L'imagerie électromagnétique est le problème de la détermination de la distribution de matériaux à partir de champs diffractés mesurés venant du domaine les contenant et sous investigation. Résoudre ce problème inverse est une tâche difficile car il est mal posé en raison de la présence d'opérateurs intégraux (de lissage) utilisés dans la représentation des champs diffractés en terme de propriétés des matériaux, et ces champs sont obtenus à un ensemble fini et non nécessairement optimal de points via des mesures bruitées. En outre, le problème inverse est non linéaire simplement en raison du fait que les champs diffractés sont des fonctions non linéaires des propriétés des matériaux. Le travail décrit traite du caractère mal posé de ce problème d'imagerie électromagnétique en utilisant des techniques de régularisation basées sur la parcimonie, qui supposent que le(s) diffracteurs(s) ne capture(nt) de fait qu'une petite fraction du domaine d'investigation. L'objectif principal est d'étudier de manière approfondie la régularisation de parcimonie pour les problèmes inverses non linéaires. Par conséquent, nous nous concentrons sur la méthode de Tikhonov non linéaire normalisée qui résout directement le problème de minimisation non linéaire en utilisant les itérations de Landweber, où une fonction de seuillage est appliquée à chaque étape pour promouvoir la contrainte de parcimonie. Ce schéma est accéléré à l'aide d'une méthode de descente de plus grande pente projetée et remplace l'opération de seuillage pour faire respecter cette contrainte. Cette approche a également été implémentée dans un domaine d'ondelettes qui permet une représentation précise de la fonction inconnue avec un nombre réduit de coefficients. En outre, nous étudions une méthode corrélée à la parcimonie qui offre de multiples solutions parcimonieuses qui partagent un support commun non nul afin de résoudre le problème non linéaire concerné. / So-called quantitative electromagnetic imaging focused onto here is the problem of determining material properties from scattered fields measured away from the domain under investigation. Solving this inverse problem is a challenging task because it is ill-posed due to the presence of (smoothing) integral operators used in the representation of scattered fields in terms of material properties, and scattered fields are obtained at a finite set of points through noisy measurements. Moreover, the inverse problem is nonlinear simply due the fact that scattered fields are nonlinear functions of the material properties. The work described in this thesis deals with the ill-posedness of the electromagnetic imaging problem using sparsity-based regularization techniques, which assume that the scatterer(s) capture only a small fraction of the investigation domain and/or can be described in sparse fashion on a certain basis. The primary aim of the thesis is to intensively investigate sparsity regularization for nonlinear inverse problems. Therefore, we focus on sparsity-regularized nonlinear Tikhonov method which directly solves the nonlinear minimization problem using Landweber iterations, where a thresholding function is applied at every iteration step to promote the sparsity constraint. This scheme is accelerated using a projected steepest descent method and replaces the thresholding operation to enforce the sparsity constraint. This approach has also been implemented in wavelet domain which allows an accurate representation of the unknown function with a reduced number of coefficients. Additionally, we investigate a method correlated with the joint sparsity which gives multiple sparse solutions that share a common nonzero support in order to solve concerned nonlinear problem.

Mon travail de thèse porte sur la modélisation électromagnétique et l'imagerie de structures périodiques désorganisées. Un certain motif dans une subdivision élémentaire (une "cellule") est répété dans les autres cellules de la structure dans certaines directions de l'espace. Cette répétition est désorganisée par un changement des propriétés des matériaux et/ou géométries des parties constitutives, dans une ou plusieurs cellules. Au premier niveau de modélisation, ces panneaux sont une succession de plaques planes l'une sur l'autre. Chacun se compose d'un agencement linéaire régulier de longs cylindres avec mêmes sections circulaires finies, tous orientés dans la même direction: nous les appelons "fibres", chaque cylindre résultant de l'hypothèse d'un faisceau de fibres de petite taille. Le matériau constitutif des fibres est différent du matériau d'enrobage (matrice) et le renforce. Chaque plaque est constituée de fibres avec différents axes pour assurer la robustesse. Il y a peu ou beaucoup de plaques, avec la répétition d'une petite pile de plaques. Pour les panneaux conducteurs (à base de carbone), l'imagerie est MHz ; pour des panneaux sans pertes ou à faibles pertes (à base de verre), l'imagerie est micro-onde (quelques dizaines de GHz, voire plus, THz). Il pourrait y avoir des cylindres manquants ou déplacés à l'intérieur d'une plaque, avec des changements conséquents dans éventuellement plusieurs cellules, adjacentes ou non. Des dommages locaux peuvent également se produire, entraînant des changements dans la forme ou les propriétés électromagnétiques d'un ou plusieurs cylindres dans une ou plusieurs cellules dans une ou plusieurs plaques. Un caractère aléatoire de la distribution des inclusions pourrait tenir compte des incertitudes de positionnement par rapport aux géométries supposées. Illuminer correctement les structures et la collecte des champs résultant (dans le champ proche espérons-le, peut-être dans le champ lointain) devraient permettre leur imagerie et concourir à leur diagnostic. Ainsi, si une structure périodique sous interrogation est désorganisée, on souhaite imager cette structure tout