and that constitutes, in our opinion, the most suited one to study the problem at the formal and mathematical levels, is based on scaled asymptotic expansions. This is for instance a quite classical technique in homogenization theory or in mechanics to derive plates and thin shell models. However, its use in the context of GIBC is quite recent, and we refer to [41] where the mathematical basis of this method has been first set in the context of the Helmholtz equation. Number of related works can be cited, among them: [37, 27, 26, 30, 39]. The principle of the method is to perform a scaling with respect to the normal coordinate to the scatterer boundary so that the problem posed on a geometry that depends on a small parameter is transformed into a parameterized family of problems posed on a fixed geometry. This latter is then analyzed using appropriate asymptotic expansions. The GIBC are derived by solving semi-explicitly the equations inside the scaled domain. We applied these ideas to the case of thin coating and in the context of electromagnetic diffraction problems. The first originality of our approach was to investigate what happens in the case of time dependent problems. Working in time domain was motivated by the study of thin ferromagnetic coatings, where the non linearity of the constitutive law prohibits harmonic regimes (next chapter is dedicated to a review of our work on this non linear case). We have shown that, independently of the non linearity of the problem, specific instability linked to time dependency of the solution may occur with the use of higher order GIBCs [A13, A15]. The second originality comes from treating Maxwell’s equations and having developed a systematic formal procedure to determine the GIBCs in this case [A13]. This is crucial to obtain higher order approximations. Error analysis is then obtained in the case of linear coatings [A15, T25]. The numerical investigation of efficient implementation of these GIBC is an important issue that has been addressed in [A13] for the 1-D case, focusing on the stability of time discretizations. The generalization to higher dimensions was partially treated in [T25] and is an ongoing work. Another situation where the use of GIBC provides a valuable alternative to mesh refine- ment is the case of highly conducting materials. The first order GIBC is the well known Leontovich (impedance) boundary condition [93]. This first approximation is now widely used and has been extensively analyzed in the literature [71, 111, 75, 38, 34]. The deriva- tion of higher order approximations was studied in [111] based on the formal asymptotic expansion given by Rytov [108]. An alternative approach based on formal expansions of 1.2. THIN LAYER APPROXIMATIONS FOR TIME DEPENDENT PROBLEMS 11 pseudo-differential operators is proposed in [35]. Our first motivation behind the investigation of this problem was to provide a rigorous analysis of error estimates. Without such analysis the notion of GIBC order would be ambigu- ous: usually linked to the truncation order in the asymptotic, this order may not coincide with the accuracy provided by the GIBC. Moreover, at a given formal order, a variety of different expressions can be derived. It turns out that the scaled asymptotic expansion can be adapted with minor modifications to give an exact characterization of the asymptotic expansion of the solution inside and outside the absorbing medium. We started by investigating a model scalar problem in the harmonic regime [A17]. A com- plete analysis is given for conditions up to order 3 (the order at which a surface wave operator appear in the expression of the GIBC). The main difficulty, beyond the technical derivation of the GIBC expressions, is to deal with the non coerciveness of the Helmholtz operator in ob- taining appropriate error estimates. The conditions involving surface wave operators require additional care regarding uniform stability with respect to the small parameter. We then generalized the analysis to the the full 3-D Maxwell [AS18]. It turned out that the “natural” third order condition involves a second order surface operator with no fix sign. This causes additional difficulties. We are for instance unable to prove, using variational techniques that the underlying boundary value problem is well posed. However, a modified GIBC can be derived with appropriate properties regarding theoretical analysis. The numerical validation of these conditions has been recently achieved [A9]. The review presented in this chapter is based on the results published in [A13, A15, A17, A9, AS18] and also on some unpublished work in the manuscript of my Phd thesis [T25]. Some of the presented results may not exactly correspond to those of these references but constitute obvious consequences or generalizations.

1.2 Thin layer approximations for time dependent problems

This section is a quick self-contained overview of results from [A13, A15] and [T25]. It is devoted to the derivation, analysis and numerical implementation of GIBCs in the context of electromagnetic time dependent scattering problem from thin coatings made of dielectric materials.

1.2.1 The model problem

Let Ω be an open bounded domain of R3 with connected complement and regular boundary Γ := ∂Ω. We assume that Ω is the union of a thin tubular domain Ωδ of small width δ, δ filled with a dielectric material, and a domain Ω0 whose boundary is a perfect conductor (see Figure 1.1). Let us assume for simplicity that the exterior domain (R3 Ω) is the vacuum and that \ the time and space scales are chosen such that the wave speed is 1 and that the thin layer material is homogeneous and isotropic (the non homogeneous case where the characteristics depend on the thickness coordinate and vary slowly with respect to tangential coordinates can be treated with minor modifications [T25]). We simulate the source terms by an initial electromagnetic pair (E , H ) that we shall assume to be compactly supported in R3 Ω. 0 0 \ 12 CHAPTER 1. ASYMPTOTIC MODELS FOR SCATTERING PROBLEMS

Dielectric coating Ωδ δ

R3 Ω \ Perfect conductor Γδ

Γ

Figure 1.1: Thin layer configuration

Denoting by (Eδ(x, t), Hδ(x, t)) the electromagnetic field at (x, t) (R3 Ω) R+, then e e ∈ \ × δ δ δ δ R3 R+ ∂tEe curlHe = 0, ∂tHe + curlEe = 0 in ( Ω) , − \ × (1.1) δ δ 3 ( (E , H ) t=0 = (E0, H0) in R Ω. e e | \ We shall use letter n as a generic notation for a regular normal field on a given regular surface. Then, Eδ n = Eδ n, Hδ n = Hδ n for (x, t) Γ R+. (1.2) e × i × e × i × ∈ × δ δ δ where (Ei , Hi ) denotes the electromagnetic field inside Ω satisfying

(ε ∂ + σ)Eδ curlHδ = 0, µ ∂ Hδ + curlEδ = 0 in Ωδ R+, t i − i t i i × δ δ R+  Ei n = 0, on Γ , (1.3)  × ×  δ δ δ (Ei , Hi )(t = 0) = (0, 0) in Ω ,  where µ, ε and σ respectively denote the relative magnetic permeability, the relative electric permittivity and the conductivity inside Ωδ. These quantities are assumed to be constant and independent of δ. Equations (1.1)-(1.3) constitute the exact transmission problem. In the situations where δ 0 the effect of the coating on the field (Eδ, Hδ) can be approximated by the use of a local → e e boundary condition on Γ (GIBC). If the error between exact and and approximate solutions is O(δk+1) than the GIBC is called to be of order k.

1.2.2 Scaled asymptotic expansion The construction the GIBCs is based on constructing first the asymptotic expansion of exact solution with respect to δ after a scaling of the thin layer with respect to the normal coordinate to Γ. More precisely, for sufficiently small δ one can uniquely associate to x Ωδ the ∈ parametric coordinates (xΓ, ν) Γ (0, δ) through ∈ × δ x = xΓ + ν n, x Ω . (1.4) ∈ where n is the normal field on Γ directed to the interior of Ω. Then it can be shown that the exact solution has the following expansions:

δ 0 1 2 2 Ee (x, t) = Ee (x, t) + δEe (x, t) + δ Ee (x, t) + , · · · (1.5) Hδ(x, t) = H0(x, t) + δH1(x, t) + δ2H2(x, t) + , e e e e · · ·

1.2. THIN LAYER APPROXIMATIONS FOR TIME DEPENDENT PROBLEMS 13 for all (x, t) (R3 Ω) R+, where E`, H`, ` = 0, 1, are functions defined on (R3 Ω) R+ ∈ \ × e e · · · \ × and independent of δ, and

δ 0 1 2 2 Ei (x, t) = Ei (xΓ, ν/δ, t) + δEi (xΓ, ν/δ, t) + δ Ei (xΓ, ν/δ, t) + · · · (1.6) δ 0 1 2 2 H (x, t) = H (xΓ, ν/δ, t) + δH (xΓ, ν/δ, t) + δ H (xΓ, ν/δ, t) + i i i i · · ·

δ R+ ` ` for all (x, t) Ω , where x, xΓ and ν satisfy (1.4) and E (xΓ, η, t), H (xΓ, η, t), ` = ∈ × i i 0, 1, are functions defined on Γ (0, 1) R+ and independent of δ. · · · × × The set of equations satisfied by the terms of these expansions can be found from equating the terms of same order with respect δ after inserting them into the equations of the problem. This identification process is based on using the expression the curl operator in parametric coordinates system (see Remark 1.2 below). The obtained set of equations allows an inductive characterization of the asymptotic ex- ` ` pansion terms. In addition, analytic formulas for Ei and Hi in terms of η and the boundary tangential values of H m on Γ, m ` 1, can be established (however this technical task e ≤ − becomes more and more involving as ` increases). These expressions are used in setting up the GIBCs. Getting the GIBC expressions. The GIBC of order k is obtained by considering the truncated expansions

k k ¯δ,k ` ` ¯ δ,k ` ` E := δ Ee and H := δ He X`=0 X`=0 δ δ as an approximation of order k of respectively Ee and He . Using the first interface condition in (1.2), one has

k δ,k ` ` + E¯ (xΓ, t) n = δ E (xΓ, 0, t) n for (xΓ, t) Γ R . (1.7) |Γ × i × ∈ × X`=0 ` Substituting the computed expressions of Ei (xΓ, 0, t) into (1.7) then leads to a boundary condition of the form

E¯δ,k n + δ,k(n (H¯ δ,k n)) = δk+1 gδ on Γ (1.8) × B × × k where gδ (t) is uniformly bounded with respect to δ < 1 and where δ,k is some local k L2(Γ) B surface operator on Γ. The boundary condition obtained by neglecting the right-hand side of (1.8) will be denoted by a GIBC of order k (1). We expect that the error between exact solutions and those computed with the use of this approximate boundary condition has the same order as neglected terms, i.e. O(δk+1). We hereafter explain the requirements of such expectation.

1.2.3 Stability and error analysis δ,k δ,k k+1 Let us denote by (Ee , He ) the desired approximation of order O(δ ) of the exact elec- δ δ tromagnetic field (Ee , He ). It is defined as satisfying the same equations in the exterior 1The definition of GIBC’s order adopted here is the same as in [A17] but is different from the one in [A15]: a condition of order k here is denoted by a condition of order k + 1 in [A15]. 14 CHAPTER 1. ASYMPTOTIC MODELS FOR SCATTERING PROBLEMS domain

δ,k δ,k δ,k δ,k 3 + ∂tEe curlHe = 0, ∂tHe + curlEe = 0 in (R Ω) R , − \ × (1.9) δ,k δ,k 3 ( (Ee , He ) = (E , H ) in R Ω. |t=0 0 0 \ whereas the interface conditions on Γ and Maxwell’s equation inside Ωδ satisfied by the exact solution are replaced by a GIBC of order k on Γ that is be written in the form

Eδ,k n + δ,k n (Hδ,k n) = 0, (1.10) e × B × e × where n denotes the normal to Γ directed to the interior of
Ω. Property (1.8) can be inter- preted as a consistency property of the GIBC. It formally guarantees that the neglected terms are of the same order as the desired accuracy. However, as we shall see in next section, this property is not sufficient to ensure that the solution to (1.9)-(1.10) gives the desired approximation. An additional property is required and is linked to the stability in time uniformly with respect to δ. This property has also close links with the well-posedness of the initial boundary value problem (1.9)-(1.10). For the exact problem, this uniform stability is a consequence of the energy estimate, d (Eδ(t), Hδ(t)) + δ δ(t) 0 (1.11) dt{E e e EΓ } ≤

1 2 2 where (E, H) := 2 E L2(R3 Ω) + H L2(R3 Ω) is the electromagnetic energy functional E k k \ k k \ in the vacuum and   1 δ 1 δ 2 δ 2 (t) := (ε E (t) + µ H (t) ) j(δν) dν dxΓ EΓ 2 | i | | i | ZΓ Z0 is the electromagnetic energy inside the coating. It implies in particular that the L2 norm of the solution in the exterior domain can be uniformly bounded in time and with respect to δ by the L2 norm of the initial data. It also enables us to prove the validity of the asymptotic expansion (1.5) (see [A15]). For instance, one can prove that for all T > 0 and all ` > 0 there exists a constant C(T, `) such that

T 2 2 Eδ(t) E¯δ,`(t) + Hδ(t) H¯ δ,`(t) dt C(T, `) δ`+1. (1.12) e 2 R3 e 2 R3 0 − L ( Ω) − L ( Ω) ≤ Z  \ \ 

It is therefore desirable that the appro ximate problem benefits from similar stability proper- ties. A GIBC of order k is said to be satisfying energy stability if there exits a positive energy function δ,k(t) such that EΓ d (Eδ,k(t), Hδ,k(t)) + δ δ,k(t) 0. (1.13) dt{E e e EΓ } ≤ Based on energy identities of this type, with appropriate expressions for δ,k(t), one can prove EΓ (see [A15]) that for all T > 0 and all ` k there exists a constant C (T, `) such that ≤ 0 T 2 2 δ,k δ,` δ,k δ,` `+1 E (t) E¯ (t) + H (t) H¯ (t) dt C0(T, `) δ . (1.14) e 2 R3 e 2 R3 0 − L ( Ω) − L ( Ω) ≤ Z  \ \ 

One then observes that com bining (1.14) with (1.12) giv es the expected error estimates. 1.2. THIN LAYER APPROXIMATIONS FOR TIME DEPENDENT PROBLEMS 15

Remark 1.1 We remark that the use of the asymptotic expansion is essential in obtaining error estimates. This is why the regularity assumption on the boundary of Ω (and on the data) is needed for their validity. We shall numerically illustrate this in section 1.3.3 (in the case of GIBCs for imperfectly conducting obstacles). Optimal required regularities have not been investigated. However, it is clear from the technical details of the derivation of the GIBCs that higher is k, higher would be the needed regularity.

1.2.4 Expression of the GIBCs The expression of the GIBC for curved boundaries and also the discussion of stability issues as indicated above are the main important results. Before giving a summary of these results we need to first introduce some notation related to surface operators on Γ. We recall that n denotes the normal field on Γ directed to the interior of Ω. We denote by Γ the surface gradient on Γ and divΓ its adjoint with respect ∇ − to L2(Γ) inner product. We then define the surface curl of a tangential vector V and the surface vector curl of a scalar function u defined on Γ by

curlΓ V := divΓ (V n) and curl~ Γ u := ( Γ u) n. × ∇ ×

The curvature tensor is defined by := Γn. We recall that is symmetric and n = 0. C C ∇ C C We denote by c1 and c2 the eigenvalues of (namely the principal curvatures associated C 1 with tangential eigenvectors τ1, τ2), then g := c1c2 and h := 2 (c1 + c2) are respectively the Gaussian and mean curvatures of Γ. To these curvatures we associate

= h IΓ and = g IΓ, H G where IΓ(xΓ) denotes the projection operator on the tangent plane to Γ at xΓ.

