Dr Georg Shnatinger

Dr Oliver Wiedenmann

Dr Robert Brem

Dr Mark Eberspäher

Christoph Eisner

Fabian Faul

Dr Gerhard Hamberger

Susanne Hipp

Bernd Hofmann

Saiullah Khan

Raimund Mauermayer

Dr Björn Möhring

Arndt Ot

Dr Alexander Paulus

Dennis Shobert

Mehmet Taygur

Simon B Adrian

For those who dwell in Munih, the word "Präkonditionen" has a negative touh as indicated by Gerhard Polt, who deines it as the "total clariication of preconditions; a transparency to the point of absurdity, " for example, as it is observed in the "x-raying of tenants by real estate agents. " his treatise is devoted to the development of so-called "preconditioners" in the context of boundary integral equation methods for that it vindicates at least partially the term "preconditions. "

Um diese shlehte Konditionierung zu verhindern, werden in dieser Dissertation die Hierarhishe-Basen-und die Calderón-Vorkonditionierung weiterentwikelt. Zuerst wird eine auf Haar-Prewavelets basierende hierarhishe Basis vorgestellt, die auf strukturierten wie auh unstrukturierten Diskretisierungsgitern verwendet werden kann. Mit dieser Basis wähst die Konditionszahl nur noh logarithmish mit der Anzahl der Unbekannten. Dazu wird die Haar-Basis sowohl für das primäre als auh für das duale Giter verallgemeinert. Um den Skalarpotentialoperator der elektrishen Feldintegralgleihung vorzukonditionieren, werden die Divergenzterme, die in der Variationsformulierung autreten, mittels inverser Laplace-Matrizen indirekt entfernt: Die Entfernung geshieht durh Anwendung iterativer Verfahren, die mitels algebraisher Mehrgitervorkonditionierer stabilisiert werden. Für den Vektorpotentialoperator der elektrishen Feldintegralgleihung wird gezeigt, dass eine angepasste inverse Transformationsmatrix der dualen Haar-Basis als Vorkonditionierer eingesezt werden kann. Dieses Ergebnis wird unter Verwendung einer diskreten Calderón-Identität erreiht, welhe den hypersingulären Randintegraloperator und den Einfahshiht-Randintegraloperator in Bezug sezt. Nebenbei ergibt sih dadurh ein Haar-Basis-Vorkonditionierer für den hypersingulären Operator, der unter anderem in der numerishen Modellierung elektrostatisher und akustisher Probleme relevant ist. ix Wenn eine Geometrie mehrfah zusammenhängend ist, müssen sogenannte globale Shleifenfunktionen zu quasi-Helmholz-Zerlegungen hinzugefügt werden. Diese Funktionen sind numerish aufwendig zu konstruieren bzw. anzuwenden. Um dennoh hierarhishe Basen als Vorkonditionierer einsezen zu können, ohne dabei die quasilineare Gesamtkomplexität des Lösungsalgorithmus zu gefährden, wird ein neues Verfahren vorgestellt, welhes erlaubt, diese Basen mit quasi-Helmholz-Projektoren zu kombinieren, wodurh die explizite Konstruktion der globalen Shleifenfunktionen umgangen wird.

Anshließend wird die Anwendbarkeit von hierarhishen Basen als Vorkonditionierer auf die kombinierte Feldintegralgleihung untersuht. Es wird gezeigt, dass die hierarhishen Shleifenfunktionen, welhe den Vektorpotentialoperator niht vorkonditionieren können, sehr wohl bei der kombinierten Feldintegralgleihung angewendet werden können. Darüber hinaus wird ein Verfahren mit quasi-Helmholz-Projektoren vorgestellt, das es erlaubt Hierarhishe-Basen-Vorkonditionierer im Rahmen der kombinierten Feldintegralgleihung auf unstrukturierten Diskretisierungsgitern zu nuzen.

Für die meisten Anwendungen sind diese hierarhishen Vorkonditionierer ausreihend. Im asymptotishen Grenzfall, wenn also die Anzahl der Unbekannten gegen Unendlih strebt, würde die Konditionszahl dennoh shrankenlos wahsen. Werden hingegen Calderón-Vorkonditionierer verwendet, so konvergiert die Konditionszahl gegen eine Konstante. In dieser Dissertation wird die Calderón-Vorkonditierung derart weiterentwikelt, dass keine dualen Basisfunktionen und daher auh kein baryzentrish verfeinertes Diskretisierungsgiter benötigt wird. Es basiert auf spektralen Äquivalenzen von Laplace-Matrizen und diskretisierten Integraloperatoren (vornehmlih des Einfahshihtpotentialoperators und des hypersingulären Operators). Insbesondere gelingt es dabei auh, den Vorkonditionierer so auszulegen, dass mehrfah zusammenhängende Geometrien ohne Stabilitätsprobleme behandelt werden können. Im Gegensaz zu existierenden Verfahren ist die entstehende Systemmatrix hermetish und positiv deinit. Dadurh ist es möglih, dass iterative Verfahren der konjugierten Gradienten anzuwenden, das Konvergenz garantiert und im Vergleih zu anderen Krylov-Unterraumverfahren geringe numerishe Zusazkosten aufweist.

Shließlih wird der neue Calderón-Vorkonditionierer auf die kombinierte Feldintegralgleihung erweitert. Diese Erweiterung ist notwendig, da die elektrishe Feldintegralgleihung bei geshlossenen Geometrien für deren Resonanzfrequenzen niht mehr eindeutig lösbar ist. Dabei gibt es vershiedene Herausforderunx gen: Zunähst ist eine direkte Anwendung des neuen Verfahrens niht möglih, weil der Teil des Vorkonditionierers für den Vektorpotentialoperator zu einer shleht-konditionierten Systemmatrix der ebenso enthaltenen magnetishen Feldintegralgleihung führen würde. Daneben würde die vorkonditionierte Matrix niht mehr hermitesh und positiv deinit sein. Besondere Aufmerksamkeit erfordern shließlih mehrfah zusammenhängende Geometrien, da der toroidale Teil des quasi-harmonishen Helmholz-Unterraums so mit der Frequenz skaliert, dass sih für den statishen Grenzfall ein Nullraum ergibt. Das neue Verfahren führt, wie shon im Fall der elektrishen Feldintegralgleihung, zu einer Systemmatrix, die hermitesh, positiv deinit und wohlkonditioniert ist-sowohl für einfah als auh mehrfah zusammenhängende Geometrien.

xi Résumé Cete dissertation présente de nouveaux paradigmes de préconditionnement d'équations intégrales pour la résolution de problèmes de difusion et de radiation électromagnétique. Les équations intégrales sont communément utilisées pour résoudre ces problèmes elles incorporent naturellement les conditions aux limites ouvertes et elles ne ne soufrent pas de problèmes de dispersion numérique. Malheureusement, certaines de ces équations intégrales, en particulier l'équation intégrale du hamp électrique et l'équation intégrale du hamp combiné sont mal conditionnées. C'est à dire que, si le nombre d'inconnues augmente ou que la fréquence diminue, le conditionnement du problème se dégrade, ce qui a pour conséquence le ralentissement ou la non-convergence de solveurs itératifs.

Pour remédier à ce mauvais conditionnement, nous nous basons sur et étendons l'état de l'art dans le domaine des bases hiérarhiques et des tehniques de préconditionnement de type Calderón. Dans un premier temps, la thèse est dédiée à l'amélioration des préconditionneurs à bases hiérarhiques. Nous présentons une base pour des maillages structurés et non structurés, dans laquelle le conditionnement croît logarithmiquement en fonction du nombre d'inconnues. Cela représente un nete amélioration par rapport à l'état de l'art, qui ateint au mieux une croissance proportionnelle à la racine carrée du nombre d'inconnues. Nous obtenons ce résultat en généralisant dans un premier temps la base de Haar aux maillages triangulaires et leur dual. Pour préconditionner l'opérateur potentiel scalaire de l'équation intégrale du hamp électrique nous supprimons les termes de divergence apparaissant dans sa forme variationnelle grâce à l'utilisation de l'inverse des laplaciens de graphes. Pour stabiliser cete inversion nous utilisons des préconditionneurs à grilles multiples déjà existant. Pour de l'opérateur du potentiel vecteur de l'équation intégrale du hamp électrique, nous montrons que la matrice de transformation inverse de la base de Haar doit être utilisée. Ce résultat est obtenu grâce à une identité de Caldérón scalaire discrétisée qui lie l'opérateur hypersingulier à l'opérateur potentiel simple couhe bien connu en électrostatique. Cete démarhe a aussi permis l'obtention d'un préconditionneur xiii à base de Haar pour l'opérateur hypersingulier, opérateur qui apparaît dans la modélisation de problèmes électrostatiques et en acoustiques.

Dans le cas de géométries à connexions multiples, les boucles globales doivent être prises en compte et ajoutées aux décompositions quasi-Helmholz. La construction et l'application de ces fonctions sont coûteuses en termes de calcul. Ain d'utiliser les préconditionneurs à base hiérarhique sans nuire à la complexité dominante engendrée par une méthode rapide, nous montrons que la base hiérarhique peut être utilisée sans la construction explicite des boucle globales. Ce résultat est obtenu grâce au développement d'un système permetant de combiner les projecteurs quasi-Helmholz à la base hiérarhique. Le principe en lui même est agnostique de la base concrète, et par conséquent peut être combiné à n'importe quelle base hiérarhique disponible.