en prenant soin au mieux de l'information préalable sur la périodicité et la désorganisation, sur les systèmes de détection, et, évidemment, à propos des besoins et des limites de l'essai. / My PhD work is about electromagnetic modeling and imaging of disorganized periodic structures. A certain pattern in an elementary subdivision (a “cell”) is repeated in the other cells of the structure into certain directions of space. This repetition is disorganized by a change of material properties and/or geometries of the constitutive parts, within one or more cells. At first level of modeling, these panels are a succession of planar plates one over the other. Each consists of a regular linear arrangement of long cylinders with same finite circular sections, all orientated into the same direction: we call them “fibers”, each cylinder resulting from the assumption of a bundle of small-size fibers. The constitutive material of the fibers differs from the embedding material (matrix) that they reinforce. Each plate is made of fibers with different axes for sturdiness. There are few or many plates, with repetition of a small stack of plates. For conductive panels (carbon-based), imaging is MHz; for lossless or weakly lossy panels (glass based), imaging is microwave (a few tens GHz, possibly more). There might be missing/displaced cylinders inside a plate, with consequent changes in possibly several cells, adjacent or not. Local damages might occur also, leading to changes in shape or electromagnetic properties of one or more cylinders in one or more cells in one or more plates. Randomness in distribution of the inclusions might account for uncertainties of positioning with respect to assumed geometries. Properly illuminating the structures and collecting the resulting fields (in the near-field hopefully, possibly in the far-field) should allow their imaging and concur to their diagnostics. So, a periodic structure under interrogation is disorganized. One wishes to successfully image the structure while taking care at best of prior information on periodicity and disorganization, on sensing systems, and obviously of needs and limitations of the testing. The PhD benefits from a grant from the Chinese Scholarship Council.

<p>This dissertation focuses on a FFT based forward EM solver and its application in inverse problems. The main contributions of this work are two folded. On the one hand, it presents the first scaled lab experiment system in the oil and gas industry for through casing hydraulic fracture evaluation. This system is established to validate the feasibility of contrasts enhanced fractures evaluation. On the other hand, this work proposes a FFT based VIE solver for hydraulic fracture evaluation. This efficient solver is needed for numerical analysis of such problem. The solver is then generalized to accommodate scattering simulations for anisotropic inhomogeneous magnetodielectric objects. The inverse problem on anisotropic objects are also studied.</p><p>Before going into details of specific applications, some background knowledge is presented. This dissertation starts with an introduction to inverse problems. Then algorithms for forward and inverse problems are discussed. The discussion on forward problem focuses on the VIE formulation and a frequency domain solver. Discussion on inverse problems focuses on iterative methods.</p><p>The rest of the dissertation is organized by the two categories of inverse problems, namely the inverse source problem and the inverse scattering problem. </p><p>The inverse source problem is studied via an application in microelectronics. In this application, a FFT based inverse source solver is applied to process near field data obtained by near field scanners. Examples show that, with the help of this inverse source solver, the resolution of unknown current source images on a device under test is greatly improved. Due to the improvement in resolution, more flexibility is given to the near field scan system.</p><p>Both the forward and inverse solver for inverse scattering problems are studied in detail. As a forward solver for inverse scattering problems, a fast FFT based method for solving VIE of magnetodielectric objects with large electromagnetic contrasts are presented due to the increasing interest in contrasts enhanced full wave EM imaging. This newly developed VIE solver assigns different basis functions of different orders to expand flux densities and vector potentials. Thus, it is called the mixed ordered BCGS-FFT method. The mixed order BCGS-FFT method maintains benefits of high order basis functions for VIE while keeping correct boundary conditions for flux densities and vector potentials. Examples show that this method has an excellent performance on both isotropic and anisotropic objects with high contrasts. Examples also verify that this method is valid in both high and low frequencies. Based on the mixed order BCGS-FFT method, an inverse scattering solver for anisotropic objects is studied. The inverse solver is formulated and solved by the variational born iterative method. An example given in this section shows a successful inversion on an anisotropic magnetodielectric object. </p><p>Finally, a lab scale hydraulic fractures evaluation system for oil/gas reservoir based on previous discussed inverse solver is presented. This system has been setup to verify the numerical results obtained from previously described inverse solvers. These scaled experiments verify the accuracy of the forward solver as well as the performance of the inverse solver. Examples show that the inverse scattering model is able to evaluate contrasts enhanced hydraulic fractures in a shale formation. Furthermore, this system, for the first time in the oil and gas industry, verifies that hydraulic fractures can be imaged through a metallic casing.</p> / Dissertation