Remark 1.2 A key point in the formal construction of the GIBCs is the expression of the curl operator in parametric coordinates. For instance, it is shown in [A15, AS18] that

1 1 curl V = (I + ν ) Γ (V˜ n) n + (I + ν ) Γ (V˜ n) n j(ν) M ∇ · × j(ν) M ∇ · ×      1 ( + ν )V˜ n ∂ (V˜ n), − j(ν) C G × − ν ×   δ for regular functions V defined on Ω , where V˜ is defined on Γ (0, ν¯) by V˜ (xΓ, ν) = × V (xΓ + ν n), and where the tangential operator is defined on Γ by = , and j(ν) := M M C G det(I + ν ) = 1 + 2νh + ν2g. C The expressions we shall give below are different from those in [T25] since we included here the conductivity σ. These expressions are deduced from the case without conductivity 1 1 by substituting to the operator (ε∂t)− the causal convolution operator It := (ε∂t + σ)− that associates to f(t) the function

t 1 σ (t τ) I f(t) = e− ε − f(τ) dτ. t ε Z0 16 CHAPTER 1. ASYMPTOTIC MODELS FOR SCATTERING PROBLEMS

One gets for k 2, ≤ δ,0 = 0, B δ,1 ~  = δ µ∂t + curlΓ ItcurlΓ , (1.15)  B  δ,2   = δ (1 δ( ))∂ + curl~ Γ (1 δh)I curlΓ . B − C − H t − t    For these conditions, the uniform stability (1.14) holds with

δ,0(t) = 0, EΓ

δ,1 1 0 δ,1 2 δ,1 2 Γ (t) = ε It curlΓ ϕ (t) + µ ϕ (t) dxΓ, (1.16) E 2 Γ | | | | Z   δ,2 1 0 δ,2 2 δ,2 δ,2 Γ (t) = (1 δh)ε It curlΓ ϕ (t) + µ(1 δ( ))ϕ (t) ϕ (t) dxΓ, E 2 Γ − | | − C − H · Z   δ,k δ,k 0 where we have set ϕ := n (He n) and I is the operator I with σ = 0 (anti-derivative × |Γ × t t operator). Higher order GIBCs involve surface derivatives of order greater that 2. We investigated the case of non curved boundary and showed that the natural conditions (the ones yielded by the construction procedure mentioned above) are unstable. For instance, in the case of planar surfaces

δ,3 2 = δ µ∂ + curl~ Γ I curlΓ 1 δ (µ∂ + curl~ Γ I curlΓ Γ I divΓ ) (1.17) B t t − t t − ∇ t     and as it is shown in [A13, A15], no energy stability like (1.14) can be obtained for this condition. Also, no convergence of the approximate solution to the exact one, as δ 0, can → be expected in this case. This instability is specific to the time dependent problem and can be easily analyzed for the corresponding 1-D problem. Assume for instance that Γ is the plane x1 = 0 and that the domain of propagation is x 0 and the solutions are invariant with respect to x and x . 1 ≤ 2 3 Assume also for simplicity that µ =  = 1 and σ = 0. Then, the 3-D problem can be reduced to a scalar one dimensional boundary value problem (with x = x1)

∂2 uδ ∂2 uδ = 0, x < 0, tt − xx 2 (1.18) ( ∂ uδ + δ∂2 (1 δ ∂2 )uδ = 0, at x = 0, x tt − 3 tt coupled with appropriate initial conditions, where u stands for the second or the third com- ponent of the magnetic field. Performing a plane wave analysis of these equations, one easily checks the existence of solutions of the form

δ κ (x+t) u (x, t) = c e δ ,

where c is an constant and κ is the positive real solution to 3+3κ κ3 = 0. One then observes − that although the L2 norms of uδ( , t) and ∂ uδ( , t) are bounded with respect to δ for t = 0, · t · these norms become infinite as δ 0 for all t > 0. Of course this is sufficient to show that one → cannot obtain convergence results between exact and approximate solutions. More precisely, if we consider problem (1.18) where the right hand side of the boundary condition is replaced 1.2. THIN LAYER APPROXIMATIONS FOR TIME DEPENDENT PROBLEMS 17 by a given causal data f δ(t) (that plays the role of the consistency error in the right hand side δ of (1.8)) and with zero initial data, then for all T > 0, u L2(R− (0,T ) cannot be bounded δ × by C(T ) f 2 with a constant C(T ) independent of δ . However, if we limit ourselves L (0,T ) to harmonic excitations f δ(t) = fˆδ(ω)eiωt then the solutions propagating to are uδ(x, t) = uˆδ(x, ω)eiωt where −∞ ˆδ δ f (ω) iωx uˆ (x, ω) = 2 e . iω δω2(1 + δ ω2) − 3 For instance uˆδ(x, ω) 1 fˆδ(ω) which means that at a fixed frequency, one has uniform | | ≤ ω | | stability with respect to δ (which is synonym of error estimates in concordance with the consistency property).

Stable third order condition. We also gave in [A15] a procedure to derive a stable third order GIBC. The key idea is similar to the technique used for higher order absorbing boundary conditions [63, 52] (also used in [39] in the context of GIBCs). It consists in using Pad´e expansions (with respect to δ) to replace instable parts in the operator δ,3 with stable B expressions, and making sure that the obtained operator coincides with expression (1.17) up to O(δ4) error. The latter point formally guarantees the consistency property (1.8). We refer to [A15] for the expression in the 3-D planar case. In the 1-D case, the situation is much simpler and obtaining a stable condition can be done using the formal identity

2 δ2 2 2 δ2 2 1 4 δ∂ (1 ∂ ) = δ∂ (1 + ∂ )− + O(δ ), tt − 3 tt tt 3 tt which consists into writing the boundary condition in the form

δ2 2 δ 2 δ (1 + 3 ∂tt)∂xu + δ ∂ttu = 0, at x = 0. (1.19)

One can verify that no unstable plane waves satisfy this condition. Let us finally notice that in the case of ferromagnetic coating, deriving stable third order conditions leads to non local GIBC (i.e. cannot be expressed in terms of local differential operators as in [A13]). The numerical efficiency of these boundary conditions is therefore not clear for this case.

1.2.5 Numerical Efficiency The third fundamental point in the design of GIBCs is their numerical efficiency. This part has been studied in [A13] for the 1-D case, and the treatment of the 3-D case can be found in [T25]. The main issue discussed is the stability in time of explicit discretization in time. For instance, it is important to guarantee that the CFL of the scheme does not depend on the small parameter. We shall give in the following a quick summary of its main results. One already notices that the GIBCs have the advantage of being variational, which means that they can be easily incorporated into a wide variety of numerical methods. As an illus- trative example, let us consider for simplicity the case σ = 0. Introducing the rotated current density ϕ = n (H n) on Γ, × × 18 CHAPTER 1. ASYMPTOTIC MODELS FOR SCATTERING PROBLEMS as an auxiliary variable (this a classical trick in handling boundary conditions requiring extra regularity for the boundary values), then Maxwell’s equations in R3 Ω coupled with second \ order GIBC on Γ can be variationally written in the form: 3 3 Seek (H(t), E(t), ϕ(t)) H(div, R Ω) H(curl, R Ω) H(curlΓ , Γ) such that ∈ \ × \ ×

(i) ∂tH(t) H0 + curlE(t) H0 dx = 0, R3 Ω · · Z \

(ii) ∂tE(t) E0 curlE0 H(t) dx = E0 n, ϕ(t) , (1.20) R3 Ω · − · × Z \

δ 2 ϕ ϕ δ ϕ ϕ ϕ (iii) ∂tt (t) 0 + χ curlΓ (t)curlΓ 0 dxΓ = ∂tE(t) n, 0 , Γ R · − × Z   3 3 for all t > 0 and (H , E , ϕ ) H(div, R Ω) H(curl, R Ω) H(curlΓ , Γ), with appropriate 0 0 0 ∈ \ × \ × initial conditions. For short notation we have set δ := µδ(1 δ ( )), χδ := δ(1 δ h)/ε 1 1 R− 2 − C −− 2H − and , denoting the duality product between H (divΓ , Γ) and H (curlΓ , Γ). Also notice h· ·i that equation (iii) corresponds to the derivative in time of the second order GIBC. The existence of an energy-like functional associated with this variational formulation pro- vides a straightforward analysis of the existence and uniqueness of solutions to this problem. It also allows the use of standard techniques to prove the convergence of conforming finite element discretizations. 3 3 Let X H(div, R Ω), V H(curl, R Ω), Y H(curlΓ , Γ) be the discrete sub- h ⊂ \ h ⊂ \ h ⊂ spaces where the numerical solution (Eh, Hh, ϕh) is sought. Then the semi-discrete system associated to (1.20), can be written in the following matrix form: d H (t) + E (t) = 0, MH dt h K h d ?  E dt Eh(t) Hh(t) = ϕh(t), (1.21)  M − K B  δ d2 δ ? 2 ϕ (t) + ϕ (t) = ∂ E (t), MΓ dt h KΓ h −B t h  δ δ with obvious expressions for the matrices H , E, , Γ, Γ and in terms of basis M M K M K B elements of Xh, Vh and Yh. The upscript ? denotes transposition. The less obvious task is related to time stepping, especially stability and dispersion issues. One notices that the Maxwell-GIBC system (1.20) couples two waves equations: the first one is 3-D and formed by (i) (ii) and the second one is 2-D, formed by (iii). − Rapidity of the schemes stipulates the use of explicit schemes for both, which constituted the framework of our numerical analysis. We confined ourselves to second order discretizations by using the leap-frog scheme: introducing the standard notation, where ∆t denotes the time step, 1 n+ 2 n H H ((n+ 1 )∆t), E E (n∆t) h ' h 2 h ' h this scheme can be written in the form 1 1 n+ 2 n− 2 n H Hh Hh + ∆t Eh = 0 M − K 1 1 (1.22) n+ n+ En+1 En ∆t t H 2 = ϕ 2 ME h − h − K h B h n+ 1 2 1
where ϕ ϕ ((n+ 2 )∆t). It is well known that using N´ed´elec’s lower order elements in h ' h a uniform mesh with step h requires the CFL stability condition

√N ∆t < N h, (1.23) 1.3. SCATTERING FROM IMPERFECTLY CONDUCTING BODIES 19

where N denotes the space dimension. It is desirable that the coupling with the GIBC preserves this condition. Two possible ways of coupling have been explored. We started with the full explicit scheme (after the use of appropriate mass-lumping). This scheme is obtained by associating to (1.22) the following discretization of the GIBC:

3 1 1 1 n+ n+ n− n+ δ ϕ 2 2ϕ 2 + ϕ 2 + ∆t2 δ ϕ 2 = ∆t t En+1 En . (1.24) MΓ h − h h KΓ h − B h − h  
Unfortunately, it turned out that, even if it is conservative (i.e. it admits a stable discrete energy functional), this scheme does not preserve the CFL (1.23). Moreover, the 1-D analysis shows that this CFL deteriorates as δ 0: it is of the form : ∆t < f(δ) h, where f(δ) √2δ → ∼ as δ 0. Of course this is a quite bad news for the efficiency of the GIBC since this → requirement increases the computational cost and accentuates the dispersion in the numerical scheme (1.22). This observation lead us to explore a different way of coupling by introducing the following alternative

n+ 1 ϕ 2 = (ϕn+1 + ϕn)/2, h h h (1.25) δ n+1 n n 1 2 δ n ∆t t n+1 n 1 ϕ 2ϕ + ϕ − + ∆t ϕ = E E − . MΓ h − h h KΓ h − 2 B h − h Both (1.22) and (1.25) are explicit,
but their coupling leads toan implicit sc
heme (even with the use of mass lumping). However, the size of the sparce matrix to invert at each time step is small since it concerns only the boundary terms. This scheme has also the advantage of being conservative and stable under the CFL (1.23). We also remark that since the coupling is implicit, one can use without increasing the cost of the numerical computations a fully implicit scheme on the boundary. We refer to [A13] and [T25] for some numerical examples in the case of (non linear) ferromagnetic coatings.

1.3 Scattering from imperfectly conducting bodies

The second type of problems where GIBC can be of interest is the case of diffraction by strongly absorbing obstacles. Compared to the previous case, here we assume that the con- ductivity σ is “very high” with regard to the width of the coating... More precisely, we assume that the large conductivity is such that it induces a sharp (exponential) decay of the electromagnetic field inside the coating, so that this field almost vanishes before reaching the interior boundary of the coating. In this type of configuration the asymptotic analysis of the solution is almost the same as if we consider that Ω is entirely filled with the absorbing material, which we shall assume in the sequel. To define the inherent small scale of such problems, it is instructive to adopt an engineering point of view. Assume for instance that Ω is the half space x1 > 0 and that the wave is a monochromatic plane wave with pulsation ω propagating inside this medium, then the expression of the normal component of the electric field inside Ω is of the form

σµ iω(√εµ+i x1 t) u(x, t) = αe− ω − 20 CHAPTER 1. ASYMPTOTIC MODELS FOR SCATTERING PROBLEMS where α is a constant and the complex square-root is the one with negative imaginary part. We observe that for large σ,

√ ω εµ + i σµ 2 √ωµσ. = ω ∼ − 2 n q o √2 Hence the wave inside the medium decays like exp( √ωµσx1). The quantity √2/√ωµσ, − 2 referred in the literature to skin-depth, therefore constitutes an indicator of the width of the penetrable region inside the conductive medium. Given the above considerations, the asymptotic regime that we shall study is the one where

δ := 1/√ωµσ

is small as compared with the wavelength 2π/ω (and the width of the coating). One already observes that this parameter depends itself of the frequency content of the wave. This causes difficulties in defining its counterpart for non harmonic regimes. In the following summary, we restricted ourselves to the harmonic regimes where some key results from [A17, AS18, A9] are exposed. The study of the transient regime is a still ongoing research.

1.3.1 The Helmholtz case We started in [A17] by studying a simplified 3-D case where the wave is a scalar unknown. Some technical details related to this problem are only present in [R19]. The underlying studied model is in fact a mimic of the 2-D Maxwell equations for TM polarizations. For instance, it does not correspond to an acoustical model, where the trans- mission conditions for normal derivatives depend on the small parameter δ. We believe that the mathematical study of the latter case can be deduced without major difficulties. However, the expression of the GIBC’s would be different. To be in the continuity of the previous section, we choose to rather present the results of the 3-D Maxwell equations [AS18, A9].

1.3.2 The Maxwell case The model problem We maintain here the notation of Section 1.2. We further assume that the time dependence is harmonic with a frequency ω > 0, i.e. the electromagnetic field is of the form

Eδ(x, t) = Re Eδ(x) exp(iωt) , Hδ(x, t) = Re Hδ(x) exp(iωt) .

δ δ  δ δ  Denoting by (Ee , He ) = (E , H ) R3 Ω, then | \ iωEδ curl Hδ = 0, iωHδ + curl Eδ = 0, in R3 Ω, (1.26) e − e e e \

where this total field can be decomposed into the sum of an incident field (Einc, Hinc) and a δ δ scattered one (Ee,s, He,s) that satisfies the Silver-Muller¨ radiation condition:

lim xˆ (E xˆ) H xˆ = 0 (1.27) x × × − × | |→∞ 1.3. SCATTERING FROM IMPERFECTLY CONDUCTING BODIES 21

uniformly with respect to xˆ := x/ x . Let n denotes a regular normal field on Γ. Then, | | Eδ n = Eδ n, Hδ n = Hδ n on Γ, (1.28) e × i × e × i × δ δ where (Ei , Hi ) denotes the electromagnetic field inside Ω. As indicated in the introduc- tory discussion, the conductivity inside Ω depends on δ and is set to σ := σδ = 1/(µωδ2). Therefore, 1 δ δ (iωε + 2 )Ei curl Hi = 0, in Ω, µωδ − (1.29) δ δ ( iωµHi + curl Ei = 0, in Ω.

Derivation of the GIBCs The procedure is similar to the one developed for thin coatings. The only difference consists into isolating the region where the skin-effect is present, i.e. in the vicinity of Γ, and to perform a scaling in that region. More precisely, for a sufficiently small given positive constant e¯ we define Ωe¯ = x Ω ; dist(x, ∂Ω) < e¯ , and to any x Ωe¯ we uniquely associate the { ∈ } ∈ parametric coordinate (xΓ, ν) Γ (0, e¯) satisfying ∈ × e¯ x = xΓ + ν n, x Ω . (1.30) ∈ The asymptotic expansion for the exterior field is the same as (1.5) while one needs to specify the asymptotic expansion of the interior field only in the domain Ωe¯. More precisely,

δ 0 1 2 2 e¯ Ei (x) = Ei (xΓ, ν/δ) + δEi (xΓ, ν/δ) + δ Ei (xΓ, ν/δ) + for x Ω , · · · ∈ (1.31) δ 0 1 2 2 e¯ H (x) = H (xΓ, ν/δ) + δH (xΓ, ν/δ) + δ H (xΓ, ν/δ) + for x Ω , i i i i · · · ∈

` ` + where x, xΓ and ν satisfy (1.30) and where E (xΓ, η), H (xΓ, η) : Γ R C such that i i × 7→ ` lim E (xΓ, η) = 0 for a.e. xΓ Γ, η i ∈ →∞ ` (1.32) lim H (xΓ, η) = 0 for a.e. xΓ Γ. η i ∈ →∞

` ` These conditions ensure that Ei and Hi are exponentially decreasing inside the obstacle. The general considerations exposed in Section 1.2.4 are basically the same, the only dif- ference at this stage is purely technical (see [A17]).

Expression of the GIBCs and error analysis

δ,k δ,k k+1 We denote by (Ee , He ) the desired approximation of order O(δ ) of the exact electro- δ δ magnetic field (Ee , He ). It satisfies the standard Maxwell equations in the exterior domain

iωEδ,k curl Hδ,k = 0, iωHδ,k + curl Eδ,k = 0, in R3 Ω, (1.33) e − e e e \

and is the sum of the incident field (Einc, Hinc) and a scattered one that satisfies the radiation condition (1.27). The interface conditions on Γ and Maxwell’s equations inside Ω satisfied by the exact solution are replaced by a GIBC of order k on Γ that can be written in the form

Eδ,k n + i(µω) δ,k n (Hδ,k n) = 0, (1.34) e × D × e ×
22 CHAPTER 1. ASYMPTOTIC MODELS FOR SCATTERING PROBLEMS where n denotes the normal to Γ directed to the interior of Ω of and where δ,k is a local D boundary operator acting on tangential vector fields on Γ. The order of this operator increases with the desired order of accuracy. As in the case of coatings, we shall restrict ourselves to GIBCs leading to surface operators of order 2. In his case we have the following expressions ≤ for k = 0, 1, 2, 3, (see [AS18])

δ,0 = 0, D  δ,1 = √2 i √2 δ,  D 2 − 2  δ,2 δ,1 2 (1.35)  =  + i δ ( ),  D D C − H δ,3 δ,2 √2 √2 3 2 2 2 = + i δ + (εµ)ω + Γ divΓ + curl~ Γ curlΓ .  4 4  D D − C − H ∇      As one can notice from a comparison with the case of coatings, the limit problem is the same (perfect conductor boundary conditions), however, the expression and nature of δ,k D greatly differ for higher order approximations. For instance, in the case of thin coating, a surface wave operator appears starting from the first order. But it is only at the third order that one can see an effect of the wave propagation along the scatterer boundary in the case of high absorption. One can also notice that the second order operator appearing in the expression of δ,3 D has no fix sign. This causes difficulties in the mathematical study of the well-posedness of the approximate problem (1.33)-(1.34) for k = 3. For that reason, by using techniques similar to those exposed at the end of section 1.2.4, we modified the expression of δ,3 by introducing suitable perturbations of order O(δ4). Our D goal is to obtain a different expression more suited to the study of the problem via variational techniques. More precisely, we used the operator

δ,3 2 √2 3 2 2 2 m = iδ ( ) δ (1 + i) + ε ω D C − H − 4 C − H r √2 2 1 2 i δ((2 δ Γ divΓ ) + δ curl~ Γ curlΓ )
(1.36) − 4 − ∇ − √2 2 1 2 + δ((2 + δ curl~ Γ curlΓ ) δ Γ divΓ ), 4 − − ∇ as a substitute to δ,3 in the error analysis. One can check that the two expressions differ by D terms of order O(δ4). In order to state the main results we introduce the Hilbert spaces X := E H (curl, R3 Ω) ; E n = 0 on Γ , 0 { ∈ loc \ × } X := E H (curl, R3 Ω) ; (E n) L2(Γ) , 1 { ∈ loc \ × |Γ ∈ t } X := E H (curl, R3 Ω) ; (E n) H1(Γ) , 2 { ∈ loc \ × |Γ ∈ t } and Y := X H (curl, R3 Ω), Y = Y := X X , Y := X X . 0 0 × loc \ 1 2 1 × 1 3 1 × 2 Following a two step procedure as in the case of linear coatings (i.e. making use of the truncated asymptotic expansions) and assuming that the boundary Γ is sufficiently regular, one obtains the following theorem [AS18].