Ensuite nous nous intéressons à l'application du préconditionneur à base hiérarhique à l'équation intégrale du hamp combiné. Nous démontrons que les boucles hiérarhiques, qui ne peuvent pas résoudre la détérioration de la solution due à la discrétisation dense pour la partie du potentiel vecteur de l'équation intégrale du hamp électrique, peuvent être efectivement utilisées pour l'équation intégrale du hamp combiné. De plus, nous proposons une nouvelle méthode permetant l'utilisation de préconditionneurs à bases hiérarhiques à la fois sur des maillages structurés et non-structurés dans le contexte de l'équation intégrale du hamp combiné. Nous utilisons le fait que l'équation du hamp combiné est bien conditionnée pour le sous-espace de Helmholz solénoïdal et ainsi nous utilisons les projecteurs quasi-Helmholz : ces projecteurs sont metent en évidence une base hiérarhique non-solénoïdale permetant de préconditionner la partie de l'opérateur du potentiel scalaire dans l'équation intégrale du hamp combiné.

Les nouveaux préconditionneurs à base hiérarhique suisent pour la majorité des applications. Cependant le conditionnement des problèmes devrait tendre asymptotiquement vers l'inini. C'est pourquoi dans un second temps, nous surmontons cete limitation en introduisant le préconditionneur multiplicatif de Caldéron sans rainement (RF-CMP) qui, contrairement aux autres méthodes, ne nécessite pas une seconde discrétisation de l'équation intégrale du hamp électrique avec les fonctions de base duales. Tout d'abord, nous dérivons cete tehnique pour l'équation intégrale du hamp électrique en nous basant sur l'équivalence spectrale entre les laplaciens (de graphe) et les opérateurs intégraux discrétisés (les opérateurs simple couhe et hypersingulier). Une atention particulière a été portée à l'obtention d'une formulation stable dans le cas des xiv géométries à connexion multiple. Contrairement à d'autres préconditionneurs, le système matriciel résultant est Hermitien, déini positif ce qui permet l'utilisation de la méthode des gradients conjugués. A la diférence des autres méthodes des sous-espaces de Krylov, la méthode des gradients conjugués a une convergence garantie et a un surcoût calculatoire minimal.

Enin, le RF-CMP est étendu à l'équation intégrale du hamp combiné. Nous obtenons ainsi une formulation sans résonances artiicielles. Il y a là plusieurs déis : premièrement l'application directe du RF-CMP est impossible puisque la partie du préconditionneur pour l'opérateur du potentiel vecteur rendrait l'équation intégrale du hamp magnétique mal conditionnée. Deuxièmement, nous n'aurions pas un système matriciel hermitien déini positif. Et enin nous n'obtiendrions pas une formulation stable pour des géométries à connexions multiples. Cela est du au fait que l'équation intégrale du hamp magnétique n'est pas entièrement bien conditionnée puisqu'elle possède un noyau statique associé à la partie toroïdale du sous-espace de Helmholz quasi-harmonique. La nouvelle formulation est construite de façon à ce que le système préconditionné soit, comme pour l'équation intégrale du hamp électrique, hermitien déini positif et reste stable pour des géométries à connexions multiples. W hen I plunged into my dissertation research, I did not know what hallenges and issues I would encounter, and what the outcome of all my eforts would be. While I would never dare to compare these eforts with, let's say, the efort it takes to move a ship across a mountain ridge from one river system into another, I found the existence as doctoral student quite suitable for me. ‡ his is not only due to the fun I had whilst exploring the unknown, but also due to the people who I have met in that quest.

he researh of this thesis was carried out at the Chair of High-Frequency Engineering at the Tehnical University of Munih and the Computational Electromagnetics Researh Laboratory at the École nationale supérieure Mines-Télécom of Atlantique Bretagne-Pays de la Loire. During my Lehrjahre, I have met many wonderful people both in my academic and my private life and I would like to express my sincere gratitude to all of them.

First, I would like to thank my advisors professor homas Eibert and professor Francesco Andriulli. his work would not have been possible without their guidance, counsel, encouragement, and the many fruitful discussions we † W. Herzog. Conquest of the Useless: Relections from the Making of Fizcarraldo. 1st ed. Ecco: New York, 2009, p. 248. ‡ As a side note for (future) doctoral students, I found that the ilm "Burden of Dreams" is a great source of motivation helping to master the (occasional) periods of frustration.

xxi Preface and Aknowledgment had-a guidance whih was, in fact, not limited to this work alone. heir (not only scientiically) inspiring nature will always remain an example for me. Von ganzem Herzen danke ich professor Eibert for introducing me to the world of high-frequency engineering and for giving me a large measure of freedom in my teahing. Io ringrazio di cuore Francesco for introducing me to Calderón and hierarhical basis preconditioning, leting me use his boundary element code, the myriad Skype sessions, and all the adventures we had together.

Moreover, I am grateful to professor Romanus Dyczij-Edlinger and professor Ralf Hiptmair for serving on my thesis commitee and for their invaluable feedbak, and to professor Ulf Shlihtmann for hairing the commitee.

My advisors enabled me to visit many conferences, whih allowed me to get feedbak on my researh early on by meeting researhers from all over the world. he many interesting discussions, not necessarily limited to my own researh, made the conference visits so priceless. To name but a few, I would like to thank professors Hakan Bağci, Shanker Balasubramaniam, Amir Boag, Kristof Cools, Leslie Greengard, Ludger Klinkenbush, Eric Mihielssen, Andrew Peterson, Sebastian Shöps, Felipe Vico, Karl Warnik, homas Weiland, Daniel Weile, and Dr. Yves Beghein for the many inspiring conversations.

A special thanks goes to professor Yaniv Brik, who honored me with his friendship since we meet at IEEE International Symposium on Antennas and Propagation in Orlando. I am deeply grateful not only for our discussions on common researh interests, but also for introducing me to the IEEE Education Commitee and to the Japanese cuisine! hat my stays at the end of the earth § have been so enjoyable, I owe very muh to my friends and co-workers professor Rajendra Mitharwal, Dr. Abdelrahman Ijjeh, Dr. Maxim Abalenkov, Dr. Maxim Dr. Lyes Rahmouni, Dr. Axelle Pillain, Dr. John Erik Ortiz Guzman, Adrien Merlin, and Alexandre Dely not only for many fruitful discussions, but also for showing me around in Britany and only thanks to them I did not get "lost in translation" when I had to deal with Frenh red-tape. A special thanks goes to Alexandre and Adrien, who "helped" me in writing the Frenh abstract. Merci beaucoup ! I would also like to thank all my friends and colleagues from Munih, who made the Chair of High-Frequency Engineering an ideal working place. First, I would Introduction Kinder, shat Neues! Richard Wagner S even score and twelve years ago, James C. Maxwell published twenty equations whih describe the behavior of electromagnetic ields [START_REF] Maxwell | A Dynamical heory of the Electromagnetic Field[END_REF] and whih we refer today as "Maxwell's equation" (though the equations as they are taught in university nowadays are fewer in number thanks to the introduction of vector calculus and the omission of Lorenz force law [START_REF] Sengupta | Maxwell, Herz, the Maxwellians, and the Early History of Electromagnetic Waves[END_REF][START_REF] Arthur | he Evolution of Maxwell's Equations from 1862 to the Present Day[END_REF]). 1 Despite the many decades whih have passed since their publication, inding a solution to Maxwell's equations for an arbitrary problem remains a hallenge. Analytic solutions are known only for a few canonical geometries suh as a sphere [START_REF] Mie | Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen[END_REF], but even for a geometry as simple as a torus no analytic solution is known [Lau67; [START_REF] Bibby | A High Order Numerical Investigation of Electromagnetic Scatering From a Torus and a Circular Loop[END_REF][START_REF] Venkov | Low-Frequency Electromagnetic Scatering by a Perfectly Conducting Torus. he Rayleigh Approximations[END_REF].

Yet, there is a need for obtaining these solutions: if we were not be able to solve Maxwell's equations, the design process of any device with electromagnetic properties would be limited to extensive prototyping and experimentation, a process whih is both expensive and time consuming. Given the number of quotidian electromagnetic devices and systems suh as television, radio, microwave ovens, satellite communication systems, radar systems, electrical motors and generators, medical imaging systems, it is evident that we need methods that allow us to predict the (electromagnetic) behavior of these devices and systems at an early stage in the design process.

Introduction Chapter 1 a) Fragmentary 2 Review of Numerical Tehniques

If there is no analytic solution, one must resort to a numerical tehnique, whih yields an approximate solution. A plethora of tehniques have been developed, and the hoice of the method depends on the problem to be solved. For example, if a scaterer spans several millions of wave lengths, then asymptotic tehniques based on geometrical or physical optics are required [Kel62; KP74; Ui07]. hese methods, however, cannot always guarantee convergence to the physical solution.

In particular for electrically small or midsize problems with non-smooth surfaces, this problem becomes critical.

In order to guarantee convergence, numerically exact methods must be used. hey are typically divided into local and global methods. A local method discretizes in most cases a (partial) diferential operator commonly resulting in a sparse linear system. Typical examples are the inite diference time domain (FDTD) method [START_REF] Yee | Numerical Solution of Initial Boundary Value Problems Involving Maxwell's Equations in Isotropic Media[END_REF], the inite integration tehnique (FIT) [START_REF] Weiland | A Discretization Model for the Solution of Maxwell's Equations for Six-Component Fields[END_REF], the transmission line matrix method (TLM) [JB71; Hoe85], or the inite element method (FEM) [START_REF] Jin | he Finite Element Method in Electromagnetics. hird edition[END_REF]. While these methods are well-suited for many applications, scattering or radiation problems are not their strength: local methods require the discretization of the entire domain, whih is impossible to ahieve for open problems (i.e., where the domain is unbounded). In order to minimize the computational resources, the domain must be truncated artiicially. his can be ahieved by using, for example, absorbing boundary conditions [START_REF] Bayliss | Radiation Boundary Conditions for Wavelike Equations[END_REF] or the perfectly mathed layer tehnique [START_REF] Berenger | A Perfectly Mathed Layer for the Absorption of Electromagnetic Waves[END_REF]. While good results can be obtained, there is still a trade-of between accuracy and computational efort.

