High-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time. / text

Eine Standardformulierung der Randwertaufgabe für die Beschreibung zeitharmonischer elektromagnetischer Phänomene hat die Vektor-Helmholtzgleichung für das elektrische Feld zur Grundlage. Bei niedrigen Frequenzen führt der große Nullraum des Rotationsoperators zu einem instabilen Lösungsverhalten. Wird die Randwertaufgabe zum Beispiel mit Hilfe der Methode der Finiten Elemente in ein lineares Gleichungssystem überführt, äußert sich die Instabilität in einer schlechten Konditionszahl ihrer Koeffizientenmatrix. Eine stabilere Formulierung wird durch die explizite Berücksichtigung der Kontinuitätsgleichung erreicht. Zur numerischen Lösung der Randwertaufgaben wurde eine Finite-Elemente-Software erstellt. Sie berücksichtigt unter anderem unstrukturierte Gitter, räumlich variable, anisotrope Materialparameter sowie die Erweiterung der Maxwell-Gleichungen durch Perfectly Matched Layers. Die Software wurde anhand von Anwendungen in der marinen Geophysik erfolgreich getestet. Insbesondere demonstriert die Einbeziehung von Seebodentopographie in Form einer stetigen Oberflächentriangulierung die geometrische Flexibilität der Software. / The physics of time-harmonic electromagnetic phenomena can be mathematically described by boundary value problems. A standard approach is based on the vector Helmholtz equation in terms of the electric field. The curl operator involved has a large, non-trivial kernel which leads to an instable solution behaviour at low frequencies. If the boundary value problem is solved approximately using, e. g., the finite element method, the instability expresses itself by a badly conditioned coefficient matrix of the ensuing system of linear equations. A stable formulation is obtained by taking the continuity equation explicitly into account. In order to solve the boundary value problem numerically a finite element software package has been implemented. Its features comprise, amongst others, the treatment of unstructured meshes and piecewise polynomial, anisotropic constitutive parameters as well as the extension of Maxwell’s equations to the Perfectly Matched Layer. Successful application of the software is demonstrated with examples from marine geophysics. In particular, the incorporation of seafloor topography by a continuous surface triangulation illustrates the geometric flexibility of the software.

L'un des enjeux majeurs pour les systèmes d'imagerie par diffraction acoustique et micro-onde, est l'amélioration des performances obtenues au moment de la reconstruction des objets étudiés. Cette amélioration peut s'effectuer par la recherche d'algorithmes d'imagerie plus performants d'une part et par la recherche d'une meilleure configuration de mesures d'autre part. La première approche (recherche d'algorithmes) permet d'améliorer le processus d'extraction de l'information présente dans un échantillon de mesures donné. Néanmoins, la qualité des résultats d'imagerie reste limitée par la quantité d'information initialement disponible. La seconde approche consiste à choisir la configuration de mesures de manière à augmenter la quantité d'information disponible dans les données. Pour cette approche, il est nécessaire de quantifier la quantité d'information dans les données. En théorie de l'estimation, ceci équivaut à quantifier la performance du système. Dans cette thèse, nous utilisons la Borne de Cramer Rao comme mesure de performance, car elle permet d'analyser la performance des systèmes de mesures sans être influencé par le choix de la méthode d'inversion utilisée. Deux analyses sont proposées dans ce manuscrit. La première consiste en l'évaluation de l'influence des facteurs expérimentaux sur la performance d'inversion. Cette analyse a été effectuée pour différents objets le tout sous une hypothèse de configuration bidimensionnelle. La seconde analyse consiste à comparer les performances de l'estimateur obtenu avec l'approximation de Born aux valeurs de la borne de Cramer Rao (BCR); l'objectif étant d'illustrer d'autres applications possibles de la BCR. / Improving the performance of diffraction based imaging systems constitutes a major issue in both acoustic and electromagnetic scattering. To solve this problem, two main approaches can be explored. The first one consists in improving the inversion algorithms used in diffraction based imaging. However, while this approach generally leads to a significant improvement of the performance of the imaging system, it remains limited by the initial amount of information available within the measurements. The second one consists in improving the measurement system in order to maximize the amount of information within the experimental data. This approach does require a quantitative mean of measuring the amount of information available. In estimation problems, the {appraisal of the} performance of the system is often used for that purpose. In this Ph.D. thesis, we use the Cramer Rao bound to assess the performance of the imaging system. In fact, this quantity has the advantage of providing an assessment which is independent from the inversion algorithm used. Two main analysis are discussed in this thesis. The first analysis explores the influence on the system's performance, of several experimental conditions such as the antennas positions, the radiation pattern of the source, the properties of the background medium, etc. Two classes of objects are considered: 2D homogeneous circular cylindrical objects and 2D cylindrical objects with defect. The second analysis studies the performance of an estimator based on Born approximation with the Cramer Rao Bound as reference. The aim of this second analysis is to showcase other possible applications for the Cramer Rao Bound.