Theorem 1.1 For k = 0, 1, 2 or 3, there exists δk such that for δ δk, the boundary value δ,3 δ,3 ≤ δ,k δ,k problem ((1.33), (1.34)), where is replaced with m , has a unique solution (Ee , He ) D D ∈ Yk. Moreover, if B is a bounded domain containing Ω, then there exists a constant Ck, independent of δ, such that δ δ,k k+1 Ee Ee H(curl,B Ω) Ck δ . (1.37) k − k \ ≤ 1.3. SCATTERING FROM IMPERFECTLY CONDUCTING BODIES 23

Let us finally mention that there are other possible GIBCs of order k, that differs from the ones given above only by O(δk+1) terms. Even if they formally lead to the same accuracy, their adaptability to a given numerical solver may be different (see [T25] for time dependent problems and [39, 114] for integral method in frequency regimes). It turns out that the expres- sions given in (1.35) are suitable to a finite element implementation [A9]. These expressions are those used in the following numerical experiments (the second order surface operator in δ,3 seems to not induce troubles for the tested axi-symmetric configurations). D 1.3.3 Numerical validation We present in this section some numerical results discussing the accuracy of the GIBCs and their validity when the boundary contains corner type singularities. A more detailed discussion can be found in [A9]. The examples are restricted to axi-symmetric obstacles which authorizes the use of the 3-D axisymmetric Hcurl spectral isoparametric Q` finite elements [51]. The used numerical method is based on a variational formulation and a Fourier expansion in the azimuthal variable. To simulate the radiation condition at infinity we used the integral representation of the solution on an artificial boundary at a finite distance from the scatterer (see [94] for more details). In the following examples, the approximate solution E δ,k (k refers to the order of the GIBC) is computed with a polynomial order ` = 7 of the finite element approximation. The mesh step size is fine enough so that the discretization error is negligible. We compute a reference solution Eδ after having meshed the interior of the obstacle. The mesh is constructed in such a way that the skin-effect is correctly taken into account. The accuracy of the GIBC is tested by representing in log-log scale the error functional δ δ,k δ Error := curlE curlE L2(D Ω¯ )/ curlE L2(D Ω¯ ) k − k \ k k \ in terms of δ, where D is the bounded domain of computations. Validation of the GIBCs for smooth boundaries. We choose a (non convex) peanut geometry as an example of C 1 domain: see Figure 1.2-left (in each picture, we shall superpose to the obstacle geometry, the distribution of the current density modulus for a given scattering experiment). We compute the diffraction of an incident plane wave propagating along the axis of revolution of the obstacle (In this case only two Fourier mode are needed to compute the solutions). This wave propagates from top to bottom, according to Figure 1.2-left. In this simulation a moderate frequency is used: ω = 0.2 π, which corresponds to a wavelength two times smaller than the height of the scatterer. As shown by the error curves in Figure 1.2-right, one gets a convergence rate that roughly corresponds to the theoretical O(δk+1) for a GIBC of order k. More precisely one gets O(δ3.8) for k = 3, O(δ2.9) for k = 2 and O(δ2.5) for k = 1. In the second example shown in Figure 1.3, we increased the frequency: ω = π. In this case one observes that the GIBC of order 1 and order 2 give similar precision (till 3 digits): an improvement between k = 1 and k = 2 would be observed only with smaller values of δ. However, the condition of order 3 improves significantly the precision. This is (more or less) expected since when the wavelength is very small as compared with the smallest radius of curvature of the surface, the corrections due to geometrical terms are not significant: the wave “locally sees” the obstacle as a flat boundary, for which the curvature is 0 (therefore the conditions of order 1 and order 2 are the same in this case). The improvement observed in the case of the third order GIBC is due to the surface wave operator. 24 CHAPTER 1. ASYMPTOTIC MODELS FOR SCATTERING PROBLEMS

0 Order 1 Order 2 −0.5 Order 3

−1

−1.5 (error) 10 −2 log

−2.5

−3

−3.5 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 log (ω δ) 10

Figure 1.2: Left: Re(J) on the boundary Γ. Right: error curves in terms of δ (log-log scale). | |

−0.5 Order 1 Order 2 −1 Order 3

−1.5

(error) −2 10 log −2.5

−3

−3.5 −1.5 −1 −0.5 log (ω δ) 10

Figure 1.3: Left: Re(J) on the boundary Γ. Right: error curves in terms of δ (log-log scale). | |

The treatment of singular boundaries. When the boundary is not smooth, for instance when it contains corners, the error estimate in (1.1) is no longer valid (one has even to be cautious in the definition of the GIBC of order 3 for non smooth boundaries [112]). To overcome this difficulty we have proposed to combine the use of local mesh refinement around the singularity and the use of GIBC in the region with regular boundary. The coupling between the two is done by introducing a fictitious regular boundary inside the absorbing medium at the singularity regions that links the regular parts of Γ so that their union gives a C1 surface Γ.˜ The GIBC is then applied on Γ˜ and the small region around the singularity is treated as a part of the computational volume domain. In Figure 1.4-right, we compare the results obtained from a naive treatment of the singu- larity, consisting in applying (at the discrete level) the GIBC on Γ as in the case of regular surfaces, with the results obtained after applying the numerical treatment explained above. The error curves associated to this treatment are labeled by adding the word “modified”. The considered object is a “sharp ring” generated by rotating a diedron-disk around the z axis (see Figure 1.4-left). The incident plane wave propagates as in section 5.2. The pulsation 1.4. PERSPECTIVES AND RELATED DEVELOPMENTS 25

−0.5

−1

−1.5

−2 (error) 10 −2.5 log

−3 Order 1 Order 3 −3.5 Order 1 modified Order 3 modified −4 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 log (δ) 10

Figure 1.4: Left: Re(J) on the boundary Γ. Right: error curves in terms of δ (log-log scale). | |

is chosen equal to ω = 1, so that the wavelength is two times smaller than the height of the scatterer. As shown by the error curves in Figure 1.4-right, the GIBC of order 1 or 3 fails to give the same convergence rate as in the case of smooth boundary. Let us notice, that both conditions give a convergence rate O(δ0.9), and no improvement is observed when the ∼ GIBC of order 3 is used instead of order 1, for δ small enough. However, after applying the numerical treatment of the singularities, one significantly improves the accuracy: we get a convergence rate O(δ2.7) for the GIBC of order 1, and a convergence rate O(δ4.4) for ∼ ∼ GIBC of order 3.

Remark 1.3 The small region between Γ˜ and the exterior domain enforces the approximate solution to locally have the same singular behavior as the exact one. We noticed that its presence is essential in retrieving the order of accuracy. This observation is somehow in concordance with the results in [115] on the asymptotic behavior of the solutions near the corners.

1.4 Perspectives and related developments

There are several perspectives of this part and shall I start by the ones we are currently investigating. The first one is in connection with the numerical treatment of GIBCs for time depen- dent problems. The study presented in section 1.2 only concerns second order discretization in time and the use of lower order N´ed´elec’s finite elements. We would like to extend this work to discretization schemes with higher order of accuracy based on discontinuous Galerkin methods. The investigation of higher order time discretizations is also an interesting issue that we would like to study (using for instance simplectic schemes). This work is pursued in collaboration with J. Hesthaven. We are investigating at the same time the situations where the layer is an interface between two dielectric media. First results showed that this case cannot be a straightforward generalization of the coatings one. For instance, obtaining uni- form stability in time of first order conditions is not obvious for curved geometries. Designing efficient numerical implementations is also an non straightforward task! 26 CHAPTER 1. ASYMPTOTIC MODELS FOR SCATTERING PROBLEMS

The second perspective is in connection with the study of the problem in section 1.3 for time dependent problems. As already mentioned, since the small parameter depends itself on the frequency, the error analysis would require a special care in the general case where the solution spectrum contains all frequencies. Also the expressions of the GIBCs are expected to involve non local operators in time (like fractional derivatives) which raises new interesting challenges in the study of the continuous problem and also for time discretization. The first investigations on this issue were initiated by H.M. Nguyen and we are in the process of recruiting a student to continue working on this problem in our group. The third perspective is not in direct connection with this work, but shares the same philosophy... It concerns the derivation of wire models when the cross section diameter is relatively small. The major difference with the problems above is that the asymptotic models one would like to derive have not a conventional functional setting. We would like to couple a 3-D wave equation for the exterior domain with a 1-D model for the wire. Classically, the wires are modeled in harmonic regimes by using integral equation techniques (Pocklington, or Hallen equations). For time domain, the heuristic numerical model of Holland is the reference. However only few mathematical justifications are available in the literature and the case of finite wire is an open problem. The Phd thesis of X. Claeys, that I am supervising in collaboration with P. Joly aims at using matched asymptotic procedure to rigorously derive wire models. Their numerical implementation in the framework of higher order Discontinuous Galerkin method should then be studied. This part will be carried out in collaboration with F. Collino. We end this section by indicating three future possible perspectives. The first one con- cerns the theoretical justification of the corner singularities and their numerical treatment, which would be in the continuation of the numerical discussion given at the end of this chap- ter. The second perspective, suggested (and would be supported) by Onera, is the study of asymptotic models associated with thin dielectric curved layers with periodic grating of perfect conductors. Several works in the littrature have treated similar cases The third perpective is somehow the orientation that I would like to give to my research in the mid-future. It is related to the derivation and validation of approximate models for very rough (random) surfaces or interfaces. This would constitute an opening toward the study of non-deterministic models in wave propagation problems... Chapter 2

Asymptotic Models and Ferromagnetic Materials

Contents

2.1 Introduction ...... 27 2.2 Study of the forward problem ...... 31 2.3 Approximate models for thin ferromagnetic coatings ...... 34 2.3.1 Expression and stability of GIBCs ...... 34 2.3.2 Theoretical validation ...... 36 2.3.3 Numerical scheme and validation ...... 36 2.4 Homogenized model for periodic ferromagnetic media ...... 36 2.4.1 Description of the mathematical model ...... 37 2.4.2 Description of the homogenized model ...... 38 2.4.3 Convergence results ...... 40 2.4.4 Applications and validation ...... 41 2.5 Perspectives ...... 41

This chapter is dedicated to an overview of some work on ferromagnetic materials mostly done during my Phd thesis at INRIA in collaboration with my Phd supervisor P. Joly. It has strong connections with the previous chapter since the main objective is to derive and study asymptotic models for special configurations. I choose to present it as a separate unity due to the specificity of the underlying non-linear physical model. The self-contained summary below is based on materials from [A13, A10, A14, A15, A16].

2.1 Introduction

Depending on their relative permeability µr, materials can be divided into three basic types: diamagnetic, paramagnetic and ferromagnetic. For the first class, µr < 1 and for most of these diamagnetic materials (like silver, water, copper, ...) µ 1 and they can be assumed r ' as nonmagnetic. In the second class (air, aluminum, palladium, ...) the relative permeability is slightly larger than 1 and unlike diamagnetic materials, these materials posses a spontaneous

27 28 CHAPTER 2. ASYMPTOTIC MODELS AND FERROMAGNETIC MATERIALS

magnetization. However, the smallness of this magnetization is such that it is neglected in most applications. Ferromagnetic materials, whose name derive from iron (ferrum) as the most common of the ferromagnetic materials, have a relative permeability much larger than 1 (can be thou- sands higher). Some typical ferromagnetic materials are iron, cobalt, nickel. They have the property of retaining a large magnetization after an external magnetic field has been applied, and some of them are hard to demagnetize (alnico, ferrite, ...). An additional important property of ferromagnetic materials is the dependence of magnetization on the level of ap- plied external magnetic field. If the magnitude of this field is sufficiently high, then the magnetization modulus reaches a peak called a saturation modulus. In our study we were concerned only by ferromagnetic materials that are at the saturation stage. In most common applications of ferromagnetic materials, it is rather the state of magnetization or its orien- tation in a given ferromagnetic region that is exploited (recording media, magnetic tapes, magnetic strips on credit cards, magnetic resonance imaging, ...). The applications we are interested in are different and rather exploit the absorbing property of these materials with respect to electromagnetic waves. They are for instance largely used in stealth technology and also used as protective coatings for antennas. Our investigations of these applications were subsequent to the Phd thesis of O. Vacus within the project ONDES at INRIA [100], in collaboration with Dassault Aviation, devoted to the study of efficient numerical discretiza- tion of electromagnetic wave propagation inside ferromagnetic materials. My Phd thesis, that can be seen as a continuation of this work, was mainly oriented towards asymptotic models adapted to numerical simulation of scattering problems from thin ferromagnetic coatings. From the mathematical point of view, one of the main characteristics of ferromagnetic materials lies in the fact that their constitutive law, namely the relationship between the magnetic field H and the magnetization M, is non linear and non local with respect to time. The most common model for this interaction is the so-called Landau-Lifshitz-Gilbert (LLG) law [92, 65]: α ∂ M = γ M HT + M ∂ M (2.1) t × M × t | | where γ is a universal negative constant called the gyromagnetic factor and α is a positive constant called the damping factor. The second term in this equation is the one responsible of absorbing properties of the ferromagnetic medium (see [66] for a revisited physical derivation of this model). The function HT is the total electromagnetic field inside the medium and can be expressed as

HT = H + Heff (M) where Heff (M) is the effective magnetic field that derives from some potential electromagnetic energy proper to the ferromagnetic material. This energy contains different contributions. The one retained in the case of our applications are anisotropic energy (due to crystalline structure of the ferromagnetic material) and the one due to the presence of an external static

field Hs. For mono-axial anisotropy along a direction p, the expression of Heff (M) can be written in the following form

Heff (M) = H M + K (p M) p, s − a × × where Ka is a positive constant. A major simplification has been made here by neglecting the so-called exchange energy that would have added a term proportional to the ∆M. In practice, it turns out that the constant in front of this term is in general very small so that the exchange 2.1. INTRODUCTION 29

energy plays a significant role only at a micro-scale (for instance, this contribution is important to explain the natural repartition of the magnetization into regions of uniform magnetization separated by nano transition walls: the Weiss walls). One has therefore to keep in mind that when we shall consider situations where the ferromagnetic medium occupies a small region, the dimensions of this region are assumed to be at a scale largely bigger than the micro one. From the mathematical point of view, this simplification has also a major importance since it changes the nature of (2.1): it becomes a non linear ODE. This implies for instance that no control on the spatial derivatives of the magnetization can be guaranteed in general. This is why the mathematical tools involved in the study of the problem with and without exchange terms are very different. This issue was discussed in [T25]. For the study of the problem with exchange energy, we refer to [116, 43, 50] and references therein. Let us already mathematically express the two physical properties hereabove mentioned (that are independent of the presence of exchange field or not). The first one concerns the magnetization: one easily see, taking the dot product of (2.1) by M that

M(x, t) = M (x) | | | 0 | where M0 denotes the initial magnetization. This implies in particular that the model is only valid for so-called saturated ferromagnetic materials, for which the magnetization mod- ulus is constant. One also notices from this property how the magnetization field cannot be propagative, contrarily to the electromagnetic field. The second property concerns the ab- sorption with respect to electromagnetic waves. The magnetic density flux induction inside a ferromagnetic material is B = µ0(H + M) where µ0 is the magnetic permeability in vacuum. Hence, Maxwell’s equations can be written as

ε ∂ E + σE curlH = 0, t − (2.2) µ ∂ (H + M) + curlE = 0,  0 t where ε is the electric permittivity and σ is the conductivity. Now if we set 1 (E, H) = ε E 2 + µ H 2 dx and, E 2 | | 0 | | Z   µ0 2 2 eff (M) = H M + K (M p)p M dx, E 2 | s − | a | · − | Z then it can be proved that a solution to (2.2) and (2.1) satisfies

d 2 µ0 α 2 (E, H) + eff (M) = σ E dx ∂ M dx. (2.3) dt {E E } − | | − γ M | t | Z | | Z | | The second term at the right hand side explicitly shows why α is denoted by damping factor. The analysis of the forward problem was among the investigated questions in our study of ferromagnetic materials [A10]. The main contribution was to complement previous works, in the 1-D case [80] and the 3-D one [79] by giving an existence and uniqueness result in a general 2-D setting. The used technique were inspired from the pioneering work [79], with large simplifications of the proof arguments due to the 2-D setting of the problem. We already mentioned that the study of electromagnetic scattering from ferromagnetic coatings was the original motivation behind our study of GIBCs (Chapter 1). These type of coatings are commonly used in stealth technology, and therefore designing efficient numerical 30 CHAPTER 2. ASYMPTOTIC MODELS AND FERROMAGNETIC MATERIALS

methods that correctly handle these type coatings is very important. Let us also point out that the usual way to take into account the ferromagnetic absorbing effect is based on a linearization of the LLG law. This is done assuming that the magnitude of the incident field is small compared to the magnitude of the static field. Under this assumption, a ferromagnetic material behaves like a linear material with anisotropic permeability depending on the wave frequency (see for instance [101, 117] and also [100] and [T25]). In our study we considered the non linear regime described by the LLG law. The formal methodology in deriving GIBCs associated with thin coatings is similar to the case of linear coatings. However, obtaining the effective expressions and estabilishing stability results are much more complicated [A13, A14, A15]. A quick summary is provided in Section 2.3.