Another issue, whih plagues local methods is numerical dispersion, that is, if the frequency is increased while keeping the ratio of the average edge length of the mesh and the wavelength constant, then the phase becomes polluted and is eventually lost [START_REF] Babuška | Is the Pollution Efect of the FEM Avoidable for the Helmholz Equation Considering High Wave Numbers?[END_REF]. Remedies for this problem usually comprise the use of higher order basis functions, whih leads to a densely populated linear system in the asymptotic limit (and thus to the loss of one of the most harming properties of local methods).

Section 1.a Fragmentary Review of Numerical Tehniques Integral equation based methods, whih belong to the class of global methods, only demand the discretization of the scaterer or the antenna, respectively: open boundary conditions are modeled within the analytical formulation so that no additional efort is required. If the obstacle is homogeneous (or consists of a several homogeneous subdomains), a further reduction in the computational efort is possible since it suices to discretize the surface resulting in a surface integral equation that has to be solved. In addition, integral equation based methods do not sufer from numerical dispersion, though of course, the linear system emerging from a standard discretization of an integral equation is densely populated.

he dense system matrix has been one of the main reasons why integral equations were not in the focus of the scientiic community for a long time: since the matrix is fully populated, the computational complexity to obtain a solution becomes at least quadratic both in memory and in time consumption. Given the computational resources available in the last century, researh was mostly limited to, for example, thin-wire approximations or 2-D problems [START_REF] Harrington | Time-Harmonic Electromagnetic Fields[END_REF][START_REF] Sarkar | A Study of the Various Methods for Computing Electromagnetic Field Utilizing hin Wire Integral Equations[END_REF]. he issue has been overcome with the advent of fast methods that allowed to reduce the cost of a single matrix-vector product as well as the memory consumption to quasilinear complexity, that is, if is the number of unknowns, the complexity is O( log ), where ∈ R + is a constant depending on the respective method. Among the most popular methods are the multilevel fast multipole method (MLFMM), multilevel matrix decomposition algorithm (MLMDA), hierarhical matrices, or adaptive cross approximation (ACA) [GR87; HN89; SLC97; Dar00; MB96; ZVL05; Beb00; BR03]. he physical scenario considered in this thesis is the scatering of electromagnetic waves by perfectly electrically conducting (PEC) objects, that is, a known electromagnetic wave impinges on an obstacle and the scatered wave shall be computed. As a variant of this scenario, we also take radiation problems into account, where we solve for the wave radiated by an antenna whih is fed by a voltage-gap source.

Given these scenarios, the focus is on surface integral equation methods, where we do not solve directly for the electric or magnetic ield, but instead electric and magnetic currents are obtained on the surface of the scaterer by applying the surface equivalence principle [START_REF] Straton | Difraction heory of Electromagnetic Waves[END_REF][START_REF] Peterson | Computational Methods for Electromagnetics[END_REF]. Depending on the formulation, these currents can have a physical meaning or can be purely artiicial.

For solving a PEC problem, there are in essence two integral operators available, the electric ield integral equation (EFIE) and the magnetic ield integral equation (MFIE) operator, whih lead to the EFIE [START_REF] Rumsey | Reaction Concept in Electromagnetic heory[END_REF] and the MFIE [START_REF] Maue | Zur Formulierung eines allgemeinen Beugungsproblems durh eine Integralgleihung[END_REF]. While both of them yield the physical electric surface current density, their properties difer in many regards: most prominently, the EFIE works for both closed and open surfaces (e.g., an ininitely thin metal sheet would be an example of an open surface), while the MFIE is only applicable to closed surfaces. hey do share, however, a common problem: if we have an electrically large object with a closed surface (i.e., the size of the scaterer spans several wavelengths), then both the EFIE and the MFIE sufer from spurious modes; at resonance frequencies, the operators have a null space and the electric current is not uniquely determined anymore. his issue usually results in a loss of accuracy and slow convergence if an iterative solver is used. he classic remedy is to combine the EFIE and the MFIE to the combined ield integral equation (CFIE) [START_REF] Mauz | H-Field, E-Field, and Combined-Field Solutions for Conducting Bodies of Revolution[END_REF], where we still solve for the physical current. Alternatively, a combined source integral equation (CSIE), as for example proposed in [START_REF] Mauz | A Combined-Source Solution for Radiation and Scatering from a Perfectly Conducting Body[END_REF] or analogously to [START_REF] Brakhage | Über das dirihletshe Außenraumproblem für die Helmholzshe Shwingungsgleihung[END_REF], can be used. In this case, however, the solution is not the physical current anymore. Whatever strategy is hosen, the resulting combined integral equation operator will inherit properties of the EFIE and the MFIE operator. Hence, both operators must be studied and dealt with.

b) Computational Complexity and Ill-Conditioned Integral Equations

It was mentioned that fast methods can compress the system matrix so that the memory consumption and the cost of a single matrix-vector product scale only quasilinearly in the number of unknowns. he overall numerical costs for obtaining the solution of the scatering problem, however, have not been discussed so far. To obtain the solution, two approahes are possible: a direct inversion of the system matrix or the usage of an iterative solver, typically one belonging to the class of Krylov subspace methods. he advantage of a direct inversion is its robustness: we are guaranteed to obtain the inverse matrix, whereas iterative solvers can converge slowly or, in the worst case, do not converge at all. In general, the direct inversion of the system matrix is of limited practicality since the numerical costs scale cubically in . Even if the inverse matrix is obtained, for multiple right-hand sides it is still impractical since the cost of single matrix-vector product scales quadratically in . Recently, advances have been made in the ield of fast direct methods in the static [START_REF] Martinsson | A Fast Direct Solver for a Class of Elliptic Partial Diferential Equations[END_REF] and in the dynamic regime [GHM13; BLB14; GJM14; Guo+16], where in the later case [START_REF] Guo | A Buterly-Based Direct Integral Equation Solver Using Hierarhical LU Factorization for Analyzing Scatering from Electrically Large Conducting Objects[END_REF] it is observed that the memory requirement scales as O( log 2 ) and the computational costs for obtaining the inverse as O � 1.5 log � . Since the inverse is compressed, a single matrix-vector product has quasilinear complexity, whih makes these methods particularly for multiple right-hand side problems interesting. For large problems, however, any growth exceeding quasilinear complexity can be prohibitive.

For iterative methods, the computational costs scale as O � iter log � , where iter is the number of iterations that the iterative solver needs to converge. he decisive question is whether there is an upper bound for iter independent of . In the case of an integral equation of the second kind-the MFIE is a typical example [START_REF] Yla-Oijala | Error-Controllable and Well-Conditioned MoM Solutions in Computational Electromagnetics: Ultimate Surface Integral-Equation Formulation[END_REF]-there is suh an upper bound (apart from the resonance frequencies, where the MFIE operator has a null space). hus if a mesh is reined, that is, → ∞, then iter → , where ∈ N is some constant.

For an integral equation of the irst kind, the EFIE is a typical example [START_REF] Yla-Oijala | Error-Controllable and Well-Conditioned MoM Solutions in Computational Electromagnetics: Ultimate Surface Integral-Equation Formulation[END_REF], the situation is more complicated. For most Krylov subspace methods no sharp bounds on iter can be given. For illustration purposes and for the sake of simplicity, we assume that the conjugate gradient (CG) method is used (whih is actually not applicable since it requires a Hermitian, positive deinite (HPD) matrix [START_REF] Hestenes | Methods of Conjugate Gradients for Solving Linear Systems[END_REF], whih neither the EFIE nor the MFIE are), then the number of iterations iter used by the CG method can be bounded by the square root of the condition number of the system matrix [START_REF] Shewhuk | An Introduction to the Conjugate Gradient Method Without the Agonizing Pain[END_REF][START_REF] Axelsson | Error Norm Estimation and Stopping Criteria in Preconditioned Conjugate Gradient Iterations[END_REF]. he condition number is the ratio of the largest over the smallest singular value and in the case of the EFIE the condition number scales linearly in and so the computational costs are O � 1.5 log � for the EFIE and O � 1.25 log � for the CFIE or the CSIEs.

Clearly, if iter grows with increasing , then the quasilinear complexity set by the respective fast method is lost. Given the considerations on the CG method, if the leading complexity set by the fast method shall be maintained, the system matrices of the EFIE or CFIE must be transformed suh that the condition number of the resulting system matrix has an upper bound independent of .

Whenever the growth of the condition number is unbounded in , the problem is ill-conditioned in and this growth is referred to as the dense-discretization breakdown. his terminology is necessary since other forms of ill-conditioning can exist, in particular, the EFIE sufers from the so-called low-frequency breakdown: the condition number grows when the frequency is decreased.

Historically, in fact, when no fast method was available and the computational resources were more limited than today, the low-frequency breakdown was, with respect to ill-conditioning, the pressing problem (and not that muh the dense-discretization breakdown). he low-frequency breakdown originates from the fact that the EFIE operator is composed of two operators, the vector and the scalar potential operator, whih scale with and 1/, respectively, where is the wavenumber. Since the scalar potential has a null space associated with solenoidal functions, the system matrix possess two branhes of singular values and these branhes are driven apart when the frequency is decreased.

To resolve these issues or their practical impact at least partially, there are essentially three means: (i) developing a new formulation whih does not sufer from these defects, (ii) domain decomposition approahes, and (iii) construct a suitable preconditioner. A preconditioner can be understood as one or two matrices whih are multiplied let and right to the system matrix, respectively, and whih should yield a lower condition number of the overall matrix. Examples for new formulations are the augmented EFIE [START_REF] Qian | An Augmented Electric Field Integral Equation for High-Speed Interconnect Analysis[END_REF] or the decoupled potential integral equation [START_REF] Vico | he Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scatering[END_REF]. hese approahes resolve the problems, however, only partially [START_REF] Qian | An Augmented Electric Field Integral Equation for High-Speed Interconnect Analysis[END_REF] or are limited to smooth objects [START_REF] Vico | he Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scatering[END_REF].