Eine Standardformulierung der Randwertaufgabe für die Beschreibung zeitharmonischer elektromagnetischer Phänomene hat die Vektor-Helmholtzgleichung für das elektrische Feld zur Grundlage. Bei niedrigen Frequenzen führt der große Nullraum des Rotationsoperators zu einem instabilen Lösungsverhalten. Wird die Randwertaufgabe zum Beispiel mit Hilfe der Methode der Finiten Elemente in ein lineares Gleichungssystem überführt, äußert sich die Instabilität in einer schlechten Konditionszahl ihrer Koeffizientenmatrix. Eine stabilere Formulierung wird durch die explizite Berücksichtigung der Kontinuitätsgleichung erreicht. Zur numerischen Lösung der Randwertaufgaben wurde eine Finite-Elemente-Software erstellt. Sie berücksichtigt unter anderem unstrukturierte Gitter, räumlich variable, anisotrope Materialparameter sowie die Erweiterung der Maxwell-Gleichungen durch Perfectly Matched Layers. Die Software wurde anhand von Anwendungen in der marinen Geophysik erfolgreich getestet. Insbesondere demonstriert die Einbeziehung von Seebodentopographie in Form einer stetigen Oberflächentriangulierung die geometrische Flexibilität der Software. / The physics of time-harmonic electromagnetic phenomena can be mathematically described by boundary value problems. A standard approach is based on the vector Helmholtz equation in terms of the electric field. The curl operator involved has a large, non-trivial kernel which leads to an instable solution behaviour at low frequencies. If the boundary value problem is solved approximately using, e. g., the finite element method, the instability expresses itself by a badly conditioned coefficient matrix of the ensuing system of linear equations. A stable formulation is obtained by taking the continuity equation explicitly into account. In order to solve the boundary value problem numerically a finite element software package has been implemented. Its features comprise, amongst others, the treatment of unstructured meshes and piecewise polynomial, anisotropic constitutive parameters as well as the extension of Maxwell’s equations to the Perfectly Matched Layer. Successful application of the software is demonstrated with examples from marine geophysics. In particular, the incorporation of seafloor topography by a continuous surface triangulation illustrates the geometric flexibility of the software.

Les équations de Maxwell en régime harmonique comportent plusieurs difficultés lorsque la fréquence est élevée. On peut notamment citer le fait que leur formulation variationnelle n’est pas définie positive et l’effet de pollution qui oblige à utiliser des maillages très fins, ce qui rend problématique la construction de solveurs itératifs. Nous proposons une stratégie de solution précise et rapide, qui associe une discrétisation par des éléments finis d’ordre élevé à des préconditionneurs de type décomposition de domaine. La conception, l’implémentation et l’analyse des deux méthodes sont assez difficiles pour les équations de Maxwell. Les éléments finis adaptés à l’approximation du champ électrique sont les éléments finis H(rot)-conformes ou d’arête. Ici nous revisitons les degrés de liberté classiques définis par Nédélec, afin d’obtenir une expression plus pratique par rapport aux fonctions de base d’ordre élevé choisies. De plus, nous proposons une technique pour restaurer la dualité entre les fonctions de base et les degrés de liberté. Nous décrivons explicitement une stratégie d’implémentation qui a été appliquée dans le langage open source FreeFem++. Ensuite, nous nous concentrons sur les techniques de préconditionnement du système linéaire résultant de la discrétisation par éléments finis. Nous commençons par la validation numérique d’un préconditionneur à un niveau, de type Schwarz avec recouvrement, avec des conditions de transmission d’impédance entre les sous-domaines. Enfin, nous étudions comment des préconditionneurs à deux niveaux, analysés récemment pour l’équation de Helmholtz, se comportent pour les équations de Maxwell, des points de vue théorique et numérique. Nous appliquons ces méthodes à un problème à grande échelle qui découle de la modélisation d’un système d’imagerie micro-onde, pour la détection et le suivi des accidents vasculaires cérébraux. La précision et la vitesse de calcul sont essentielles dans cette application. / The time-harmonic Maxwell’s equations present several difficulties when the frequency is large, such as the sign-indefiniteness of the variational formulation, the pollution effect and the problematic construction of iterative solvers. We propose a precise and efficient solution strategy that couples high order finite element (FE) discretizations with domain decomposition (DD) preconditioners. High order FE methods make it possible for a given precision to reduce significantly the number of unknowns of the linear system to be solved. DD methods are then used as preconditioners for the iterative solver: the problem defined on the global domain is decomposed into smaller problems on subdomains, which can be solved concurrently and using robust direct solvers. The design, implementation and analysis of both these methods are particularly challenging for Maxwell’s equations. FEs suited for the approximation of the electric field are the curl-conforming or edge finite elements. Here, we revisit the classical degrees of freedom (dofs) defined by Nédélec to obtain a new more friendly expression in terms of the chosen high order basis functions. Moreover, we propose a general technique to restore duality between dofs and basis functions. We explicitly describe an implementation strategy, which we embedded in the open source language FreeFem++. Then we focus on the preconditioning of the linear system, starting with a numerical validation of a one-level overlapping Schwarz preconditioner, with impedance transmission conditions between subdomains. Finally, we investigate how two-level preconditioners recently analyzed for the Helmholtz equation work in the Maxwell case, both from the theoretical and numerical points of view. We apply these methods to the large scale problem arising from the modeling of a microwave imaging system, for the detection and monitoring of brain strokes. In this application accuracy and computing speed are indeed of paramount importance.

Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface. We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem.

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