Getting convergence results for the GIBCs is complicated, especially for the second order one, where more than L2 regularity is needed for the magnetization, which is not naturally yielded by the equations for the exact solution (the analysis of the forward problem only requires M L2 L )). The derivation of error estimates is still an open problem for ∈ ∩ ∞ dimensions 2. A first step is done by analyzing the one dimensional case, where there is no ≥ surface derivatives. We refer to [A14] for technical details.

Numerical issues related to GIBCs in the framework of ferromagnetic coatings are dis- cussed in [A13] for the one dimensional case and in my Phd thesis [T25] for higher dimensions. As in the case of linear coating (see Chapter 1) stability is the central discussed point for the derived numerical discretizations in time. A second major requirement, due to the LLG law, is maintaining at the discrete level the pointwise conservation of the magnetization modulus. Discretizations that satisfy this property were extensively studied in [100, 96]. We were largely inspired from these methods in our proposed numerical schemes. A second main feature of this type of dicretizations, for the exact problem (without GIBC), is that they preserve the local character of the LLG law in space. We showed that this property is preserved for the first order condition. However, since this character is already not true for the continuous setting of the GIBC of order 2, it is also lost for discrete case. But the conservation of the magnetization modulus is guaranteed.

We also investigated another asymptotic configuration of ferromagnetic materials, namely a periodic distribution with a periodicity relatively small as compared to the wavelength of incident electromagnetic waves. Once again, the motivation of the asymptotic analysis is to provide a simplified model that enables us to compute an approximation of the scattered electromagnetic wave without using a mesh step constrained by the small parameter (which is here the periodicity size δ). In this case, the limit problem as δ 0, called the homogenized → model, is generally sufficiently rich to already give satisfactory approximations.

This work was motivated by the observation that in common situations, ferromagnetic coatings are made of periodically stratified homogeneous layers (separated by a linear dielec- tric, that can be considered as a special case of a ferromagnetic material). This corresponds to what is called a laminar configuration, and is used, for instance, to combine the absorbing properties of different ferromagnetic layers. Other type of coatings are made of small fer- romagnetic inclusions within dielectric substrates in order to reduce the conductivity of the resulting material, and therefore reducing reflections at the interface vacuum-coating. In this first study, we considered the case where the periodicity is also very small compared to the width of the coating [A16]. The results related to this work are summarized in Section 2.4. 2.2. STUDY OF THE FORWARD PROBLEM 31

2.2 Study of the forward problem

The analysis of the forward problem was among the investigated questions in our study of ferromagnetic materials. The main contribution published in [A10] was to complement previous works, in the 1-D case [80] and the 3-D one [79] by giving an existence and uniqueness result in a general 2-D setting. The used technique were inspired from the pioneering work [79], with large simplifications of the proof arguments due to the 2-D setting of the problem. In order to give a flavor of the encountered difficulties and specificities of each dimension, the uniqueness proof of strong solutions is very representative. We shall summarize here the main ingredients (the technical details and the proofs are present in [T25]). Let us first introduce the mathematical setting of the physical model. We consider the Maxwell equation in Rn R+ × ε ∂ E + σE curlH = 0, µ (∂ H + ∂ M) + curlE = 0, t − 0 t t (2.4) (E, H) t=0 = (E0, H0),  | where ε > 0, µ > 0 and σ 0 (the permittivity ε can be a positive definite function, 0 ≥ however the permeability µ0 is assumed to be constant, which corresponds to most physical situations). The initial electromagnetic field (E0, H0) models possible source terms. This system is coupled with the non linear LLG equation for the magnetization that we shall write in an abstract form as Rn R+ ∂tM = (M, HT; x) with HT = H mΦ(M; x), in L − ∇ Rn × (2.5) M t=0 = M0, in  | where M is the initial magnetization, the functional represents the ferromagnetic law and 0 L Φ is the effective energy density (without exchange terms). The analysis of system (2.4)-(2.5) can be carried out under the following hypothesis on and Φ. L Hypothesis 2.1 For almost all x Rn, the function (m, h) (m, h; x) is continuous ∈ 7−→ L from R3 R3 into R3 and satisfies × (m, h; x) m = 0 for all (m, h) R3 R3. (2.6) L · ∈ × In addition, for all R > 0 there exists a constant C(R) such that,

(i) (m, h; x) C(R) h , |L | ≤ | | (2.7) (ii) (m, h; x) (m , h ; x) C(R) ( m m h + h h )  |L − L 0 0 | ≤ | − 0| | | | − 0| for all m, m satisfying m R and m R and all h, h in R3. 0 | | ≤ | 0| ≤ 0 Hypothesis 2.2 For almost all x Rn, the function m Φ(m; x) is of class C 1 from R3 ∈ 7−→ into R+. In addition, for all R > 0 there exists a constant C(R) such that,

Φ(m; x) C(R), |∇m | ≤ (2.8) Φ(m; x) Φ(m ; x) C(R) m m  |∇m − ∇m 0 | ≤ | − 0| for all m, m satisfying m R and m R. 0 | | ≤ | 0| ≤ The following hypothesis on ensures the absorbing properties of the medium. L 32 CHAPTER 2. ASYMPTOTIC MODELS AND FERROMAGNETIC MATERIALS

Hypothesis 2.3 For almost all x Rn and for all m R3 and h R3 ∈ ∈ ∈ (m, h; x) h 0. (2.9) L · ≥ This hypothesis is not needed in the existence and uniqueness proof of strong solutions. Let us first notice that the LLG law corresponds to

(m, h; x) = θ(x) (h m + β(x) m (h m)) (2.10) L × × × γ α(x) where θ(x) = | | 2 and β(x) = if M0(x) = 0 and β(x) = 0 if not. This law satisfies 1+α(x) M0(x) | | 6 Hypothesis 2.1 and 2.3. The effectiv| e field| corresponds to 1 Φ(m; x) = H (x) m 2 + K (x) m (m p(x)) p(x) 2 (2.11) 2 | s − | a | − · |   with p = 1, and Hypothesis 2.2 is satisfied as soon as H and K are bounded. | | s a The main result is the following where we have set

1 2 2 (E, H) = ε E + µ0 H dx and eff (M) = µ0 Φ(M; x) dx. E 2 Rn | | | | E Ω Z   Z n 2 n 3 Theorem 2.1 Assume that (E0, H0) H(curl; R ) , M0 L∞(R ) with compact support, n ∈ ∈ such that div(M0 + H0) = 0 in R . Then, under Hypothesis 2.1 and 2.2, the Cauchy problem 1, 2 n 3 3 (2.4)-(2.5) has an unique strong solution (E, M, H) W ∞(0, ; L (R ) ) . satisfying for ∈ loc ∞ all t 0, ≥ M(x, t) = M (x) p.p. x Rn. (2.12) | | | 0 | ∈ If we further assume that Hypothesis 2.3 is satisfied, then

(E(t), H(t)) + eff (M(t)) (E , H ) + eff (M ). (2.13) E E ≤ E 0 0 E 0 One difficulty inherent to the study of this problem comes from the fact that the variational study of equation (2.4) requires H(t) to be in H(curl; Rn) while the differential equation (2.5) is naturally well posed for H(t) L (Rn)3, and these two spaces are disjoint. Moreover, the ∈ ∞ latter cannot be suited to variational techniques. This difficulty is solved differently for each dimension. As said before, an illustrative example can be given by the uniqueness proof that we shall give the outline hereafter. Assume for instance that there exist two strong solutions (E1, M1, H1) and (E2, M2, H2). Then if we denote by (δE, δM, δH) the difference between the two solutions and set

2 2 2 (t) = δE(t) 2 + δH(t) 2 + δM(t) 2 , E k kL k kL k kL   one can prove from (2.4) and (2.5) that for all T > 0, there exists a constant C(T ) such that for all t T , ≤ d 2 (t) C(T ) δM(t) H (t) 2 δH(t) 2 + δH(t) 2 . (2.14) dtE ≤ k · 1 kL k kL k kL   1 (a) In 1-D, thanks to the embedding of H (R) into L∞(R), one can easily prove that H L ((0, T ) R)3. Therefore d (t) C (T ) (t) for some positive constant C (T ), which 1 ∈ ∞ × dt E ≤ 0 E 0 simply implies, (t) exp(C (T )t) (0) = 0. E ≤ 0 E 2.2. STUDY OF THE FORWARD PROBLEM 33

(b) In higher dimensions the used embedding does not hold anymore... but is in fact almost true in 2-D. More precisely, let ϕ C (Rn), satisfying ϕ(ξ) = 0 for ξ 2, ϕ(ξ) = ∈ ∞ | | ≥ 1 for ξ 1 and 0 ϕ 1, and denote by Sλ the Fourier multiplier of symbol ϕ(ξ/λ). Then | | ≤ ≤ ≤ one can check, using Fourier transform, that Sλ is an approximation of unity on L2 and that 2 n n it is continuous from L (R ) into L∞(R ) with a continuity modulus that explodes like λ. However, for u H1(R2), ∈

λ 1 2 λ u H1 S u u H1 ln(1 + 4λ ) and u S u k k , (2.15) L∞ ≤ 2 k k − L2 ≤ √1 + λ2

which means in particular that Sλu u Sλu goes to zero as λ . This is specific L∞ − L2 → ∞ to dimension 2. Using property (2.15) one can prove the existence of a function H λ L (R2)3 1 ∈ ∞ and a positive constant C indep enden t of λ and t T such that ≤

λ λ C H1 C ln(λ) and H1(t) H1 L∞ ≤ − L2 ≤ λ

for sufficiently large λ and t T . Therefore, inequalit y (2.14) yields the existence of two ≤ constants C1 > 0 and C2 > 0 such that d C (t) C ln(λ) (t) + 2 (t) dtE ≤ 1 E λ E p for all integer N > 1 and t T , which implies, ≤

[C1T/N] 1 (t) N C λ − . E ≤ 2 p One concludes that (t) = 0 by choosing N such that (C T )/N 1 < 0, then letting λ . E 1 − −→ ∞ (c) In 3-D, the first (type of) estimate in (2.15) is not true anymore, however one has a slightly different version of it for u solutions of

3 + ∂ttu ∆ u = f, in R R − R3 × u t=0 = u0 and ∂tu t=0 = u1, in  | | Using Strichartz like estimate [67], it is shown in [79] that there exists a constant c independent of u0, u1, f, such that for all λ > 1, and T > 0,

Sλu c ln(1 + λT ) L2(0,T ;L∞(R3)) ≤ u0 p1 + u1 2 + f 1 2 R3 . k kH k kL k kL (0,T ;L ( ))   With additional technical complications, one can use this estimate to conclude the proof of uniqueness following similar procedure as in 2-D. On the existence of solutions. In all dimensions, the existence of solutions is proved through an approximation procedure. In 2-D we used the following regularized system

λ 1 λ λ ∂tE = ε curlS H λ 1 λ λ λ λ λ λ  ∂tH = curlS E (M , S H mΦ(M ) − µ0 − L − ∇   ∂ M λ = (M λ, SλHλ Φ(M λ) t L − ∇m   34 CHAPTER 2. ASYMPTOTIC MODELS AND FERROMAGNETIC MATERIALS

coupled with the initial conditions

λ λ Rn 2 λ Rn 3 (E , H ) t=0 = (E0, H0) H(curl, ) , M t=0 = M0 L∞( ) | ∈ | ∈ This formulation has the advantage to be adapted to the inhomogeneous case. It constitutes a standard Cauchy problem for the unknowns

λ λ λ 2 n 3 2 n 3 2 n 3 E (t), H (t), M (t) L (R ) L (R ) (L∞ L )(R ) =: X ∈ × × ∩ For which the Cauchy-Lipschitz theorem can be applied to prove the existence of a solution (Eλ, Hλ, M λ) C1(0, T ; X). This solution is shown to be weak-? convergent (as λ ) in 1, 2 ∈n 9 → ∞ W ∞(0, T ; L (R )) towards (E, H, M) satisfying Maxwell-LLG system (2.4)-(2.5).

2.3 Approximate models for thin ferromagnetic coatings

We already mentioned that the study of electromagnetic scattering from ferromagnetic coat- ings was the original motivation behind our study of GIBCs (Chapter 1). These type of coatings are commonly used in stealth technology, and therefore designing efficient numerical methods that correctly handle these type of coatings is very important. Let us also point out that the usual way to take into account the ferromagnetic absorbing effect is based on a linearization of the LLG law. This is done assuming that the magnitude of the incident field is small compared to the static field. Under this assumption, a ferromagnetic material behaves like a linear material with anisotropic permeability depending on the wave frequency (see for instance [101, 117] and also [100] and [T25]). In our study we considered the non linear regime described by the LLG law. The formal methodology in deriving GIBCs associated with thin coatings is similar to the case of linear coatings. However, obtaining the effective expressions and establishing stability results are much more involved. We shall summarize hereafter the main results concerning these type of coatings that has been published in [A13, A14, A15]. The notation of Section 2.2 is maintained here.

2.3.1 Expression and stability of GIBCs The derivation of the GIBCs was done under the assumption that the magnetic characteristics of the thin layer are independent of its width: i.e., there exist three functions , Φ, and M0, δ L defined on Γ ]0, 1[ and independent of δ, such that: for x Ω ; x = xΓ + δν n, × ∈ δ 3 (m, h; x) = (m, h; xΓ, ν), m, h R , Lδ L 3∈ Φ (m; x) = Φ(m, h; xΓ, ν), m R , δ ∈ M0 (x) = M0(xΓ, ν). In the remaining of this section and in order to simplify the notation, we will not indicate

explicitly the dependence of and Φ on the variables (xΓ, ν). The expression of the GIBCs L can be written in a general form like

Eδ,k n + δ,k n (Hδ,k n) + δ,k n (Hδ,k n) = 0, (2.16) e × B × e × A × e × where n denotes the normal to Γ directed to the
interior of Ω, the operator
δ,k is the same B as in the case of linear coatings and the operator δ,k is a non linear one that takes into A 2.3. APPROXIMATE MODELS FOR THIN FERROMAGNETIC COATINGS 35

account the contribution of the ferromagnetic law. Let us notice that as in the case of linear coatings the limit problem consists in ignoring the thin coating and therefore δ,0 = 0. Let us A remind the reader that the definition of order adopted here is different from the one in [A15]: a condition of order k here is denoted by a condition of order k + 1 in [A15] (see Chapter 1). First order GIBC: This condition has a relatively simple expression: for all ϕ L2(Γ), ∈ t 1 δ,1 + ϕ(xΓ, t) = δµ M (xΓ, ν, t) dν ; (xΓ, t) Γ R , (2.17) A T ∈ × Z0 where M := n (M n) and the function M(xΓ, ν, t) satisfies T × × + ∂ M = (M, HT) on Γ (0, 1) R t L × × (2.18) M t=0 = M0 on Γ (0, 1)  | × with

HT = ϕ + [(M M) n] n Φ(M). (2.19) 0 − · − ∇m Second order GIBC: Contrary to the case of linear coating, the second order condition does not only contain corrective geometrical terms. It also contains first and second order surface derivatives. More precisely,

1 δ,2 (1 δν ) + ϕ = δµ − C M δ (1 ν) Γ (M n) dν on Γ R (2.20) A (1 + 2δνh) T − − ∇ · × Z0  

where the function M(xΓ, ν, t) satisfies (2.18) with

δ ν δ(1 ν) HT = (1 δ ν )ϕ + Γ ψ(xΓ, ξ, t) dξ + ψ + − divΓ ϕ n mΦ(M) − C (1 + 2δνh) 0 ∇ (1 + 2δνh) − ∇ Z   (2.21) where we have set

δ 1 ψ(xΓ, ν, t) = [M M] n divΓ ((M M) )(xΓ, ξ, t) dξ. (2.22) 0 − · − (1 + 2δνh) 0 − T Zν Let us mention that this is not the natural expression that one gets from the formal procedure (for more details, we refer to [T25]). The “natural condition” has in fact been modified so that uniform stability results can be derived for the coupled Maxwell-GIBC system. For instance, one can show that the solution to (1.9)-(2.16) satisfies an energy estimate of the form d (Eδ,k(t), Hδ,k(t)) + δ δ,k(t) + δ δ,k(t) 0, dt{E e e EΓ Eeff } ≤ where δ,k has the same expression as in the linear case (see (1.16)), and where, EΓ δ,0(t) = 0, Eeff 1 δ,1 µ 2 (t) = (M M(t)) n 2 + µ Φ(M(t)) dxΓ dν, Eeff 2 k 0 − · kL Z0 ZΓ 1 δ,2 1 2 (t) = µ ψ(t) + Φ(M(t)) (1 + 2δνh) dxΓ dν. Eeff 2 | | Z0 ZΓ   36 CHAPTER 2. ASYMPTOTIC MODELS AND FERROMAGNETIC MATERIALS

The study of the 1-D case showed that the third order condition may be numerically expensive, since the proposed procedure to remove the instability inherent to the natural condition, requires the inversion of wave operator living in the scaled domain. This would be penalizing in dimensions higher than 1. We refer to [A13] for a complete study of this condition in the one dimensional case.