Domain decomposition methods are well-suited if the object is electrically large or discretized with many unknowns. hey divide the domain into smaller subdomains, where in eah subdomain the problem is solved independently from the other subdomains. Transmission conditions are required to prevent unphysical relections from the boundary of the subdomains. While these approahes work well for FEMs, whih are local in nature, one needs more elaborate transmission conditions in the case of integral equations: some methods perform decently for simple problems [START_REF] Braunish | Tapered Wave with Dominant Polarization State for All Angles of Incidence[END_REF] or speciic problems suh as the scatering from antenna arrays [SM00; PM03; MLV07], they fail, however, when confronted with intricate problems. A possible remedy comprises the use of second order transmission conditions [RL10; PWL11], though at the price of the introduction of auxiliary unknowns. More recent methods avoid the introduction of auxiliary unknowns by introducing a penality term [START_REF] Peng | Domain Decomposition Preconditioning for Surface Integral Equations in Solving Challenging Electromagnetic Scatering Problems[END_REF].

Section 1.c

Review of Preconditioning Tehniques c) Review of Preconditioning Tehniques

Preconditioners, to be efective, must be carefully designed. For example, to cure the low-frequency breakdown a simple Jacobi preconditioner is not suicient. Instead, the low-frequency breakdown has been cured by using quasi-Helmholz decompositions suh as the loop-star and the loop-tree decomposition [WG81; WGK95; BK95; Vec99; ZC00; Eib04]. Here, the loop functions are a solenoidal basis and the star or tree functions complementing the loop functions are non-solenoidal. By scaling these functions in frequency, the low-frequency breakdown is resolved. hese preconditioners, however, cannot cure the dense-discretization breakdown. In fact, the loop-star preconditioner is even more sensible to the dense-discretization breakdown [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF]. Another approah to overcome the low-frequency breakdown is the augmented EFIE [START_REF] Qian | An Augmented Electric Field Integral Equation for High-Speed Interconnect Analysis[END_REF], whih does not require an explicit quasi-Helmholz decomposition and whih has the advantage that for multiply connected geometries the global loops are not needed (whih are otherwise costly to obtain). Recently, quasi-Helmholz projectors have been presented, whih can cure the low-frequency breakdown and do not require a searh for the global loops [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF]And+13].

One cure for the dense-discretization breakdown are near-zone preconditioners. Typically, fast methods suh as the MLFMM or the ACA distinguish between near-and far-interactions. he near-interactions can be extracted from the system matrix as sparse matrix with O( ) elements. Since the near matrix is the dominant part of the system matrix, its inverse is a good candidate as preconditioner. his inverse, however, must be obtained. A classic approah is the incomplete LU factorization [START_REF] Benzi | Preconditioning Tehniques for Large Linear Systems: A Survey[END_REF][START_REF] Saad | Iterative Methods for Sparse Linear Systems[END_REF], whih was invented by Varga [START_REF] Varga | Factorization and Normalized Iterative Methods[END_REF]. he LU factorization decomposes a matrix into the product of two triangular matrices, whih, however, do not maintain the sparsity of the original matrix rendering it a costly preconditioner. he incomplete LU factorization discards some of the elements of the LU decomposition, whereby the computational costs are small at the price of a less efective preconditioner [START_REF] Meijerink | An Iterative Solution Method for Linear Systems of Whih the Coeicient Matrix Is a Symmetric -Matrix[END_REF]. While various strategies have been proposed in the past for obtaining an incomplete LU decomposition, it remains a hallenge to get an incomplete LU decomposition whih works suiciently for any given problem without the need to optimize its parameters [SV00; CDG00; LLB03; MG07; Car09].

Another near-zone preconditioner approah is based on using a sparse approximate inverse (SAI): it is assumed that the inverse can be approximated by Introduction Chapter 1 a sparse matrix up to a certain error (an error measured typically with respect to the Frobenius norm) [Che+01; LZL04; Car+05; CB12; PS14]. he construction of the SAI preconditioner is an optimization problem and because of the use of the Frobenius norm, SAI preconditioners are usually easier to parallelize than incomplete LU decompositions [START_REF] Malas | Incomplete LU Preconditioning with the Multilevel Fast Multipole Algorithm for Electromagnetic Scatering[END_REF].

A third near-zone preconditioner approah is based on an iterative inversion of the near-zone matrix in an inner-outer solver sheme. Compared with the incomplete LU or SAI approahes, no additional setup time is required; this advantage comes at the price of increased computational costs per iteration step [START_REF] Eibert | Iterative Near-Zone Preconditioning of Iterative Method of Moments Electric Field Integral Equation Solutions[END_REF][START_REF] Wiedenmann | he Efect of Near-Zone Preconditioning on Electromagnetic Integral Equations of First and Second Kind[END_REF].

What all these near-zone preconditioning shemes have in common is that it is diicult to assess how the condition number depends on the number of unknowns. Nonetheless, it can be said that, while they usually improve the conditioning, they cannot cure the dense-discretization breakdown in the asymptotic limit → ∞, diferent from shemes whih take into account the underlying mathematical nature of these operators.

he two most popular strategies in this regard are Calderón preconditioning tehniques and hierarhical basis preconditioners. Calderón preconditioning leverages the fact that the EFIE operator applied to itself equals the identity operator plus a compact operator. In other words, it becomes an integral equation of the second kind. hus there is an upper bound for iter independent from . he discretization of the Calderón identity is by no means trivial: a simple multiplication of the system matrix with itself would just square the condition number. Instead an inverse Gram matrix is needed to link the two EFIE operators [START_REF] Steinbah | he Construction of Some Eficient Preconditioners in the Boundary Element Method[END_REF]. he EFIE is commonly discretized with Rao-Wilton-Glisson (RWG) functions; if the second EFIE operator is discretized with RWG basis functions as well, then the Gram matrix is singular [START_REF] Christiansen | A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas[END_REF].

To bypass this problem, diferent strategies were proposed. In [START_REF] Christiansen | A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas[END_REF] the authors resorted to a saddlepoint formulation. he contributions of [Ada04; AC04] split the product of the EFIE operator with itself by considering that the operator is composed of the vector and the scalar potential operator and by discretizing eah of the products of the scalar and the vector potential operator separately. In particular, the vanishing of the term, where the scalar potential operator is applied to itself, is enforced. he contribution [START_REF] Contopanagos | Well-Conditioned Boundary Integral Equations for hree-Dimensional Electromagnetic Scatering[END_REF] follows a similar approah compared with [START_REF] Adams | A Numerical Implementation of a Modiied Form of the Electric Field Integral Equation[END_REF] in the sense that here as well the diferent products of the vector and the scalar potential operator are treated separately, Section 1.c Review of Preconditioning Tehniques where they are simpliied by using Stokes' theorem. [START_REF] Borel | A New Well-Conditioned Integral Formulation for Maxwell Equations in hree Dimensions[END_REF] focuses on the CSIE and uses a localized admitance operator for combining the EFIE with the MFIE. In order to link this operator to the domain of the EFIE operator, projections using the Gram-Shmidt orthogonalization are used, whih increases the computational costs. In [START_REF] Bruno | Electromagnetic Integral Equations Requiring Small Numbers of Krylov-Subspace Iterations[END_REF], the CFIE and CSIE are preconditioned by regularizing the EFIE operator with the vector potential in the context of a Nyström discretization. Similarly to [START_REF] Vico | he Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scatering[END_REF], a smooth surface is required.

What the approahes for the EFIE, mentioned in the previous paragraph, have in common is that open surfaces need a special treatment-if they can be treated at all. Also, signiicant hanges of an existing code are required. With the introduction of Bufa-Christiansen (BC) functions a direct discretization of the Calderón identity became possible [START_REF] Bufa | A Dual Finite Element Complex on the Barycentric Reinement[END_REF] so that this tehnique could be easily integrated into an existing code [And+08].3 he price for the increased comfort is, however, that the number of unknowns is increased by a factor of six since the EFIE is discretized on the barycentrically reined mesh. In addition, (minor) modiications of the existing solver code are still necessary since half-RWGs have to be added on the boundary of open surfaces in order to represent the BC functions. Further issues related to multiply connected geometries are discussed in Chapter 8.

Hierarhical basis preconditioners are usually explicit quasi-Helmholz decompositions similar to a loop-star or loop-tree basis, yet with the diference that the basis itself is ill-conditioned in suh a way that the ill-conditioning of the EFIE is counteracted. he roots of hierarhical basis preconditioners come from wavelet analysis. Since the irst wavelets have been one-dimensional functions (the Haar wavelets as one of the most famous examples [START_REF] Haar | Zur heorie der orthogonalen Funktionensysteme: Erste Miteilung[END_REF]), these methods were irst applied to thin-wire approximations or two-dimensional problems [Alp+93; SL93; GCC95; WC95; Sar97; SS02; Ala+03; Ger+06]. he focus was less on preconditioning, but on compressing the system matrix and accelerating the matrix-vector product [And+05]. In the inite element community, various hierarhical bases were presented for preconditioning diferential operators [START_REF] Yserentant | On the Multi-Level Spliting of Finite Element Spaces[END_REF][START_REF] Dahmen | Wavelet and Multiscale Methods for Operator Equations[END_REF][START_REF] Shwab | Fully Discrete Multiscale Galerkin BEM[END_REF].

One of the key properties of wavelets is that they form a set of orthogonal functions (if only functions of diferent levels are orthogonal to eah other, they are called prewavelets). For preconditioning purposes it turns out that suh an or-

thogonality is desirable if the goal is to cure the dense-discretization breakdown. Early hierarhical bases for the EFIE did not necessarily have this property or only had this property for the non-solenoidal basis [VPV05; BSJ05; VVP07; ATV07]. hey did improve the conditioning compared with a conventional loop-star or loop-tree preconditioner, but when the non-solenoidal basis is complemented with loop or hierarhical loop functions (the later based on Yserentant's hierarhical nodal basis [START_REF] Yserentant | On the Multi-Level Spliting of Finite Element Spaces[END_REF]), the overall system remained ill-conditioned. In [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF], a hierarhical basis preconditioner for the EFIE was presented that yielded a logarithmic growth of the condition number (see also the analysis in [START_REF] Hiptmair | Stable Multilevel Splitings of Boundary Edge Element Spaces[END_REF]).