2.3.2 Theoretical validation Getting convergence results for the GIBCs is complicated, especially for the second order one, where more than L2 regularity is needed for the magnetization, which is not naturally yielded by the equations for the exact solution (the analysis of the forward problem only requires M L2 L )). The derivation of error estimates is still an open problem for dimensions ∈ ∩ ∞ 2. A first step is done by analyzing the one dimensional case, where there is no surface ≥ 1 derivatives. The continuous embedding of H into L∞ for this space dimension also simplifies the analysis of this case. The main difficulty consists in obtaining uniform bound, with respect to δ, of the non linear term of the GIBC. This requires for instance obtaining estimates on the derivative in time of the solutions. We refer to [A14] for technical details.

2.3.3 Numerical scheme and validation Numerical issues related to GIBCs in the framework of ferromagnetic coatings are discussed in [A13] for the one dimensional case and in my Phd thesis [T25] for higher dimensions. As in the case of linear coating (see Chapter 1) stability is the central discussed point for the derived numerical discretizations in time. A second major requirement, due to LLG law, is maintaining at the discrete level the pointwise conservation of the magnetization modulus. Discretizations that satisfy this property were extensively studied in [100, 96]. We were largely inspired from these methods in our proposed numerical schemes. A second main feature of this type of discretizations, for the exact problem (without GIBC), is that they preserve the local character of the LLG law in space. We showed that this property is preserved for the first order condition. However, since this character is already not true for the continuous setting of the second order GIBC, it is also lost for the discrete case. But the conservation of the magnetization modulus is guaranteed. We reported in [A13] extensive validating numerical experiments for the 1-D model, dis- cussing for instance different regimes (linear, non-linear) of the ferromagnetic material behav- ior. The numerical discretization of the 3-D case is discussed within the framework of finite element discretization in [T25] and numerical validation of the conditions expressed above was done in 2-D case (non curved boundary). We showed for instance the importance of the second order condition in taking into account the anisotropy caused by the ferromagnetic layer.

2.4 Homogenized model for periodic ferromagnetic media

We investigate here a different asymptotic configuration of ferromagnetic materials, namely a periodic distribution with a periodicity relatively small as compared to the wavelength of incident electromagnetic waves. Once again, the motivation of subsequent asymptotic analysis is to provide a simplified model that enables us to compute an approximation of the scattered electromagnetic wave without using a mesh step constrained by the small parameter (which 2.4. HOMOGENIZED MODEL FOR PERIODIC FERROMAGNETIC MEDIA 37

is here the periodicity size δ). In the present case, the limit problem as δ 0, called the → homogenized model, is sufficiently rich to already give satisfactory approximations. This work was motivated by the observation that, in common situations, ferromagnetic coatings are made of periodically stratified homogeneous layers (separated by a linear dielec- tric, that can be considered as a special case of a ferromagnetic material). This corresponds to what is called a laminar configuration, and is used, for instance, to combine the absorbing properties of different ferromagnetic layers. Other type of coatings are made of small fer- romagnetic inclusions within dielectric substrates in order to reduce the conductivity of the resulting material, and therefore reducing reflections at at the interface vacuum-coating. In this first study, we considered the case where the periodicity is also very small compared to the width of the coating. Our goal is then to derive the equivalent nonlinear problem (namely the homogenized problem) as the periodicity of the ferromagnetic medium approaches 0. Of course the main difficulty is to establish the convergence for the non linear terms. The lack of control for the space derivatives of the magnetization is the hard part of the problem (as compared with models with the exchange-field [70]). However, there are benefits from the local character of the LLG law: under suitable assumptions on the initial magnetization, the spatial dependence of the magnetization at the micro-scale is “almost unchanged”. This property is exploited to establish strong two-scale convergence (this notion has been introduced in [29]), which is sufficient to identify the limit of the non-linear terms. The obtained nonlinear homogenized law is a nonstandard one. It is written using a two-scale framework, coupling microscopic and macroscopic variations. Moreover, some ad- ditional nonlocal corrector terms appears at the microscopic scale, breaking with the original local aspect. The convergence is only proved under the laminar assumption. We shall sum- marize here the main results of [A16] and take advantage of this summary to include some numerical validating examples. Let us notice that some interesting application to directional susceptibilities can be found in [T25].

2.4.1 Description of the mathematical model We denote in the sequel by n 3 the space dimension and by Y = [0, 1]m the dimensionless ≤ periodicity cube of Rm with m n. For x Rn, we denote by x the first m components ≤ ∈ m of the vector x in the canonical basis of RN . The ferromagnetic material is supposed to be confined into a bounded regular domain Ω of Rn and has periodic characteristics with period δ Y , where δ is the small parameter of our problem. For T > 0 we set = Rn (0, T ) and QT × Ω = Ω (0, T ). T × The problem unknowns are, Eδ(x, t), Hδ(x, t), M δ(x, t) R3, and respectively denote, ∈ the electric, magnetic and magnetization fields. They satisfy Maxwell’s system

δ δ δ δ δ δ ε ∂tE curlH = 0, µ0 ∂t(H + M ) + curlE = 0 on T δ δ− Rn Q (2.23) (E , H ) t=0 = (E0, H0) on  | coupled to the ferromagnetic nonlinear law

δ δ δ δ ∂tM = (M , H mΦ(M ; x, xm/δ); x, xm/δ) on T δ L δ R−n∇ Q (2.24) M t=0 = M0 on .  |

In these equations, the magnetic permeability µ0 > 0 is constant while the electric permittivity εδ > 0 may vary periodically in space. More precisely, we assume the existence of a constant 38 CHAPTER 2. ASYMPTOTIC MODELS AND FERROMAGNETIC MATERIALS

m ε0 > 0 and a scalar positive function εr(ξ) defined on R , Y -periodic, such that

δ ε (x) = ε0(1 + εr(xm/δ) 1Ω(x)), (2.25)

where 1Ω indicates the characteristic function of Ω. The field (E0, H0) stands for the incident δ electromagnetic wave and is independent of δ. The initial magnetization M0 is such that: δ M0 (x) = M0(x, xm/δ), (2.26) where the vectorial function M (x, ξ) defined on Rn Rm, is supported in Ω Rm, and is 0 × × Y -periodic with respect to ξ. The function (m, h; x, ξ) : R3 R3 Rn Rm R3, is L × × × → Y -periodic with respect to ξ and satisfies Hypothesis 2.1 and 2.3 uniformly with respect to ξ. The functional Φ is defined on R3 Rn Rm with values in R+ such that ξ Φ( ; , ξ) is × × 7−→ · · Y -periodic and satisfies Hypothesis 2.2 (where x is replaced by (x, ξ))

Remark 2.1 We have neglected conduction effects in the ferromagnetic material by taking σ = 0 in Maxwell’s equations. It is well known that when σ = 0, the homogenized problem 6 associated to Maxwell’s equations is of an integro-differential type in time (see for example [109]). Its study does not enter into the framework of the present paper. However, we do not see any fundamental additional difficulty in generalizing our results to this case.

2.4.2 Description of the homogenized model Before introducing the homogenized model, let us make precise some further notation. For any function u(x, ξ), (x, ξ) Rn Rm, Y -periodic with respect to ξ, we shall denote ∈ × u : x u(x, ξ) dξ, (2.27) h i 7−→ ZY k the average of u over Y . Concerning functional spaces, we denote by C#(Y ), for k integer or k = , the space of Ck(Rm) functions that are Y -periodic, and for 1 p , we set p ∞ p Rm ≤ ≤ ∞ L#(Y ) the subspace of Lloc( ) constituted by Y -periodic functions. For k integer, the space k k H#(Y ) will denote the closure of C#∞(Y ) functions in H (Y ). We shall prove that the solution (Eδ, Hδ, M δ) to (2.23)-(2.24) approaches, in some sense that will be made precise later, a limit (Eh, Hh, M h), as the periodicity length δ approaches 0. The set of equations satisfied by the limit field, that have to be identified, constitutes what we call the homogenized problem associated to (2.23)-(2.24). In our case, the general form of the limit problem cannot be written only in terms of the (macroscopic) unknowns Eh, Hh and M h. We need to introduce an intermediate unknown (microscopic magnetization) M(x, t, ξ) from which we have

M h(x, t) = M(x, t, ) . h · i Then we can write the homogenized equations for (E h, Hh, M)

h h h h h ε ∂tE curlH = 0, µ0∂t(H + M ) + curlE = 0 on T h h− n h i Q (2.28) (E , H ) t=0 = (E0, H0) on R  | h # ∂tM = (M, H Pk M mΦ(M; x, ξ); x, ξ) on T Y L −Rn − ∇ Q × (2.29) M t=0 = M0 on Y  | × 2.4. HOMOGENIZED MODEL FOR PERIODIC FERROMAGNETIC MEDIA 39

where, as usual, εh is no longer a scalar function but a matrix of R3 R3, called the matrix × of homogenized coefficients (or H-limit) associated to the sequence (εδ), and where, which # is really new here, the operator Pk is pseudo-differential of order 0 and is only acting on the ξ variable (x being a parameter). We recall that εh is determined by solving m cellular problems (see [109]): if we consider for j = 1, , m the solution ϕ H1 (Y ) to · · · j ∈ # div ((1 + ε ) (ϕ ξ )) = 0 on Y ξ r ∇ξ j − j (2.30) ϕ = 0  h ji

then, with the usual meaning of the Kronecker symbol δi,j,

h ε (x) = ε0 (1 + εr)(δi,j ∂ξ ϕj) 1Ω(x) + ε0(1 1Ω(x)) if 1 i, j m, i,j h − i i − ≤ ≤ (2.31) εh (x) = ε (1 + ε 1 (x)) δ if not. ( i,j 0 h ri Ω i,j

# On the other hand, the operator Pk is defined by

(P #M)(x, ) = div ( M(x, )) (2.32) k · ∇ξ ξ N# · where is the periodic Newtonian potential defined by: N# Definition 2.1 For χ L2(Y ), χ H1 (Y ) is the unique solution to ∈ N# ∈ # ∆ χ = χ χ in Y, N# − h i (2.33) χ = 0.  hN# i # 2 m Remark 2.2 Note that Pk is nothing but the orthogonal projection of L (Y ) into the curlξ # 2 Rn 2 3 free functions. It can be easily seen (using Fourier’s transform) that Pk : L ( ; L#(Y )) 2 Rn 2 3 → L ( ; L#(Y )) is a continuous operator.

The laminar case. The laminar case corresponds to m = 1 and is important here for two reasons:

It is for this case that convergence proofs will be carried out.  The homogenized problem takes a simpler form. First, the tensor εh becomes diagonal  and is given by:

εh(x) = ε 1 (x) εh ~e ~e + εh (~e ~e + ~e ~e ) + (1 1 (x)) , (2.34) 0 Ω x ⊗ x y ⊗ y z ⊗ z − Ω   1   where we have set, εh = (1 + ε ) 1 − and εh = (1 + ε ) . Second, when m = 1, r − h r i

P #M = ∂ (∂ ( M) )~e = (M M )~e = Π (M M ) , k ξ ξ N# x x x − h xi x x − h i where Π v = (v ~e )~e for v R3. We therefore see how the orientation of the laminar x · x x ∈ medium appears explicitly in the homogenized model. System (2.29) becomes, h ∂tM = (M, H + Πx ( M M) mΦ(M; x, ξ); x, ξ) on T Y, L n h i − − ∇ Q × (2.35) M t=0 = M0 on R Y.  | × 40 CHAPTER 2. ASYMPTOTIC MODELS AND FERROMAGNETIC MATERIALS

Remark 2.3 Contrarily to the case m = 1, we do not have explicit expression of the operator P # for m 2. However, when M L2(Rn; L2 (Y ))3 is a function of separated variables: k ≥ ∈ # L M(x, ξ) = M `(x) χ (ξ) a.e. (x, ξ) Rn Rm, (2.36) ` ∈ × X`=1 for a given integer L, where for each `, M ` L2(Rn)3 and χ L2 (Y ), we remark that ∈ ` ∈ # L L P #M = div M ` χ = M ` χ . (2.37) k ∇ξ ξ N# ` ∇ξ · ∇ξN# ` X`=1 X`=1   # Hence, Pk M is also a function of separated variables and one needs to solve only L cellular # problems of type (2.33) in order to compute Pk M everywhere with respect to x. We believe that this property can be exploited to generalize the convergence proof to non laminar cases.

2.4.3 Convergence results Classically [109, 42], we proceed in two steps:

(i) We first derive the homogenized problem through a formal two-scale asymptotic expan- sion with respect to δ.

(ii) We then prove the convergence using, for example, energy estimates.

Frequently, we can carry out step (i) with few restrictions, but are able to prove convergence (step (ii)) only in particular cases (which will be the subject of next sections). Our technique is based on the two-scale convergence theory introduced in [98] and devel- oped by Allaire in the foundational paper [29] (we gave in [A16, T25] an overview of some fundamental concepts related to this method with small adaptations useful to our case). This method has the advantage of combining the two steps and directly gives the weak limit of the linear equations. The difficult part is to carry out convergence for the non linear terms. Only strong two-scale convergence is suited for this part and requires a particular care on the notion of admissible functions. The technicality of this latter point has confined us to treat only the laminar case. Also, only dimensions n 2 has been investigated, since no available results on strong solutions ≤ to non homogeneous 3-D problem are available in the literature. The main results concerns existence and uniqueness of strong solutions to the homogenized problem (Theorems 5.5 and 5.6 of [A16]) and the following convergence results under the hypothesis that : m = 1, ε L (Y ), M L (Rn; L (Y )3) and the functions M , and Φ are ξ-piecewise constant r ∈ #∞ 0 ∈ ∞ #∞ 0 L functions (i.e. are of the form (2.36) where the χ`’s are characteristic functions of intervals).

Theorem 2.2 Assume that E and H are in H(curl, Rn) with div M δ + H = 0, δ > 0. 0 0 0 0 ∀ Then, for every T > 0, the sequence (Eδ, Hδ, M δ) converges in W 1, (0, T ; L2(Rn)3)3 weak ? ∞
− to (Eh, Hh, M h) where M h = M and where (Eh, Hh, M) is the unique strong solution of h i (2.28)-(2.29) on . In addition, QT lim M δ (M)δ = 0 (2.38) δ 0 − L∞(0,T ;L2(Rn)) →

2.5. PERSPECTIVES 41

lim Hδ Hh + Π ( M (M)δ) = 0, (2.39) x 2 δ 0 − h i − L ( T) →   Q where (M)δ(x, t) := M(x, x1 , t). δ

2.4.4 Applications and validation The homogenized problem can serve to characterize the absorption properties of the medium through the study of so-called directional permeabilities. For a detailed discussion of this issue we refer to Chapter 10 of [T25]. It can also be used to numerically compute the ap- proximate solution to the exact solution of the scattering problem. We report hereafter some validating numerical results that have not been published. Consider the experience described in Figure 2.1 where the plane of propagation if (x, z), the laminar part is made of parallel vacuum and ferromagnetic layers perpendicular to the x direction.

Homogeneous ferromagnetic layers

λ

δ S

Vacuum

Figure 2.1: Description of the numerical simulation

The incident wave is simulated by adding a point source term at S in the first equation of (2.28). We used a Ricker’s signal of the form d2 A (exp( π2(ct/λ 1)2)) 0 dt2 − − where λ is a parameter that plays the role of the central frequency of the signal and c is the light speed (= 1 here). The initial magnetization is uniform, oriented along the y axis. Figure 2.2 exhibits snapshots of the solutions at a fixed time and show how the convergence occurs as δ goes to zero (compare the strong convergence for the y-component of the electric field with the weak convergence for the x-component of the magnetization).