What all of these approahes for three-dimensional problems have in common is that they are limited to structured meshes. Suh a mesh is obtained by starting from a coarse mesh that captures (suiciently) the geometrical details of the scaterer. Assuming that a triangulation is used, a reined mesh is generated by connecting the midpoints of the edges of the triangles; this so-called dyadical reinement is repeated until the desired resolution is reahed [START_REF] Andriulli | A Multiresolution Approah to the Electric Field Integral Equation in Antenna Problems[END_REF]. For these meshes, existing hierarhical preconditioners can yield a condition number that grows only logarithmically in . When a scaterer is smooth, these approahes would result, in order to be practical, in a loss of geometrical details and thus a possibly large geometrical error can occur. his problem was partially alleviated by a hierarhical basis preconditioner generalized to unstructured meshes [AVV08]-partially, because only the non-solenoidal basis was generalized and so the condition number still grows with O( 1/2 ). Compared with Calderón preconditioning, the implementation is easier (no dual functions are required) and the computational costs are smaller. In the asymptotic limit → ∞, in particular for unstructured meshes, the hierarhical basis preconditioner can, however, not prevent the dense-discretization breakdown. d) Scope and Outline of the hesis T his dissertation presents new paradigms for the preconditioning of the EFIE and the CFIE for solving electromagnetic scatering and radiation problems, that is, we advance and extend the state of the art both in hierarhical basis and in Calderón preconditioning tehniques. To this end, Part I presents Section 1.d Scope and Outline of the hesis the bakground material and equations, whih will be used throughout the thesis: Chapter 2 introduces fundamental concepts from diferential geometry and Sobolev space theory. Chapter 3 covers the electromagnetic theory and equations required to model scatering problems and Chapter 4 discusses the discretization of these equations.

Part II is dedicated to the advancement of hierarhical basis preconditioners. Chapter 5 presents a basis whih is deined on structured and on unstructured meshes. hereby, a condition number is obtained for the EFIE that grows only logarithmically in ; this improves the state of the art [START_REF] Andriulli | Hierarhical Bases for Nonhierarhic 3-D Triangular Meshes[END_REF], where the growth is O � 1/2 � .4 his result is obtained by irst generalizing the Haar basis, both for the primal triangular and for its dual mesh. To precondition the scalar potential part of the EFIE, the divergence terms appearing in its variational formulation are removed by using inverse graph Laplacians. his inversion is performed implicitly by solving the associated linear system iteratively, where for the stabilization algebraic multigrid preconditioners are employed. For the vector potential part of the EFIE, it is shown that an inverse Haar basis transformation matrix must be used. his result is derived by leveraging a discretized scalar Calderón identity that relates the hypersingular to the single layer operator known from electrostatics. As an interesting auxiliary result, we thus get a Haar basis preconditioner for the hypersingular operator, an operator not only used in electrostatics but also, for example, in acoustics.

When a geometry is multiply connected, so called global loops have to be added to the quasi-Helmholz decompositions. hese functions are numerically expensive to construct and/or to apply. 5 In order to use hierarhical basis preconditioners without destroying the leading complexity set by a fast method, Chapter 6 shows how the hierarhical basis can be applied without the need to explicitly ind the global loops. hereby, the complexity of a fast method can be maintained. his result is obtained by developing a framework that allows to combine quasi-Helmholz projectors, whih were irst introduced in [And12a], with the hierarhical basis. he sheme itself is agnostic to the concrete basis and thus can be combined with any hierarhical basis available.

Chapter 7 analyzes the applicability of hierarhical basis preconditioners to the CFIE. It shows that the hierarhial loops, whih cannot cure the dense-discretization breakdown of the vector potential part of the EFIE, can efectively be applied to the CFIE. In addition, we propose a sheme that allows to use the hierarhical basis preconditioners on both structured and unstructured meshes in the context of the combined ield integral equation: by employing quasi-Helmholz projectors, the fact that the CFIE is well-conditioned on the solenoidal Helmholz subspace is used. hese projectors are accompanied by a hierarhical non-solenoidal basis for preconditioning the scalar potential part of the CFIE.

he new hierarhical basis preconditioners are suicient for most applications. Still, in the asymptotic limit → ∞, the condition number would grow to ininity. Part III overcomes this limitation by introducing the reinement-free Calderón multiplicative preconditioner (RF-CMP) that diferent from existing Calderón preconditioners does not require a second discretization of the EFIE operator with dual basis functions. Chapter 8 derives this tehnique for the EFIE. It is based on spectral equivalences between (graph) Laplacians and discretized integral operators (namely the single layer and the hypersingular operator). Particular care was devoted to multiply connected geometries to obtain a formulation that remains stable on these. In contrast to other preconditioners, the resulting system matrix is HPD whih allows the application of the CG method. Diferent from other Krylov subspace methods, the CG method guarantees convergence and has the least computational overhead among the Krylov methods.

Chapter 9 extends this method to the CFIE resulting in a formulation that is well-conditioned and free from spurious resonances. here are several hallenges: irst of all, a direct application of the preconditioner from Chapter 8 is not possible since the part of the preconditioner for the vector potential operator would render the MFIE ill-conditioned. Secondly, it would not yield an HPD system matrix, and thirdly, the formulation would not be stable on multiply connected geometries. he reason is that the MFIE is not entirely well-conditioned since it possesses a static null space associated with the toroidal part of the quasi-harmonic Helmholz subspace [START_REF] Bogaert | Low Frequency Stability of the Mixed Discretization of the MFIE[END_REF]. he new formulation is constructed suh that the preconditioned system is-as in the case of the EFIE-HPD and remains well-conditioned on multiply connected geometries.

Finally, Chapter 10 concludes this thesis and outlines possible future researh directions.

Chapter 2

Mathematical Preliminaries

A vanced mathematical concepts concerning function spaces will be frequently used in the subsequent hapters. he most important results are reviewed here. he reader may ind deinitions of commonly used terms suh as "open" or "closed" in [START_REF] Rudin | Functional Analysis. 2nd ed. International series in pure and applied mathematics[END_REF]. Similarly, we presume that the reader is familiar with concepts suh as normed vector spaces. a) Notation "Physical vectors", that is, vectors that are situated in R 2 or R 3 suh as the electric ield , the position vector , or the surface current density , are writen in a bold, italic, and serif font. We use for non-physical vectors or matrices a bold, italic, and sans-serif font (e.g., A ∈ R × is a matrix), and we distinguish matrices from vectors by using capital leters for matrices and minuscules for vectors (this rule does not apply to "physical vectors", where we follow standard conventions). Scalar integral operators are denoted with a calligraphic font suh as the single layer operator V, and vector integral operators with a bold calligraphic font suh as the EFIE operator T . here is an exception to every rule: this thesis follows the convention used in most textbooks that the space of test functions D() and the Shwarz space S() are denoted with a calligraphic leter. b) Surfaces and Mathematical Operators his thesis deals with the solution of electromagnetic scatering problems by three-dimensional obstacles. Mathematically, suh an obstacle can be considered as a subset ⊂ R 3 . In order to obtain results on the existence and regularity of a Mathematical Preliminaries Chapter 2 solution, this subset cannot be arbitrary. he less regular suh a set is, the more diicult it becomes to obtain rigorous mathematical statements. First of all, we require that is a domain. Since the focus is on the modeling of realistic scatering problems, it is presupposed in the following that is a bounded domain. In order to deine function spaces, in whih we can seek for a solution of the scatering problem, we need to make statements on the regularity of the surface. 6To impose more constraints on , we need to introduce the classical -function spaces. As a concise way of denoting derivatives, we deine a derivative operator and use the multi-index notation: a vector α = ( 1 , 2 , … , ) for ∈ N with ∈ N 0 is called a multi-index. Its absolute value is|α| = 1 + … + . hen we deine the derivative operator

D α ∶= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ � ∂ ∂ � 1 � ∂ ∂ � 2 () , for = 2 , � ∂ ∂ � 1 � ∂ ∂ � 2 � ∂ ∂ � 3 () , for = 3 , (2.1)
where, in a slight abuse of notation,

= � (, ) T ∈ R 2 , for = 2 , (, , ) T ∈ R 3 , for = 3 , (2.2)
is the position vector. In addition, we deine7

α ∶= � 1 2 , for = 2 , 1 2 3 , for = 3 . (2.3) Section 2.b
Surfaces and Mathematical Operators he set denotes the space of all continuous functions where is -times continuously diferentiable. We equip this space with the norm

‖‖ () ∶= ∑ |α|≤ sup ∈ � � D α () � � .
(2.4) he space ∞ denotes the space of functions whih have ininitely many continuous derivatives and we refer to these as smooth functions.

For a scatering problem, more relevant than the properties of are the properties of it's boundary: the surface . To describe it, we follow closely the treatment in [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF].