2.5 Perspectives

Several open questions have been quoted in the above summary, and some of them seems to be reasonably accessible. The first one is extending the mathematical justification of the homogenized model to non laminar cases. We believe that Remark 2.3 gives the first key point for that perspective. The second ingredient would be the use of suitable regularization (in the same spirit of those used in studying existence of solution and exposed in Section 2.2) for the non local operator appearing at the micro-scale. It would also be interesting to study the asymptotic models associated with thin ferro- magnetic periodic coatings where the periodicity and the thickness have the same order of 42 CHAPTER 2. ASYMPTOTIC MODELS AND FERROMAGNETIC MATERIALS

magnitude. The formal study of first order conditions in non curved situations has been started in Chapter 11 of my Phd thesis. As mentioned in the introduction, this configuration is more representative of practical situations. The issues related to the numerical validation of the non linear GIBCs in curved situations have not been explored, so far. Also the study of the underlying boundary value problem is an open problem. A more difficult part would be proving error estimates for orders greater than 1 and dimensions greater than 2.

δ = λ/4

δ = λ/6

δ = λ/12

Homogenized solution

Figure 2.2: Snapshots of the solutions at a fixed time. Left: y-component of the electric field. Right: x-component of the magnetic field. Only the upper half of the propagation domain in Figure 2.1 is represented. Chapter 3

Imaging techniques and inverse problems

Contents

3.1 Introduction ...... 43 3.2 The Linear Sampling Method ...... 44 3.2.1 A brief (historical) review ...... 44 3.2.2 A model problem: Maxwell’s equations for anisotropic media . . . . 45 3.2.3 The linear sampling method ...... 48 3.2.4 Numerical validation ...... 50 3.2.5 Interior Transmission Problems ...... 52 3.3 Inversion methods using the Reciprocity Gap principle . . . . . 54 3.3.1 Inverse planar screen problem ...... 55 3.3.2 A sampling method based on RG ...... 56 3.4 Iterative Methods ...... 59 3.4.1 A result in connection with the Newton Methods ...... 59 3.4.2 Electrostatic Imaging via Conformal Mapping ...... 59 3.5 Perspectives ...... 61

3.1 Introduction

This chapter is dedicated to the presentation of my research in connection with inverse prob- lems. Most of this research is concerned with the so-called inverse scattering problem at a fixed frequency: determine the shape of an obstacle or an inclusion from the knowledge of scattered electromagnetic (or acoustic) waves. The orientation of my work in this area was more focused on the development of methods rather than solving specific applications. It was therefore natural to divide this summary according to the investigated methods. A large part of my involvement in this field was dedicated to the development of the so- called Linear Sampling Method [A7, A24, A8, A5] (Section 3.2) and the closely related study of so-called interior transmission problem [A2, A24, P4, A11] (Section 3.2.5). My collaborators for this part are F. Cakoni, D. Colton, F. Collino, M’B. Fares and P. Monk.

43 44 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS

More recently I was interested in the applications of the so-called Reciprocity Gap principle in solving the above problem [A6, A1, AS3] (Section 3.3). This is done in collaborations with A. Ben-Abda, D. Colton, F. Cakoni, F. Delbary and M’B. Fares. In collaboration with R. Kress, we established the differentiability of the far-field map for the electromagnetic scattering problem with impedance boundary condition [A20]. This work has connections with the use of Newton Methods in solving the inverse scattering problem (Section 3.4.1). We are currently investigating the use of conformal mapping in the context of electrostatic 2-D imaging problem [A21] (Section 3.4.2).

3.2 The Linear Sampling Method

3.2.1 A brief (historical) review The Linear Sampling Method (LSM) was first introduced by Colton-Kirsch in 1996 [54] as a simple method to solve the following basic inverse scattering problem: determine the shape of an obstacle 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 its principle, with classical methods in handling this inverse problem. These classical methods are based in general on an iterative procedure, and also require some a priori knowledges on the physical properties of the inclusion. The LSM does not require this a priori knowledge and is not iterative. This is why it is in general a relatively quick algorithm. The counterpart is that an important amount of data is needed in general to achieve reasonable accuracy. It is also worth mentioning that even if the principle of the method is innovative, LSM was inspired by previously developed methods, like the dual space method introduced by Colton- Monk [56] and also the closely connected iterative method by Kirsch-Kress [85], where the use of an exact solution to the forward problem is avoided. It has also close connections with the uniqueness proofs based on the use of singular solutions [86, 77, 89]. The methods later developed by R. Potthast [104, 105], that are based on these uniqueness proofs, have also strong similarities with LSM. Quite surprisingly, the mathematical analysis of LSM is rather pessimistic on its perfor- mances. First, this method relies on the spectral data of an integral linear operator with (very) smooth kernel, which may be severely unstable. Second, the “formal justification” is mathematically incorrect. However, it numerically exhibits unexpected good performances! One challenge faced at the very beginning of this method was therefore to fix or justify this incompatibility. A. Kirsch has answered this challenge with the so-called Factorization method [82]: the philosophy behind his procedure is to modify the compact operator used by the LSM, so that the mathematical characterization of the inclusion with the spectral content of the new one holds. He shows for instance that in the case of acoustical obstacle scattering the “square root” of the LSM operator exactly characterizes the shape of the obstacles. This method is known as the first version of the factorization method. More general versions have been introduced since that time [68, 83]. At the university of Delaware, and by the impulse of D. Colton, the philosophy was different since the goal was to keep the LSM operator but try to mathematically analyze this incompatibility. The proved results essentially show, by using a density argument, that the LSM is arbitrarily close to an exact characterization of the obstacle shape. We shall clarify this point in section 3.2.3. Two key works for the scalar problem are [57] and [45] (see [53] and the recent book by Cakoni-Colton [47] for a review). 3.2. THE LINEAR SAMPLING METHOD 45

The extension to the 3-D Maxwell case has been initiated in [A7] for perfect scatterers and continued in a series of papers [A24, A11, A5], [48] for different kind of targets. A review of these developments is provided in [A8]. Two schools have then emerged developing these parallel visions, and each one has its advantages and its drawbacks. The factorization method is very attractive from the mathe- matical point of view and also more difficult. For instance, the justification of this method in the case of obstacle scattering for Maxwell’s equations is still an open question, despite the efforts of many authors. Recent advances has been made by A. Kirsch for the inverse medium problem [84]. This method is also numerically less flexible than the LSM, especially in han- dling limited aperture data. For the LSM, its attractive part is its simplicity and practicality: independently from what you are trying to image, and the amount of data you have (limited or full aperture, near-field or far-field) the integral equation you solve is straightforwardly constructed. However, the mathematical justification that has been developed is less sharp than for the Factorization. The proved results are usually partial ones. In fact the difficult part in complementing existing results is still an open problem, and the work by Arens [36] has partially answered it in the case of the Helmholtz equation. His result roughly says that, whenever the first version of Factorization method holds, one can make the link between the numerical regularized solution and the indicator function predicted by the theory (see section 3.2.3). I started working on LSM when I arrived to the University of Delaware for a one year post-doc in 2001. F. Cakoni was already there since one year forming with P. Monk and D. Colton the Delaware’s LSM team. That was the beginning of a fruitful and very enjoyable collaboration that conjugated both, agreeable environment and exciting challenges. Our col- laboration started with the LSM applied to inverse electromagnetic scattering problem [A7] and the inverse electromagnetic medium problem [A24]. We also investigated the interior transmission problem for anisotropic acoustic medium [A2, P4]. More recently our collabora- tion focused on the use of the reciprocity gap functional in the context of a sampling method [A6, AS3]. Arriving back to Inria, it was also with a F. Collino and M’B Fares from Cerfacs that my work on LSM has been continued, mainly on the numerical valorization of the potentialities of this method [A5].

3.2.2 A model problem: Maxwell’s equations for anisotropic media

My involvement in the development of the LSM was mainly concerned with the Maxwell equations, and this is why I choose to focus in this quick (self-contained) summary of the 3-D electromagnetic inverse problem. The model problem I shall consider is the inverse scattering from anisotropic medium. This choice gives me the occasion to present some work related to the so-called interior transmission problem. The material contained here is mainly based on [A11, A5].

The direct problem

Let D R3 be a bounded domain with connected complement such that ∂D is Lipschitz ⊂ continuous with unit outward normal ν. We denote by ( , ) the L2(D) scalar product and · · D 46 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS consider the Hilbert spaces

H(curl , D) := u L2(D)3; curl u L2(D)3 , { ∈ ∈ } H (curl , D) := u H(curl , D); u ν = 0 on ∂D , 0 { ∈ × }

equipped with the scalar product (u, v)curl = (u, v)D +(curl u, curl v)D and the correspond- ing norm , and define k·kcurl (D) := u H(curl , D); curl u H(curl , D) , U { ∈ ∈ } (D) := u H (curl , D); curl u H (curl , D) , U0 { ∈ 0 ∈ 0 } equipped with the scalar product (u, v) = (u, v)curl + (curl u, curl v)curl and the corre- sponding norm . We also denote by U (R3) the Fr´echet space of functions u L2 (R3) k·k Uloc ∈ loc such that u (KU) for all compact sets K R3. ∈ U ⊂ Let N be a 3 3 symmetric matrix whose entries are bounded complex-valued functions × in R3 and such that N = 1 in R3 D where 1 denotes the 3 3 identity matrix. Let E i be an \ × entire solution to Maxwell’s equations:

curl curl Ei k2Ei = 0 in R3, − where k > 0 is the wave number. Then the direct problem for the scattering of a time harmonic electromagnetic waves by an anisotropic medium D with an index of refraction N, can be formulated as the problem of finding an electric field E (R3) such that ∈ Uloc (i) curl curl E k2 N E = 0 in R3, − s i  (ii) E = E + E , (3.1)   (iii) lim (curl Es x ik x Es) = 0 uniformly w.r. to xˆ = x/ x . r →∞ × − | | | |  Let ε, σ and µ = µ0 be respectively the electric permittivity, the electric conductivity and the magnetic permeability of the medium. If ω denotes the frequency of the incident wave, 2 then k = ε0 µ0 ω and ε(x) σ(x) N(x) = + i . ε0 ε0ω Problem (3.1) has been studied variationally in [87, 95] from which one can deduce for instance that under the hypothesis,

(Nξ, ξ) γ ξ 2, ξ C3 and a.e. in R3, (3.2) < ≥ | | ∀ ∈ for some γ > 0, and (N) > 0 in D, the problem (3.1) is well posed. Finally let us recall = that using the Stratton-Chu formula it follows from (3.1-(iii)) that E s has the asymptotic behavior [55], eik x 1 Es(x) = | | E (xˆ) + O (3.3) 4π x ∞ x | |  | | uniformly with respect to xˆ, where the vector field E is the electric far field pattern and ∞ satisfies E (xˆ) xˆ = 0. Since we are always assuming that k is fixed, we have suppressed the ∞ · dependence of E on k. ∞ A slight extension of this result is needed to construct a theoretical justification of LSM. For instance, the above results define a scattered wave (and its far field pattern) associated 3.2. THE LINEAR SAMPLING METHOD 47

with entire solutions of Maxwell’s equations. We shall extend this definition to a more general class of incident waves, dictated by the space of solutions to the interior transmission problem (see Section 3.2.5). This is the first step towards the justification of the LSM. This class is defined by the space:

H (D) = E L2(D)3, such that, curl curl E k2 E = 0 in D . inc { 0 ∈ 0 − 0 } A scattered wave Es (R3) associated to E H (D), is defined as the unique solution ∈ Uloc 0 ∈ inc (if it exists) to

s 2 s 2 3 (i) curl curl E k N E = k (N 1)E0 in R , − − (3.4)  (ii) lim (curl Es x ik x Es) = 0 uniformly in xˆ = x/ x . r  →∞ × − | | | | We then introduce the operator

S : (S) H (D) L2 (R3) (3.5) D ⊂ inc −→ loc defined by S(E ) = Es where Es (R3) and satisfies (3.4). (S) denotes the domain of 0 ∈ Uloc D S, i.e. the set of functions E H (D) for which there exists a solution Es (R3) to 0 ∈ inc ∈ Uloc (3.4). From the above considerations, (S) contains the restrictions to D of entire solutions D to Maxwell’s equations. By proving the denseness of (S) into H (D) and using a continuity D inc property of S under the condition

(Nξ, ξ) γ ξ 2, ξ C3 and a.e. in D, (3.6) = ≥ | | ∀ ∈ for some γ > 0, one can prove the following theorem (see [A11]).

Theorem 3.1 Assume that N satisfies properties (3.2) and (3.6). Then the operator S can be extended to a bounded linear operator S : H (D) L2 (R3). Moreover SE satisfies inc −→ loc 0 (3.4-i) in the distributional sense and satisfies the following integral representation formula: e e (SE )(x) = k2 GT (x, y) (N 1) (SE + E ) dy, for all x R3 D, (3.7) 0 − 0 0 ∈ \ ZD where G(x, ye) is the Green tensor associated withe Maxwell’s equations defined by 1 G(x, y) = Φ(x, y) I + div (Φ(x, y) I) k2 ∇x x where I is the 3 3 identity matrix and Φ(x, y) be the fundamental solution associated with × the Helmholtz equation and given by: Φ(x, y) = eik x y /(4π x y ). | − | | − | s The operator S˜ enables us to define an extended far field operator: since E = SE0 satisfies (3.7), then it also satisfies the asymptotic expansion (3.3). Moreover e 2 ikxˆ y E (xˆ) =: ( E0)(xˆ) = k e− · (xˆ (N 1)(SE0 + E0)) xˆ) dy. (3.8) ∞ F × − × ZD Using the results related to the interior transmission problem,e we prove the following prop- erties of , where S2 denotes the unit sphere and L2(S2) denotes the space of tangential L2 F t fields on S2. 48 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS

Theorem 3.2 Assume that D is a C 3 domain and that N satisfies the requirements of The- orem 3.1. Then the far field operator : H (D) L2(S2) defined by (3.8) is compact, F inc → t injective and has dense range.

The inverse problem. We consider incident plane waves defined by

i ik x dˆ E (x; dˆ, p) = p e− · (3.9) where dˆis the direction of propagation, dˆ = 1, and p R3 is the polarization vector satisfying | | ∈ p dˆ = 0. We denote by E (xˆ; dˆ, p) the far field associated with Ei(x; dˆ, p). Remark that Ei · ∞ depends linearly on p, so does E . ∞ Let Γ and Σ be two non empty parts of S2 with positive surface measures. The inverse problem we want to solve is to determine the domain D from the knowledge of E (xˆ; dˆ, p) for all xˆ Γ and all (dˆ, p) Σ R3 such that dˆ p = 0. The uniqueness of the ∞ ∈ ∈ × · reconstruction of D from such measurements is proved by Cakoni-Colton in [46].

3.2.3 The linear sampling method In order to solve the inverse problem stated above the principle of the LSM is to characterize the domain D using the range of the far field operator with kernel E (xˆ; dˆ, p). More precisely ∞ we introduce the operator F : L2(Σ) L2(Γ) defined by t −→ t

(F g)(xˆ) = E (xˆ; dˆ, g(dˆ)) ds(dˆ), (3.10) ∞ ZΣ where the subscript t refers to spaces of tangential fields. This operator is linear, and also compact, due to the analyticity of its kernel [55]. Remark also that evaluating this operator does not require any a priori knowledge on the direct problem (for instance on N). Using the linearity of the forward problem, one easily check that F g is nothing but the far field pattern associated with the incident field,

ik dˆ x ( g)(x) := g(dˆ) e− · ds(dˆ), x D. (3.11) H ∈ ZΣ called Herglotz wave function with kernel g. We immediately deduce that

F = . (3.12) F ◦ H From this decomposition and Theorem 3.2, one can show that F is injective and also has dense range (see [A11], Theorem 4.3). Now let q R3 0 and z R3 be two given parameters. We introduce the electric ∈ \ { } ∈ dipole at z with polarization q, defined by 1 E (x; z, q) := curl curl Φ(x, z) q . e ik x x{ } E ( ; z, q) is a radiating solution to Maxwell’s equations in R3 z with a far field pattern e · \ { } given by ik xˆ z Ee, (xˆ; z, q) := ik(xˆ q) xeˆ − · . ∞ × × 3.2. THE LINEAR SAMPLING METHOD 49

The key idea of LSM comes from the observation that Ee, ( ; z, q) cannot be in the range of ∞ · the operator F if z is outside D. Verifying this statement is an easy exercise on the use of the Rellich lemma and the unique continuation principle: for instance, if F g = Ee, ( ; z, q) they ∞ · imply that S g and E ( ; z, q) coincides in R3 D, which is impossible in neighborhood of z. H e · \ Since the arguments of this objection do not apply for z D, it sounds plausible to try ∈ to characterize D by the set of points z for which Ee, ( ; z, q) is in the range of the operator ∞ · F . This is the LSM. The arguments above at least show that this set of points z is contained in D... But unfortunately, this set is in general empty! In the case of a sphere, and for the scattering from perfect conductor, we proved in [A5] that this set is reduced to the center of the sphere. However, we shall now explain how this characterization is nearly true. This is based on relation (3.12) and the fact that this characterization is exact if we use instead of F . F Theorem 3.3 Assume that D is of class C 3 and that N satisfies properties (3.2) and (3.6) and that (N 1) 1 is a bounded and definite positive matrix field on D, and let q R3 0 − − ∈ \ { } be a fixed vector. Then, the far field xˆ Ee, (xˆ; z, q) is in the range of if and only if 7→ ∞ F z D. ∈ The proof of this theorem is closely connected to the study of the ITP, developed in next section. Theorem 3.3 tells us that the equation

E0( ; z, q) Hinc(D) such that E0( ; z, q) = Ee, ( ; z, q) (3.13) · ∈ F · ∞ · admits a solution if and only if z D. The domain D can then be determined then as being ∈1 the support of z E ( ; z, q) −2 . Of course this is not a useful characterization of the → k 0 · kL (D) domain D, since and E ( ; z, q) 2 require a knowledge of D. F k 0 · kL (D) However, Herglotz wave functions are dense in Hinc(D) (see Theorem 4.3 of [A11]: the density is proven with Σ = S2, but using the Jacobi-Anger expansion, it is easy to deduce that Herglotz wave functions with kernel supported in Σ are also dense in Hinc(D)). Therefore, it seems reasonable to rather seek E ( ; z, q) as an Herglotz wave, so that one determines 0 · g( ; z, q) L2(S2) such that · ∈ t

g( ; z, q) Ee, ( ; z, q) F g( ; z, q) Ee, ( ; z, q), (3.14) FH · ' ∞ · ⇔ · ' ∞ · 1 and z g( ; z, q) −2 would be a good indicator for D: the values should be much larger → kH · kL (D) for z outside D than for z inside D with a transition region concentrated at ∂D. We therefore (almost) retrieve the LSM equation... But the criteria still depends on D! The last conjecture stipulates that the behavior of 1 z g( ; z, q) −2 (3.15) → k · kL (Σ) 1 would be similar to z g( ; z, q) L−2 (D). → kH · k 2 This conjecture is verified at the discrete level where the space Lt (Σ) is replaced with a subspace XN with finite dimension. Since is injective, g g L2(D) is also a norm on 2 H → kH k XN and is therefore equivalent to L norm on XN . This somehow justifies the use of (3.15) as a numerical criteria to visualize D... However, it does not tell that this is the best one (see discussion below). The missing part in this justification is to explain how to compute the approximation of E ( ; z, q), or more precisely to explicit the symbol in (3.14). One can think of using a 0 · ' 50 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS regularization of the operator F , which is in any case needed for numerical computations, giving the compactness of this operator. For instance, if one uses the Tikhonov technique, one seeks g L2(S2) such that α ∈ t

αgα( ; z, q) + F ∗F gα( ; z, q) = F ∗Ee, ( ; z, q) (3.16) · · ∞ · where α > 0 is a regularization parameter. Since F has dense range, one has F g ( ; z, q) α · → Ee, ( ; z, q) as α 0. Therefore, if z / D, gα( ; z, q) L2 (D) as α 0 (otherwise ∞ · → ∈ kH · k → ∞ → it would contradict Theorem 3.3). However, the arguments above do not guarantee that for z D, g ( ; z, q) 2 < as α 0 (which is equivalent to prove that g ( ; z, q) ∈ kH α · kL (D) ∞ → H α · → E ( ; z, q), using the fact that is injective). The latter point has been proved in [36] in the 0 · F case of scattering problem for the Helmholtz equation. The key point of the proof relies on the use of the Factorization method, which stipulates that Theorem 3.3 holds if is replaced F with (F F )1/4. The proof of this, is based on the observation that F is normal and that ∗ F itself can be factored out into where satisfies specific coerciveness properties [82, 83]. H∗D D Proving this convergence in the case of Maxwell’s equations is still an open problem (see some interesting ideas for that perspective in [97]). Let us finally quote that even if one proves that this convergence result holds, giving the fact that the operator (involved in the exact characterization), is compact and more F importantly with dense range, this theoretical justification is not a guarantee that numerically or/and with noisy data, the LSM would work. It only gives reasons to be optimistic!

3.2.4 Numerical validation The numerical validation of the LSM in the context of Maxwell’s equations was also an important part in my involvement in the development of the method. In fact this was the start point of my work on this method with [A7], where examples for perfect conductors are given. Numerical experiments with limited aperture and for obstacles with impedance boundary conditions are later done in [A5]. The case of inhomogeneities is treated in [A24]. But unfortunately, we do not have yet some examples that corresponds to the anisotropic case... The numerical reconstructions were certainly among the first results on 3-D electromag- netic inversions. However, they were all performed using synthetic data (with added random noise). In the case the LSM the noise in the data affects the operator F . Given an upper bound  of F F , the regularized parameter α in (3.16) is determined by enforcing the k − exactk Morozov-like principle

F gα( ; z, q) Ee, ( ; z, q) =  gα( ; z, q) . k · − ∞ · k k · k One key observation for the numerical results was that better reconstructions are obtained when one uses instead of (3.15) the following criteria

1 1 1 G(z) = + + (3.17) g ( , z, q ) g ( , z, q ) g ( , z, q ) k α · 1 k k α · 2 k k α · 3 k where q1 = (1, 0, 0), q2 = (0, 1, 0) and q3 = (0, 0, 1), which corresponds to the combination of results for three independent polarizations. A striking numerical demonstration of the importance of this combination in the case of complex geometries can be found in [A7]. 3.2. THE LINEAR SAMPLING METHOD 51

To visualize the 3-D reconstruction one needs first to be able to find the transition region between small and large values of G(z). Choose a value C in that region then visualize the iso-surface (z) = C, (3.18) G which would be the approximation of the sought target. Of course the choice of C is problem- atic, since the results may have been considerably smoothed out by the regularization process. One can imagine an automatic evaluation of the “best value” based on the analysis of the gradient of this criteria, but this issue has not been investigated yet 1. We showed in [A7], using analytical solutions, that the ambiguity in finding the transition region and the value of C reduces when the frequency increases. This confirms the fact that the “resolution”of the LSM increases with the used frequency. We cannot end this paragraph without giving some numerical examples. 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 3.1-3.4 and correspond to a perfect scatterer with full aperture data: Σ = Γ = S2. The synthetic data was computed using integral equation method and was corrupted with a 1% random noise. They were obtained in collaboration with F. Collino and M’B. Fares from Cerfacs. A digression. Just to quote a little anecdote related to these results: My co-authors wanted to be convinced that this (confusingly simple) method really works and therefore decided to send me the data without indicating the exact shape of the scatterer. I obtained as a reconstruction the results in Figure 3.2... but had not enough imagination to recognize a teapot! Nevertheless and fortunately, after having seen the reconstruction, they decided to not boycott the LSM. Using higher wave numbers would have avoided this risk, as attested by Figures 3.3 and 3.4.

Figure 3.2: Reconstructed geometry with Figure 3.1: Exact geometry of the perfect a wave number k = 28 and using 252 uni- scatterer. formly distributed incident directions.

1We recently discovered that the software “tecplot” offers the possibility of finding an “optimal” surface (probably by using also some topological criteria), and the result always correspond to satisfactory choices of C. An interesting inverse problem would be to understand the algorithm used by this software! 52 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS

Figure 3.3: Reconstructed geometry with Figure 3.4: Reconstructed geometry with a wave number k = 56 and using 252 uni- a wave number k = 84 and using 492 uni- formly distributed incident directions. formly distributed incident directions.

3.2.5 Interior Transmission Problems The so-called interior transmission problem (ITP) plays an important role in the study of the previous inverse problem. It gives for example a powerful tool for uniqueness issues in the case of anisotropic media [69, 46]. It also plays a fundamental role in the justification of the LSM, and more precisely in proving Theorem 3.3 in the case of inverse electromagnetic anisotropic medium problems. This problem occurs when one tries to solve equation (3.13): using the Rellich lemma and the unique continuation principle, one can show that if E0 solves (3.13) then there exits a pair (E, E ) L2(D)3 satisfying in the distributional sense, 0 ∈ (i) curl curl E k2 N E = 0 in D, − (3.19) (ii) curl curl E k2 E = 0 in D,  0 − 0 and such that E E (D) and E E satisfies − 0 ∈ U − 0 (E E ) ν = E ( , z, q) ν on ∂D, − 0 × e · × (3.20) (curl E curl E ) ν = curl E ( , z, q) ν on ∂D.  − 0 × e · × This set of equations is what we referred to as the (ITP). One can find in [55] a review of basic results related to this problem. A possible approach to solve (ITP) is the use of a volume integral formulation. For instance, this type of methods has been successfully applied to the case of an inhomogeneous medium if one assumes that the index of refraction is smooth inside the medium and has no jump across the boundary [A24]. However, as presented in [58], it gives only partial answers in the case of anisotropic media (where the index of refraction N is a matrix-valued function). One needs to further assume that the imaginary part of N is definite positive but sufficiently small (without knowing how small it should be). The originality of the study performed in [A11] is to treat the problem using a variational framework where minimal regularity for N is required and where the entries of this matrix can have a jump across the boundary of the medium. In collaboration with F. Cakoni and D. Colton we already proposed in [A2, P4] a varia- tional technique to treat (ITP) in the acoustic case. This treatment is based first on a suitable 3.2. THE LINEAR SAMPLING METHOD 53

change of unknown (inspired by mixed finite element method). The modified system is then proved to be a compact perturbation of a coercive problem. However, it turns out that in the Maxwell case, due to the specificity of H(curl , D) spaces, the modified system is no longer a compact perturbation of the coercive form. Therefore this approach does not apply any more. We mention though that a coercive version of (ITP) can be used to obtain a uniqueness result for the reconstruction of the anisotropy support [46]. We proposed an alternative method to solve (3.19-3.20) based on a reformulation of the problem as a fourth order boundary value problem. The idea of this transformation is in fact not new and has been used before in [107] to study the acoustic case of non absorbing inho- mogeneous medium. However the technique to solve the obtained boundary value problem cannot be transposed to Maxwell’s equations. In order to rewrite (3.19-3.20) as a fourth order boundary value problem one needs to assume that N 1 is invertible a.e. in D and that (N 1) 1 L (D)3 3. − − − ∈ ∞ × Let us set

u = E E , v = NE E , (3.21) − 0 − 0 to which corresponds the inverse formulas

1 1 E = (N 1)− (v u), E = (N 1)− (Nu v). (3.22) − − 0 − − Taking the difference between the two equations of (3.19) we get

curl curl u = k2v in D. (3.23)

Therefore, we have in particular,

1 2 E = (N 1)− (k− curl curl u u). (3.24) − − Substituting for E in (3.19) one obtains the following fourth order partial differential equation satisfied by u

1 2 1 curl curl (N 1)− curl curl u k curl curl (N 1)− u − − − (3.25) k2N(N 1) 1curl curl u + k4N(N 1) 1u = 0 in D, − − − − − while one gets from (3.20)

u ν = ϕ, (curl u) ν = ψ on ∂D. (3.26) × × where we have set ϕ = E ( , z, q) ν and ψ = curl E ( , z, q) ν on ∂D. e · × e · × Relations (3.21-3.23) show that the existence and uniqueness of solutions to (ITP) are equivalent to the existence and uniqueness of u (D) satisfying (3.25-3.26). ∈ U Using this new formulation of the (ITP) we prove the following theorem in [A11].

Theorem 3.4 Assume that N satisfies the requirements of Theorem 3.3 and that ∂D is of class C3. For any ϕ H3/2(∂D) and ψ H1/2(∂D), the (ITP) has a unique strong solution ∈ t ∈ t (E, E ) L2(D)3 L2(D)3. Moreover there exists a constant c independent of ϕ and ψ such 0 ∈ × that E 2 + E0 2 c ϕ 3 + ψ 1 . (3.27) k kL (D) k kL (D) ≤ k kH 2 k kH 2   54 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS

Remark 3.1 Let us remark that the assumption on the regularity on ∂D can be weakened when the data is like in (3.19-3.20) for z / ∂D. In that case, assuming ∂D to be Lipschitz ∈ continuous is sufficient. The regularity on ∂D is needed to construct a lifting function for a given pair (ϕ, ψ) (see [A10]; Lemma 3.1). In the case of (3.19-3.20) with z / ∂D, it can be ∈ easily constructed from E ( , z, q) and a suitably chosen cutoff function without requiring any e · regularity assumption on ∂D.

3.3 Inversion methods using the Reciprocity Gap principle

The use of reciprocity gap (RG) principle in inverse boundary value problems is widespread, even if quite often this terminology is not explicitly used. But it is more likely that the first use of this concept was due to Cald´eron [49] in proving uniqueness for inverse tomography problems (see [113] for a review). A. Ben Abda and S Andrieux were the first to exploit this principle in designing explicit reconstruction procedures to recover planar cracks from electrostatic Cauchy data [32]. Their pioneering work had been extended to various other problems and a review can be found in [33, 44]. Most of these methods apply when the geometry of the obstacle is “simple”: for instance planar cracks or also point sources [62, 31]. The specificity of these geometries enables the design of “quasi-explicit” reconstructions, using only few measurements. The enclosure method by Ikeheta [76] and the probe method developed by Potthast et al. (see a review in [105]) also make use of the reciprocity gap principle. My work on inverse methods based on this concept started in collaboration with A. Ben Abda and our Phd student F. Delbary. We adressed inverse scattering problems from cracks and screens. The design of explicit reconstructions has been investigated in [A1]. This work was in the continuity of the ideas developed in [32]. Then it appeared to us that this concept can be also used to design sampling-like methods from Cauchy data. This was the starting point of works developed with Colton [A6] and Cakoni-Fares [AS3]. More recently, we investigate with R. Mdimagh in his Phd thesis, a particular use of this concept in the case of point scatterers, based on high frequency-like asymptotics. Let us finally quote that RG also provides an elementary tool to study the stability of some basic inverse problems [60, 78, 61]. The following quick summary presents the use the reciprocity gap functional in the elec- tromagnetic framework. It is based on [A1, AS3]. General setting. Let us denote by Ω the bounded domain on the boundary of which Cauchy electromagnetic data are available, namely (E ν) and (curl E ν) where ν × |∂Ω × |∂Ω denotes a unitary normal directed to the exterior of Ω. Without specifying what kind of diffraction problem is satisfied inside Ω, let us denote by the set of electric fields E that U correspond to the data. If k denote the wave number (that may not be constant) of the background medium inside Ω, we introduce

(Ω) := V H(curl , Ω) ; curl curl V k2V = 0 in Ω . (3.28) V { ∈ − } The reciprocity gap functional (or Wronskian) is defined as : (Ω) C3 ; R U × V −→

(E, V ) = (ν E) curl V (ν V ) curl E ds (3.29) R { × · − × · } Z∂Ω 3.3. INVERSION METHODS USING THE RECIPROCITY GAP PRINCIPLE 55

where the integral of each dot product should be understood as a duality pairing between 1/2 1/2 H− (div, ∂Ω) and H− (curl , ∂Ω) (this convention will be kept in the remaining of this section). As one can already notice by using Stokes’ formula, if (Ω) (i.e. when no perturbation U ⊂ V is present in the background medium) then = 0. This functional is therefore a natural R indicator of the presence of some perturbation. We shall see hereafter how one can use this functional to image the shape of an inclusion without requiring to solve the forward problem.