Deinition 2.2 (Surface). Let ⊂ R 3 be a domain. We call the boundary of ,

= ∶= ⧵ (2.5) a (two-dimensional) surface.
Deinition 2.3 (Ball B ()). We deine the -dimensional ball B () ⊂ R with center and radius as

B () ∶= � ′ ∶ � � � -′ � � � < � .
(2.6) Deinition 2.4 (Regular -Surface). Let ⊂ R 3 be a simply connected domain and its surface. Moreover, we assume that we have a covering of , that is, a inite union of open sets with ∈ [0, ] and

∈ ⋃ =0 .
(2.7)

In addition, we require that

∩ 0 = ∅ , (2.8)
whih means that 0 is completely contained inside . A surface is called regular when there exists a difeomorphism8 for eah that maps the set into the unit ball B 3 1 () suh that the ∩ is mapped into the equatorial plane = 0

of the unit ball, ∩ is mapped into the region < 0 of the unit ball, and

(R 3 ⧵ ) ∩ is mapped into the region > 0 of the unit ball. A surface is of class if the difeomorphisms are of class . If ∈ ∞ , then we speak of a smooth surface. he covering of and the difeomorphisms are called atlas and the pairs ( , ) are called harts. If are Lipshiz functions, that is, there is a constant

> 0 suh that � � � -1 () --1 ( ′ ) � � � ≤ � � � -′ � � � for all , ′ ∈ B 3 1 () (2.9)
holds, then is a Lipshiz surface.

Remark. hese harts are not only useful for deining diferentiable operators, but also for introducing Sobolev spaces on surfaces.

If is multiply connected, then more than one set 0 is needed to describe its interior [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF]. Since we are interested in the deinition of surface operators and since their deinition is the same for simply and for multiply connected geometries, we assume without loss of generality that is simply connected (and thereby we do not make the mathematical framework unnecessarily complicated).

his section follows closely the deinitions in [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF]. hus, we avoid to introduce the language of diferential forms and the framework of diferential geometry. In the following, it is assumed that is a bounded domain with a smooth surface. It is possible to extend the deinitions of the mathematical operators to Lipshiz surfaces. While for scalar functions there is only litle diference when dealing with a Lipshiz surface, the treatment of vector ields becomes more complicated since, for example, the surface normal is not deined on corners and edges of the surface. his thesis deals with vector ields and their corresponding function spaces in order to justify the hoice of expansion and testing functions. For this justiication, there is virtually no diference between smooth or Lipshiz surfaces. he objective at the heart of this thesis, however, is the introduction of new preconditioning paradigms. he novel formulations depend on scalar integral equations and by using quasi-Helmholz decompositions we connect them to the electromagnetic integral equations (i.e., vector integral equations). hus the beneit of discussing the general Lipshiz case is limited, in fact, due to the more intricate framework, it might even obfuscate the objective of this thesis. he interested reader may ind more details on this general seting in [START_REF] Costabel | Boundary Integral Operators on Lipshiz Domains: Elementary Results[END_REF][START_REF] Bufa | On Traces for Functional Spaces Related to Maxwell's Equations Part II: Hodge Decompositions on the Boundary of Lipshiz Polyhedra and Applications[END_REF][START_REF] Bufa | On Traces for Functional Spaces Related to Maxwell's Equations Part I: An Integration by Parts Formula in Lipshiz Polyhedra[END_REF][START_REF] Bufa | On Traces for (curl, ) in Lipshiz Domains[END_REF][START_REF] Bufa | Trace heorems on Non-Smooth Boundaries for Functional Spaces Related to Maxwell Equations: An Overview[END_REF].

() = inf ′ ∈ � � � -′ � � � (2.10)
denotes the minimal distance of the point to the surface . We deine the tubular neighborhood of as

= � ∶ () < � . (2.11)
If is small enough, then all ∈ have a unique projection P() onto suh that � � -P() � � = () .

(2.12)

Since it is assumed that is smooth, every point on it admits a tangent plane and the vector -P() is normal to it. Furthermore, we hoose the open cover of suh that , > 0 are completely contained in . We can then deine the unit normal directed to the exterior as

(P()) = grad () , ∀ ∈ c , (2.13)
(P()) = -grad () , ∀ ∈ . (2.14)
where c is the complement of the closure of . he points ∈ can be described using the orthogonal spliting = P() + (P()) , -< < ,

(2.15)

= � +(), ∀ ∈ c , -(), ∀ ∈ . (2.16)
We deine the pieces of surface = ∩ and note that eah is parametrized by a difeomorphism -1 whih maps by deinition from the two-dimensional unit ball B 2 1 () to . Let 1 and 2 be the variables associated with this difeomorphism. hen we have the decomposition

( 1 , 2 , ) = ′ ( 1 , 2 ) + ( 1 , 2 ) , -< < , (2.17)
whih is used throughout this section. his allows to deine a liting for scalar functions deined on : we have a natural extension () into the three-dimensional space surrounding the surface by deining

() = (P()) . ( 2
Δ ∶= div grad = -curl . (2.21)
Given that is deined as the gradient of the distance function and given the vector calculus formula [START_REF]Taschenbuch der Mathematik. 7[END_REF] curl() = grad × + curl

(2.22) we obtain = grad × .

(2.23)

For deining the surface divergence and the surfacic curl, we need a liting operator for vector ields. To this end, we irst introduce the family of parallel surfaces

= � ∶ = ′ + ( ′ ); ′ ∈ � .
(2.24) Deinition 2.8 (Curvature Operator). he curvature operator is deined by

R = grad () .
(2.25)

It is a matrix-valued operator and we set, for notational convenience, R ∶= R 0 .

his allows us to deine the transport of tangential vectors ields into the tubular neighborhood by deining

� () = ( ′ ) -R ()( ′ ) .
(2.26)

Section 2.b Surfaces and Mathematical Operators Deinition 2.9 (Surface Divergence). he surface divergence of a tangent vector ield is

div = div � | (2.27)
Deinition 2.10 (Surfacic Curl). he surfacic curl of a tangent vector ield is

curl = � curl � ⋅ � | (2.28)
Deinition 2.11 (Hodge Operator). he Hodge operator (i.e., the vectorial Laplacian) of a tangent vector ield is = grad divcurl .

(2.29)

Since we have difeomorphisms with associated coordinates 1 and 2 , we can also state more explicit forms of the surface operators. To this end, we need to deine an explicit basis for the tangent plane at eah point ∈ . Suh a basis is given by

1 () = 1 ( ′ , ) = ∂ ∂ 1 + ∂ ∂ 1 = 1 ( ′ ) + R( ′ ) 1 ( ′ ) , (2.30) 2 () = 2 ( ′ , ) = ∂ ′ ∂ 2 + ∂ ∂ 2 = 2 ( ′ ) + R( ′ ) 2 ( ′ ) .
(2.31)

If is a vector in the tangent plane, it can be writen as

= 1 1 + 2 2 .
(2.32) his allows to introduce the metric tensor

= ⋅ , (2.33)
whih links the tangent with the cotangent plane9 . More precisely, the metric tensor can be writen as a two by two matrix, and thus its inverse is -1 . he entries of this matrix are denoted as and we can deine the basis vectors of the cotangent plane as

= 2 ∑ =1 .
(2.34)

hen we have

( ⋅ ) = δ , (2.35)
where δ denotes the Kroneker delta and thus

= 1 1 + 2 2 , (2.36) with = 2 ∑ =1 .
(2.37)

Using these deinitions, explicit formulas of the surface operators are available: the surface gradient of the function is

grad = ∂ ∂ 1 1 + ∂ ∂ 2 2 , (2.38)
and the surface curl is

= 1 � det � ∂ ∂ 2 1 - ∂ ∂ 1 2 � .
(2.39)

We obtain for the surface divergence of a contravariant tangential vector ield

div = 1 � det � ∂ ∂ 1 � det 1 + ∂ ∂ 2 � det 2 � , (2.40)
and the surfacic curl of a covariant tangential vector

curl = 1 � det � ∂ ∂ 1 2 - ∂ ∂ 2 1 � . (2.41)
For the Laplace-Beltrami operator, we obtain the expression

Δ = div grad = -curl = 1 � det � 2 ∑ ,=1 ∂ ∂ � det ∂ ∂ � . (2.42)
hese surfaces operators are related to eah other, in fact, the surface divergence is the adjoint operator of the surface gradient, and likewise the surfacic curl is the adjoint operator of the surface curl.

Section 2.c Sobolev Spaces heorem 2.1 ([Néd01]). Let ∈ 1 ( ) be a function and ∈ � 1 ( ) � 2 a tangent vector ield deined on the surface . We have the following Stokes identities:

∫ � grad ⋅ � d() = -∫ div d() ,
(2.43)

∫ ( ⋅ )d() = + ∫ curl d() , (2.44) div = 0 , (2.45
)

curl grad = 0 , (2.46) div ( × ) = curl .
(2.47)

c) Sobolev Spaces

We have qualiied surfaces by using the classical -function spaces (i.e., by requiring that the difeomorphisms are members of suh a space). Correspondingly, one could presume that we use these function spaces to qualify the solution of the integral equations we are faced with. his, however, is not practical since these function spaces are deined by the strong derivative;10 a irst step to broaden the domain in whih we searh for a solution is to introduce the weak derivative. he function spaces associated with the weak derivative are the Sobolev spaces. What makes the Sobolev spaces we encounter in the description of the scatering and radiation problems particularly suitable is the fact they are Hilbert spaces, that is, they come with an inner product. his inner product corresponds to the physical energy of the solution, and by searhing for a solution in a Sobolev space, we ask for a solution with inite energy [START_REF] Hsiao | Mathematical Foundations for Error Estimation in Numerical Solutions of Integral Equations in Electromagnetics[END_REF], a result one would reasonably expect from a physical point of view. his section follows closely the treatment in [START_REF] Steinbah | Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements[END_REF].

Mathematical Preliminaries

Chapter 2

) Generalized Derivatives, Distributions, and Sobolev Spaces Let be a function ∶ ⊂ R 3 → C. he support of is deined as

supp ∶= � ∈ ∶ () ≠ 0 � . (2.48) he set D() = � ∈ ∞ () ∶ supp ⊂ � (2.49)
is the space of test functions, that is, the space of smooth functions with compact support.

In order to deine the weak derivative, we need function spaces that take into account the integrability of a function; suh spaces are the ()-spaces, whih simply speaking, contain all functions with bounded norm11

‖‖ () ∶= � ∫ � � () � � d () � 1/
, for 1 ≤ < ∞.

(2.50) he 2 -space is a Hilbert space equipped with the inner product

(, ) 2 () = ∫ ()()d () , (2.51)
where () is the conjugate of (). For deining the weak derivative in the broadest possible sense, we also need the space of locally integrable functions

1 loc () = � ∶ ∫ � � () � � d () < ∞ , for all compact subsets ⊂ � . (2.52)
Deinition 2.12 (Weak derivative). A function ∈ 1 loc () is the αth weak derivative of the function if for all test functions ∈ D(), we have

∫ d () = (-1) |α| ∫ D α d () .
(2.53)

We denote the weak derivative as D α ∶= .

Section 2.c

Sobolev Spaces Not all functions possess a weak derivative, one just has to think of the step function. A further generalization of derivatives is possible by resorting to distribution theory. Deinition 2.13 (Topological Dual Space). Let be a topological vector space, that is, a vector space on whih a topology is deined.12 hen the set of all continuous and linear functionals on is denoted as ′ .

Remark. When we speak of a dual space in this thesis, we always mean the topological dual space.

Deinition 2.14 ([Ste10]). A complex valued continuous linear functional acting on D() is called a distribution. he functional is continuous on D() if → in D() always implies ( ) → (). he set of all distributions is denoted by D ′ ().

Any functional

() ∶= ∫ ()()d () for ∈ D() (2.54)
with ∈ 1 loc () is continuous and therefore a distribution. Distributions of the type are called regular. Distributions whih cannot be expressed by suh an integral are called singular. Since any function ∈ 1 loc () can be associated with a distribution , we follow the common practice and write instead of . Deinition 2.15 (Derivatives of Distributions). Let ∈ D ′ () and α be a multi-index. hen the distribution D α ∈ D ′ () is deined as

⟨D α , ⟩ = (-1) |α| ⟨ , D α ⟩ (2.55)
for all ∈ D().

Deinition 2.16 (Shwarz Space, [START_REF] Steinbah | Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements[END_REF]). he space S(R ) is the space of functions

∈ ∞ (R ) satisfying � � � �, ∶= sup ∈R � || + 1 � ∑ |α|≤ � � D α () � � < ∞ for all , ∈ N 0 .
(2.56)

Any function ∈ S(R ) and all of its derivatives decrease faster for|| → ∞ than any polynomial. his ensures that the Fourier transformation is well-deined for any function ∈ S(R ) and we denote its Fourier transform as � , whih we deine as

� ( ) ∶= (F)( ) = (2π) -/2 ∫ R e -i⋅ ()d () for ∈ R .
(2.57)

We could have deined the Fourier transformation also for functions residing in D(R ); the Fourier transformation of these functions, however, does not necessarily reside in D(R ). In contrast, the mapping

F ∶ S(R ) → S(R ) is invertible and the inverse Fourier transformation is [Ste10] � F -1 � � () = (2π) -/2 ∫ R e i⋅ � ()d ( ) for ∈ R . (2.58)
Due to the smoothness of the functions of S(R ), the derivative is well-deined for any multi-index α by

D α � F � ( ) = (-i) |α| F � α � ( ) (2.59) and α � F � ( ) = (-i) |α| F � D α � ( ) .
(2.60)

We denote its dual space S ′ (R ) following Deinition 2.14, that is, it is the space of all linear and continuous complex valued functionals over S(R ). he elements ∈ S ′ (R ) are called tempered distributions, and we deine their Fourier transformation � ∈ S ′ (R ) as

� () ∶= ( � ) for ∈ S(R ) .
(2.61) Also for tempered distributions, the mapping F ∶ S ′ (R ) → S ′ (R ) is invertible and the inverse Fourier transformation is deined as

� F -1 � ∶= (F -1 ) for ∈ S(R ) .
(2.62) he identities (2.59) and (2.60) hold for tempered distributions as well. As a inal ingredient for the deinition of Sobolev spaces, we need the Bessel potential operator, whih is deined by

� J � () ∶= (2π) -/2 ∫ R � 1 + � � � � 2 � /2 � ( )e i⋅ d ( ), ∈ R (2.63)
for ∈ R and ∈ S(R ). he operator J ∶ S(R ) → S(R ) is a bounded and linear operator. hus we have

(J )() ≡ F -1 � � 1 + � � � � 2 � /2 (F)( ) � () .
(2.64)

If we compare this equation with (2.60), we ind that J is a diferential operator of order (and since we allowed to be a real number, we thereby obtain a notion of fractional derivatives). As we did it before with the derivative operator and the Fourier transformation operator, we can generalize the Bessel operator to tempered distributions ∈ S ′ (R ) by deining

� J � () ∶= � J � for all ∈ S � R � . (2.65) he operator J ∶ S ′ (R ) → S ′ (R ) is bounded and linear.
Deinition 2.17 (Sobolev space ). he Sobolev space , ∈ R, is deined as

(R ) ∶= � ∈ S ′ (R ) ∶ ‖‖ (R ) < ∞ � (2.66)
where

‖‖ (R ) ∶= � � � � 2 2 (R ) = ∫ � 1 + � � � � 2 � � � � ( ) � � d ( ) .
(2.67)

Remark. he space is a Hilbert space, where the inner product is deined as

(, ) (R ) ∶= � J , J � 2 (R ) .
(2.68)

We note that we have 0 = 2 . Lastly, we need to deine (), where ⊂ R is some bounded domain. We obtain this space by restriction, that is, we deine

() ∶= � = | ∶ ∈ (R ) � (2.69)
and we equip this set with the norm

‖‖ () ∶= � inf ∈ (R ), |= ‖ ‖ (R ) � (2.70)
If the domain is unbounded (e.g., the exterior scatering scenario), then the appropriate function spaces are loc (), whih are deined in the same fashion as 1 loc ().

Mathematical Preliminaries

Chapter 2

) Sobolev Spaces on Surfaces

For deining the Sobolev spaces on the boundary , we have to remember that we are in the possession of an overlapping piecewise parametrization of , where

= ⋃ =1 , ∶= � ∈ R 3 ∶ = -1 ( ′ ) for ′ ∈ B 2 1 () � . (2.71)
As mentioned in Section 2.b, it is assumed that is smooth. In addition, we introduce the partition of unity of non-negative cutof functions ∈ D(R 3 ) suh that

∑ =1 () = 1 , for ∈ , (2.72) and = 0 , for ∈ ⧵ . (2.73)
For any function deined on , we can deine functions ∶= suh that

() = ∑ =1 () for ∈ . (2.74)
Using the parametrization -1 , we deine

( ′ ) ∶= ( -1 ( ′ )) = () , ∀ ′ ∈ B 2 1 () .
(2.75)

For the parameter domain B 2 1 (), we have the appropriate Sobolev spaces at hand so that

‖‖ 2 ( ) ∶= � ∑ =1 ‖ ‖ 2 2 � B 2 1 () � � 1/2
(2.76) and

‖‖ ( ) ∶= � ∑ =1 ‖ ‖ 2 � B 2 1 () � � 1/2 .
(2.77) hen the spaces 2 ( ) and ( ) contain all functions deined on the surface for whih the norm is bounded.

Section 2.c Sobolev Spaces Equivalent norms 13 , whih are easier to use since the parametrization is not required explicitly, are

‖‖ 2 ( ) ∶= � (, ) 2 ( ) (2.79)
induced by the inner product

(, ) 2 ( ) = ∫ ()()d() , (2.80)
and the Sobolev-Slobodekii norm

‖‖ ( ) ∶= ⎛ ⎜ ⎜ ⎜ ⎝ ‖‖ 2 2 ( ) + ∫ ∫ � � � () -( ′ ) � � � 2 � � -′� � 2+2 d( ′ )d() . ⎞ ⎟ ⎟ ⎟ ⎠ 1/2
(2.81)

If is not smooth, the maximum number of allowed derivatives depends on . If ∈ -1,1 , then|| ≤ . In particular for Lipshiz surfaces, we have|| ≤ 1.

) Vector Sobolev Spaces So far we have deined all the Sobolev spaces, whih we would need to model electrostatic or acoustic problems, that is, problems that can be described with scalar functions. For electromagnetic problems, however, one is faced with vector functions and we need appropriate Sobolev spaces to accommodate these functions. We deine

() ∶= � () � 3 ∶= � ∶ () ∈ () , = 1, 2, 3 � (2.82)
equipped with the norm

‖‖ () ∶= � 3 ∑ =1 � � () � � 2 () .
(2.83)

13 Two norms‖‖ and‖‖ are called equivalent if there are constants and suh that ‖‖ ≤ ‖‖ ≤ ‖‖ .

(2.78)

All norms deined on a inite-dimensional space are equivalent, but for an ininite space, this is generally not true. And as a side note: while systems resulting from a discretization are inite-dimensional, the situation becomes interesting when the discretization is reined so that the dimensionality grows. While for a certain discretization all norms are equivalent, we might not be able to ind constants independent from the discretization density.

Mathematical Preliminaries

Chapter 2 his deinition readily extends to the case of ( ), where is a surface, and in particular we note that we have for 0 ( ) = 2 ( ) the norm

‖‖ 2 ( ) ∶= � (, ) 2 ( ) (2.84)
induced by the inner product

(, ) = ∫ () ⋅ ()d() .
(2.85) he spaces containing tangential vector ields are t ( )

∶= � ∈ ( ) ∶ ⋅ = 0 � (2.86)
where is a smooth surface. Furthermore, we need the spaces

(div , ) = � ∈ t ( ) ∶ div ∈ ( ) � (2.87)
and

(curl , ) = � ∈ t ( ) ∶ curl ∈ ( ) � (2.88)
equipped with the norms

‖‖ (div , ) ∶= ‖‖ () +‖div ‖ () ,
(2.89)

‖‖ (curl , ) ∶= ‖‖ () +‖curl ‖ () .
(2.90)

(2.91)

We note that the spaces (div , ) and (curl , ) are dual with respect to the 2 -inner product [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF].