3.3.1 Inverse planar screen problem We shall illustrate here how the functional can be used to give explicit reconstruction R formulas in the case of simple geometries embedded into homogeneous medium: k is constant. We shall consider here the case of planar screens [A1]. One can find in [60] a generalization to the impedance problem and how RG can also be used to recover the impedance (in the case of the Helmholtz equation). Assume that the inclusion is a screen σ belonging to a plane Π with normal νˆ. The electric fields E satisfy ∈ U curl curl E k2E = 0 in Ω/σ − (3.30) E νˆ = 0 on σ.  × Using Stokes’ formula and the boundary condition on σ one derives

(E, V ) = [(curl E) ] (V νˆ) ds for all V (Ω), (3.31) R T Π · × ∈ V ZΠ

where [(curl E)T ]Π denotes the jump of the tangential values of curl E across Π. One can easily prove that if (E, ) is not identically 0 then σ¯ coincides with the support R · of the jump of (curl E)T across Π (Lemma 4.1 of [A1]). Therefore, an explicit formula that

gives [(curl E)T ]Π is capable of imaging σ. We shall assume in the sequel that the used measurement is such that (E, ) = 0. R · 6 Explicit formula for [(curl E)T ]Π. The idea behind getting this formula is to apply (3.31) to suitably chosen test functions V . In the context of electromagnetic diffraction problem these fields are just plane waves with “complex direction” (known as the Cald`eron fields). They are of the form

V (x; θ, pˆ) := pˆ exp(iθ x) x R3, · ∈ where (θ, pˆ) C3 R3 is a direction-polarization couple satisfying θ θ = k2 and pˆ θ = 0. It ∈ × · · is straightforwardly seen that V ( ; θ, pˆ) and curl V ( ; θ, pˆ) belong to (Ω). · · V Assume that Π is known and is defined by the equation: x νˆ = γ where νˆ is a unit normal · to Π and γ is a given constant. Let ξ R3, ξ = 0, ξ νˆ = 0, and define ∈ 6 · θ(ξ) = ξ k2 ξ 2νˆ and pˆ(ξ) = νˆ ξˆ, − − | | × p where ξˆ = ξ/ ξ and where k2 ξ 2 := i ξ 2 k2 for ξ > k. Then the couple (θ(ξ), pˆ(ξ)) | | − | | | | − | | satisfies θ θ = k2 and pˆ θ = 0. If one defines · · p p (ξ) = (E, V ( ; θ(ξ), pˆ(ξ))) and (ξ) = (E, curl V ( ; θ(ξ), pˆ(ξ))) A R · B R · 56 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS

Then one can show that [A1],

2 2 ˆ (ξ) ˆ [(curl E)T ]Π (ξ) = exp(iγ k ξ ) (ξ) ξ i B (ξ νˆ) , (3.32) F − | | A − k2 ξ 2 × !
p − | | where denotes the 2-D Fourier transform. By evaluatingp (ξ) and (ξ) and applying the F A B inverse Fourier transform one has access to [(curl E)T ]Π. This assumes that the screen plane is known. But, using similar considerations, one can also have explicit reconstruction formulas for that plane [A1]. Numerical issues. The major numerical difficulty comes from the fact that formula (3.32) is numerically unstable outside the disc of radius k. More precisely, the expression in the √ ξ 2 k2(x νˆ) right hand side of 3.32 requires an accurate evaluation of the exponential factor e− | | − · for ξ > k, which is hardly achievable for large ξ (this is a well known problem in impedance | | | | tomography for methods based on Cald`eron’s). One also notices that smaller are the values of (x νˆ), better should be the behavior of this exponential term. In other words, closer · are the measurements to the crack plane, better would be the approximation of the Fourier transform for ξ > k. However, it turns out in practice that in order to observe a significant | | improvement of the reconstruction quality, the distance between the measurements location and the crack plane should be extremely small. This is unrealistic in the case of applications we are interested in (namely non-destructive testing). However, in some other applications, like in near-field optical tomography, this situation is plausible and even constitutes the basis of accurate imaging of surface roughness and cracks. This difficulty somehow reflects the ill-posedness of the inverse problem, and more precisely the use of the evanescent modes in the data to get informations on the shape of the obstacle. For the numerical trials below, we choose to use a rather rough regularization process that consists in evaluating ([(curl E) ] ) only for ξ < k and inverting the obtained truncated F T Π | | Fourier transform to obtain an approximation of [(curl E)T ]Π. In this case the measurements location has no significant influence on the reconstruction accuracy. From the Shannon sam- pling theorem, one deduces that the theoretical spatial resolution of the reconstruction is λ/2 where λ = 2π/k is the wavelength. This is attested by the numerical examples in Figure 3.5. They correspond to the use T E polarized waves and incident plane wave propagating along the normal to the screen plane (assumed to be known). It is worth mentioning that due to the filtering properties of the Fourier transform, the procedure is robust with respect to random noise. More elaborated numerical experiments are present in [A1].

3.3.2 A sampling method based on RG We shall illustrate here recent developments on the the use the functional to design a R sampling method with Cauchy data. The key idea came from the following observation: if one consider the scattering problem with homogeneous background (take for instance the example of section 3.2.2) and measurements associated with E(x; dˆ, p) (total field corresponding to an incident plane waves Ei(x; dˆ, p) given by (3.9)), then

(E( ; dˆ, p), V (x; kx,ˆ q)) = q E (x,ˆ dˆ, p). R · − · ∞ It is therefore possible to rewrite the far field equation (3.14) in terms of . Assume for R simplicity that Σ = Γ = S2, by using the reciprocity relations (see [55]), it is also possible to 3.3. INVERSION METHODS USING THE RECIPROCITY GAP PRINCIPLE 57

Figure 3.5: Modulus of the jump of the solution across the screen’s plane. Exact on the left hand side and reconstructed on the right hand side. From the top to the bottom : k = 40, k = 25, k = 15 (the host plane is known). 58 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS show that LSM is equivalent to replacing problem (3.14) by seeking g( , z, q) L2(S2) such · ∈ t that (E( ; dˆ, p), g( , z, q)) (E( ; dˆ, p), E ( , z, q)) (3.33) R · H · ' R · e · for all dˆ S2 and p R3. One can also see this equation as an approximation of (what would ∈ ∈ be the equivalent to (3.13)) finding E ( ; z, q) H (D) such that 0 · ∈ inc (E( ; dˆ, p), E ( ; z, q)) = (E( ; dˆ, p), E ( , z, q)) (3.34) R · 0 · R · e · for all dˆ S2 and p R3. ∈ ∈ What is interesting in reformulating the LSM equation in the (3.34) is the flexibility of the offered new framework. For instance, independently from the type of scattering problem, we can write 3.34 as

(E, E ( ; z, q)) = (E, E ( , z, q)) for all E (3.35) R 0 · R e · ∈ U where as introduced before, is the set of electric fields E that correspond to the Cauchy U data on ∂Ω. Moreover, the choice of the approximative operator is no longer dictated by the sampling-operator F . For instance any injective operator with dense range in H (D) H inc would be convenient to the theoretical justification developed in section 3.2. More elaborated approximations can also be investigated. A current issue is to find the adequate procedure that guarantees the convergence of the numerical solution to E ( ; z, q), which is one of the 0 · main open theoretical questions. We investigated the application of this framework to the imaging of buried objects, i.e. for example when the domain Ω contains a different material than the outer one. The use of data associated to point sources was considered and also the use single layer potentials instead of . The 2-D case, for perfect conductors and inhomogeneous inclusions, is studied H in [A20]. We generalized the results to Maxwell’s equations in [AS3] where the restrictive assumption on the location of the sources in [A20] has been removed: It is worth noticing that the difficult part in proving the equivalent to Theorem 3.3 is not for z inside D (as it is the case for the LSM) but for z / D. The set of data plays an important role for this ∈ U proof. Another advantage of equation (3.35) setting, in comparison with LSM, is that there is no need to compute the Green tensor for the background medium [59, 99]. One only needs E ( , z, q) (Ω z ) with being singular at z. This is a decisive advantage for e · ∈ V \ { } complex backgrounds, where no analytic expressions are available. The price to pay is to have measurements of Cauchy data on the whole boundary of Ω. This restriction can be weakened if there is enough absorption inside the medium so that the data values is negligible outside a given part Γ ∂Ω. ⊂ A particular case that falls within these considerations is the imaging of objects buried in the earth using measurements at the interface air-soil. One can think of applications to mines detection (see [40] for a historical review this classical problem). Numerical examples are given in [AS3] for 3-D Maxwell’s equations, simulating detection of buried perfect scatterers. As we already quoted, one advantage of the above sampling methods lies in the fact that there is no need of a priori knowledge on the physical properties of the buried objects to visualize them. This is particularly relevant for mines detection. And this is why numerical examples with anisotropic inclusions would be more representative of the potentialities of these methods... 3.4. ITERATIVE METHODS 59

3.4 Iterative Methods

This section is devoted to the presentation of results from my recent collaboration with R. Kress. Our fruitful (and very enjoyable) collaboration started after the course he gave at INRIA, “Ecole des Ondes”, in 2003, and during my visit to the University of Goettingen that year. I hereafter reproduced some of the general presentation given in the introduction of our two works [A20] and [A21].

3.4.1 A result in connection with the Newton Methods

We began our collaboration by investigating the Fr´echet differentiability of the boundary to far field map in the context of time-harmonic electromagnetic inverse scattering problem from coated obstacles. This is an essential issue for the mathematical foundation and implementation of ap- proximate solution methods by regularized iteration schemes via linearization. In acoustic scattering, differentiability with respect to the boundary was considered by Roger [106], who first employed Newton-type iterations for inverse obstacle scattering problems. Rigorous foun- dations for the Fr´echet differentiability including characterizations of the derivative both for the Dirichlet and Neumann boundary condition, i.e., for sound-soft and sound-hard obstacles, were given by Kirsch [81] and Hettlich [72] in the sense of a domain derivative via variational methods and by Potthast [103] via boundary integral equation techniques. Alternative proofs were contributed by Hohage [74] and Schormann [110] via the implicit function theorem and by Kress and P¨aiv¨arinta [90] via Green’s theorems and a factorization of the difference of the far field for neighboring domains. Hettlich [72] also established differentiability for the impedance boundary condition. In electromagnetic obstacle scattering from perfect conductors Fr´echet differentiability was considered by Potthast [102] via boundary integral equations. Hettlich [73] treated the transmission problem for penetrable obstacles via variational methods. The technique due to Kress and P¨aiv¨arinta was extended to the Maxwell equations for the perfect conductor case in [88]. The impedance boundary condition for electromagnetic obstacle scattering has not yet been considered in the literature. The purpose of [A20] was to extend the method introduced by Kress and P¨aiv¨arinta to the case of the impedance boundary condition. We reestablished first the results of Hettlich through an alternative proof. Then we proved the Fr´echet differentiability with respect to the boundary for the electromagnetic impedance problem and provide a characterization of the derivative. In principle, one of the key ideas of the method is to extend the continuous dependence of the solution on the boundary that can be obtained via boundary integral equations to differentiability via the application of Green’s integral and Green’s representation theorems.

3.4.2 Electrostatic Imaging via Conformal Mapping

Our ongoing collaboration focuses on the use of conformal mapping in the context of 2-D inverse boundary value problems for the Laplace equation. The algorithm based on this technique have been suggested in [28] by Akduman and Kress. 60 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS

To describe the main ideas of this reconstruction algorithm, let D R2 be a doubly con- ⊂ nected bounded domain with exterior boundary Γe and interior boundary Γi. Then consider the two set of Cauchy data

(D) := (f, g) = (u, ∂ u) : u H1(D), ∆u = 0 in D, u = 0 on Γ Ud ν |Γe ∈ i  and (D) := (f, g) = (u, ∂ u) : u H1(D), ∆u = 0 in D, ∂ u = 0 on Γ Un ν |Γe ∈ ν i where ∂νu := ∂u/∂ν denotes the derivative with respect to the unit normal ν directed into the exterior of D. Note that Cauchy data (f, g) describe the potential f = u and ∈ Ud |Γe the current g = ∂ u on Γ of the electric potential u in a conducting medium containing a ν |Γe e perfectly conducting inclusion bounded by Γ , whereas Cauchy data (f, g) corresponds i ∈ Un to a non-conducting inclusion. The inverse problem we are concerned with is to determine Γ from the knowledge of one (or several) (f, g) (Dirichlet case) or one (or several) i ∈ Ud (f, g) (Neumann case). Denote by B the annulus bounded by two concentric circles C ∈ Un 1 and Cρ centered at the origin and with radii 1 and ρ < 1, respectively. In the sequel, we will identify the Euclidean space R2 and the complex plane C in the usual way. By the conformal mapping theorem, there exists a uniquely determined radius ρ and a holomorphic function Ψ that maps B bijectively onto D such that the boundaries Cρ and C1 are mapped onto Γi and Γe, respectively. The mapping Ψ is uniquely determined by prescribing the image of one point on C1. Denoting by L the length of Γe, let

γ : [0, L] Γ 7→ e be the parameterization of Γ in terms of arc length, that is, γ = 1. We now can normalize e | 0| Ψ by enforcing Ψ(1) = γ(0). If we define a function ϕ : [0, 2π] [0, L] by 7→ 1 it ϕ(t) := γ− (Ψ(e )), t [0, 2π], ∈ then, roughly speaking, ϕ describes how Ψ maps arc length on C1 onto arc length on Γe. Clearly, knowing ϕ is equivalent to knowing the restriction of Ψ onto C1. Γ e C1 ϕ

D B Γi Ψ Cρ

Figure 3.6: Illustration of the geometry

In both the Dirichlet and the Neumann case, the main idea of the reconstruction algorithm relies on first determining the restriction of Ψ onto C1 and the radius ρ from the knowledge of one pair (f, g) of Cauchy data (or several pairs). In principle, this is achieved by solving a nonlocal nonlinear ordinary differential equation for ϕ by successive approximations. This 3.5. PERSPECTIVES 61

differential equation is based on the Cauchy–Riemann equations to the harmonic function u Ψ and a conjugate harmonic and has the form ◦ dϕ A(f γ ϕ) = ◦ ◦ , (3.36) dt g γ ϕ ◦ ◦ where A is a certain Dirichlet to Neumann map on C1. Formulas for the radius ρ can be obtained by applying Green’s integral theorem to u Ψ and appropriately chosen harmonic ◦ functions in B. In a second step, the complete mapping Ψ is computed from ϕ with the aid of a regularized Laurent expansion and then Γi is found as the image Γi = Ψ(Cρ)(see Figure 3.6). In [28] only the Dirichlet case was considered and only one formula for the radius ρ was exploited. The later required g ds = 0 for the total flux through Γe. Furthermore, since Γe 6 obviously the successive approximation for solving the differential equation (3.36) for ϕ faces R difficulties when the data function g has zeros, the numerical examples in [28] were confined to cases where the normal derivative g has no zeros on Γe. We note that both these requirements are not satisfied in the Neumann case. The purpose of our work in [A21] is to widen the applicability of the method in the Dirichlet case and to extend it to the Neumann case. We developed additional formulas for the radius ρ and designed modifications of (3.36) that remedy its failure for data functions g with zeros. For this we proposed two different procedures, namely, the introduction of some regularization parameter and the use of more than one pair of Cauchy data.

3.5 Perspectives

One of the current perspectives is the validation of the LSM and also the sampling method based on the RG in the case of anisotropic inclusions. This numerical work is needed to test the accuracy and limitations of these methods in this important configuration (from the practical point of view). Also, automatic procedures to determine the reconstructed surface (choice of the cut- off value C in (3.18)) should be investigated, and may be other criteria than the one given by (3.17) can be useful... For instance by considering the derivative of the LSM equation. Another numerical issue would be to enhance the rapidity of the method by using fast- multipole like techniques. The latter point is particularly relevant for RG methods. From the theoretical point of view, of course extending the result of T. Arens to Maxwell’s case is very desirable. The case of sampling methods based on RG seems to be more accessible, profiting from the flexibility of its framework. For instance, we are currently investigating the use of approximations based on a spatial discretization of the probed domain. Another interesting perspective is to investigate possible extensions of the use of RG technique when the measurements are available only on a limited aperture. Coupling with a data completion method would be a possibility. The recent method proposed by Andrieux and Ben Abda, which is also based on RG, seems to be well suited for that. Such coupling can be also used to visualize embedded homogeneous media, when a priori knowledge of the physical properties is available. Applications in connection with some medical imaging techniques can then be considered. More generally, investigating hybrid inversion methods, coupling for instance an iterative procedure to a sampling like method is a promising direction. 62 CHAPTER 3. IMAGING TECHNIQUES AND INVERSE PROBLEMS

I would like to mention also some perspectives related to the study of the interior trans- mission problem. It seems that the method exposed in Section 3.2.5 is also well suited to the study of ITP for partially coated media (we are currently investigating this issue with F. Cakoni). However it does not seem obvious how such a technique can be used to describe the structure of so called interior transmission problem eigenvalues (when there is no absorption). These eigenvalues are important since they constitute the set of wave numbers for which the LSM fails. I end this section with the perspectives related to the use of conformal mapping in elec- trostatic 2-D imaging. The current issue is to extend the results of the Neumann and the Dirichlet cases to the case of an impedance boundary condition. We also would like to in- vestigate the theory behind the case of cracks (the developed proofs cannot cover this case). The extension of the method to impedance tomography is also a very interesting issue. Bibliography of the author

[A = Article, AS = Preprint, T = Phd Thesis, R = Technical Report, P = Extended Proceedings] [A1] A. Ben Abda, F. Delbary, and H. Haddar. On the use of the reciprocity-gap functional in inverse scattering from planar cracks. Math. Models Methods Appl. Sci., 15(10):1553– 1574, 2005. [A2] F. Cakoni, D. Colton, and H. Haddar. The linear sampling method for anisotropic media. J. Comput. Appl. Math., 146(2):285–299, 2002. [AS3] F. Cakoni, M’B Fares, and H. Haddar. Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects. Submitted, 2006. [P4] F. Cakoni and H. Haddar. Interior transmission problem for anisotropic media. In Mathematical and numerical aspects of wave propagation—WAVES 2003, pages 613–618. Springer, Berlin, 2003. [A5] F. Collino, M’B. Fares, and H. Haddar. Numerical and analytical studies of the lin- ear sampling method in electromagnetic inverse scattering problems. Inverse Problems, 19(6):1279–1298, 2003. [A6] D. Colton and H. Haddar. An application of the reciprocity gap functional to inverse scattering theory. Inverse Problems, 21(1):383–398, 2005. [A7] D. Colton, H. Haddar, and P. Monk. The linear sampling method for solving the electro- magnetic inverse scattering problem. SIAM J. Sci. Comput., 24(3):719–731 (electronic), 2002. [A8] D. Colton, H. Haddar, and M. Piana. The linear sampling method in inverse electro- magnetic scattering theory. Inverse Problems, 19(6):S105–S137, 2003. Special section on imaging. [A9] M. Durufl´e, H. Haddar, and P. Joly. High order generalized impedance boundary con- ditions in electromagnetic scattering problems. C. R. Acad. Sci. Paris S´er. I Math., To appear, 2006. [A10] H. Haddar. Etude´ du probl`eme coupl´e Maxwell–Landau-Lifschitz. Existence et unicit´e de solutions fortes en dimension deux. C. R. Acad. Sci. Paris S´er. I Math., 332(5):411– 416, 2001. [A11] H. Haddar. The interior transmission problem for anisotropic Maxwell’s equations and its applications to the inverse problem. Math. Methods Appl. Sci., 27(18):2111–2129, 2004.

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