Chapter 3

Electromagnetic heory and Integral Equation Formulations of Scatering Problems M axwell's eqations describe the scenario whih shall be solved: the scatering of electromagnetic waves from PEC obstacles in frequency domain. his hapter reviews the bakground material: Section 3.a introduces the necessary electromagnetic theory and derives the mixed potential formulas, whih allow to compute the electromagnetic ield excited by electric and magnetic current distributions. In Section 3.b, it is shown how these formulas can be used to model a scatering scenario. his hapter is closed by briely discussing the electrostatic case in Section 3.c and by showing how integral operators can be used to obtain a solution for Laplace's equation.

a) Maxwell's Equations and heir Solution

Since the focus is on time-harmonic problems, we assume and suppress the timedependency exp(-i) in the following, where is the time and the angular frequency. How the electric current density and harge distribution e excite the electric ield and the magnetic ield and how these ields interact is described by Maxwell's equations [START_REF] Harrington | Time-Harmonic Electromagnetic Fields[END_REF]

curl = i , (3.1) curl = -i + , (3.2)
Electromagnetic heory and Integral Equation Formulations

Chapter 3 div = e , (3.3)

div = 0 , (3.4)
where is the electric and the magnetic lux density. he current density and the harge distribution e are not independent since the harge must be conserved, that is, they must satisfy 14 div = i .

(3.5) he luxes and are related to and by the constitutive relations 15

= (, ) , (3.7) = (, ) , (3.8)
that is, the relationship between the ields can be non-linear, (bi-)anisotropic, and space and frequency dependent. Due to the scope of this thesis, it suices to consider piecewise homogeneous and isotropic materials, where the diference between the ields is just a multiplicative factor, so that the constitutive relations read = = r 0 , (3.9)

= = r 0 , (3.10)
where is the permitivity and is the permeability, r is the relative permitivity and r is the relative permeability, and 0 is the electric and 0 the magnetic constant. If or are piecewise constant, Maxwell's equations do not hold (in a classical sense) on the material boundary since the ields are not continuous anymore. Instead, we need to introduce continuity conditions at the interfaces whih the ields must satisfy. hese conditions are discussed in the next section. While magnetic harges and currents do not exist, or at least have not been observed in nature so far, their introduction to Maxwell's equations provides a we can write Maxwell's equations as

curl = i -, (3.12) curl = -i + , (3.13) div = e , (3.14) div = m , (3.15)
where

= 2π/ (3.16) = √ (3.17) = 2π √ (3.18)
is the wavenumber, is the wavelength, and is the frequency. In other words, if classic and classic are the classically deined magnetic ield and the electric current density, we deine new = classic and new = classic and insert 1/ new and 1/ new into Maxwell's equations. Using this normalized system, the electric and magnetic ield have the same unit, whih simpliies the formalism.

) Continuity Conditions

Let ⊂ R 3 be a domain with a smooth boundary . he ields are subject to the continuity conditions [Har01]

× + -× -= -, (3.19) × + -× -= , (3.20) ⋅ + -⋅ -= ,e , (3.21)
⋅ + -⋅ -= ,m , (3.22)
Electromagnetic heory and Integral Equation Formulations

Chapter 3

where we used the deinition

+ () = lim ′ → () , ∀ ′ ∈ c , (3.23) -() = lim ′ → () , ∀ ′ ∈ , (3.24)
and is the surface unit normal vector directed to the exterior. he symbol in the subscript of the currents and harges indicates that these are surface densities. hese densities are rather a theoretical construct and for physical applications typically zero. If one considers, however, idealized materials suh as perfect electric or magnetic conductors, then the right-hand sides do not vanish. In this thesis, describes a PEC object and thus the continuity equations simplify to

× + = , (3.25)
× + = , (3.26)
⋅ + = ,e , (3.27)
⋅ + = 0 .

(3.28)

) Equivalence Principle

Diferent source distributions can lead to the same ields, a circumstance that is the fundamental idea of the equivalence principle. Let and be source distributions inside and and the generated ields. If we assume the existence of surface current densities ) Electromagnetic Potentials, Green's Function, and Mixed Potential Formulas he introduction of electromagnetic potentials provides an elegant means to solve Maxwell's equations since instead of solving the four Maxwell's equations it suices to solve two vector Helmholz equations. To obtain the potential approah, we leverage the linearity of Maxwell's equations: we split the electric and magnetic ield into

= -× , ∀ ∈ , (3.29) = × , ∀ ∈ , ( 3
= e + m , (3.35)
= e + m , (3.36)
where the ields e and e are due to electric and m and m due to magnetic sources. his spliting of the ields is justiied by the linearity of Maxwell's equations.

In the absence of magnetic sources, the magnetic ield is solenoidal, that is, div = 0 .

(3.37) herefore, we can ind a vector potential e suh that [START_REF] Harrington | Time-Harmonic Electromagnetic Fields[END_REF] e = curl e .

(3.38)

Inserting this into (3.12) yields curl( ei e ) = , (3.39) whih means that the total vector ield ei e is conservative. Hence, there must be a scalar potential suh that ei e =grad e .

m = i m -1/(i) grad div m .
(3.48) Summarizing, we have

= i e -1/(i) grad div e -curl m , (3.49)
= curl e + i m -1/(i) grad div m .

(3.50)

A solution to (3.44) and to (3.47) can be obtained by convolving the Green's function [START_REF] Harrington | Time-Harmonic Electromagnetic Fields[END_REF]

(, ′ ) = e i � � � , ′ � � � 4π � � , ′� � (3.51) Section 3.a
Maxwell's Equations and heir Solution with the current densities resulting in

e () = ∫ R 3 (, ′ )( ′ )d ( ′ ) (3.52) and m () = ∫ R 3 (, ′ )( ′ )d ( ′ ) .
(3.53) herefore, we ind for the electric ield

() = i ∫ R 3 (, ′ )( ′ )d ( ′ ) -1/(i) grad div ∫ R 3 (, ′ )( ′ )d ( ′ ) -curl ∫ R 3 (, ′ )( ′ )d ( ′ ) , (3.54)
and for the magnetic ield

() = i ∫ R 3 (, ′ )( ′ )d ( ′ ) -1/(i) grad div ∫ R 3 (, ′ )( ′ )d ( ′ ) + curl ∫ R 3 (, ′ )( ′ )d ( ′ ) . (3.55)
hese equations for and are called mixed potential formulas.

If and are surface current densities located on a closed smooth surface , then (3.52) and (3.53) hold for all ∉ so that we may write

() = i ∫ (, ′ ) ( ′ )d( ′ ) -1/(i) grad div ∫ (, ′ ) ( ′ )d( ′ ) -curl ∫ (, ′ ) ( ′ )d( ′ ) , (3.56)
and for the magnetic ield Since for the remainder of this thesis all currents appearing will be surface currents, we omit the subscript . For the limiting case that approahes , one inds for the (rotated) tangential component of the electric and the magnetic ield 16

() = i ∫ (, ′ ) ( ′ )d( ′ ) -1/(i) grad div ∫ (, ′ ) ( ′ )d( ′ ) + curl ∫ (, ′ ) ( ′ )d( ′ ) .
× ± = T + (∓I/2 + K) ,
(3.58)

× ± = (±I/2 -K) + T , (3.59)
where I is the identity operator,

K = -× curl ∫ (, ′ )( ′ )d() , (3.60)
and

T ∶= iT A + 1/(i)T Φ (3.61)
is the EFIE operator composed of the vector potential operator 17

T A ∶= × ∫ (, ′ )( ′ )d( ′ ) (3.62)
and the scalar potential operator 18

T Φ = -× grad ∫ (, ′ ) div ′ ( ′ )d( ′ ) . (3.64)
he derivation is rather cumbersome and we want to spare the reader the details. he interested reader may ind it in, for example, [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF].

In this thesis, we distinguish operators that map from a function space deined on to a function space deined on c and/or from those that map from and to a function space deined on . he irst type of operator will be called potential while the second type will be called potential operator. For example, � (, ′ )( ′ )d( ′ ) in (3.56) is the vector potential and T A the vector potential operator. In some cases, we will omit the expression "potential" in "potential operator" for the sake of brevity. For example, instead of speaking of the single layer potential operator, we speak of the single layer operator.

Where we implicitly used that

∫ div � (, ′ )( ′ ) � d( ′ ) = ∫ grad (, ′ )( ′ )d( ′ ) = -∫ grad ′ (, ′ )( ′ )d( ′ ) (2.43) = ∫ (, ′ ) div ′ ( ′ )d( ′ ) . (3.63) Section 3.b
Scatering by or Radiation from a PEC Object b) Scatering by or Radiation from a PEC Object his thesis deals with the scatering of electromagnetic of waves by or the radiation from a PEC object described by the domain ⊂ R 3 embedded in a homogeneous medium with permitivity and permeability . For a scatering scenario, the tuple ( i , i ) describes an incident time-harmonic electromagnetic wave impinging on , and the scatered wave, whih is the quantity to solve for, is described by the tuple ( s , s ). For a radiation scenario, it is typically assumed that

� � � i � � � = 0 / (3.65)
between two antenna terminals, where is the width of the gap and 0 the voltage [START_REF] Gibson | he Method of Moments in Electromagnetics[END_REF].

In the exterior region c , the total ields (, ) = ( i + s , i + s ) are subject to the Maxwell's equations

∇ × = +i , (3.66)
and In the following sections, we discuss surface integral equation formulations commonly used in literature.

∇ × = -i , (3.67

Electromagnetic heory and Integral Equation Formulations

Chapter 3

) Electric Field Integral Equation For PEC objects, the tangential component of = i + s on must vanish. Equation (3.19) implies that = and using (3.58), we obtain the EFIE

-× i = T . (3.71)
Once this equation is solved for , (3.56) and (3.57) may be used to compute s and s everywhere in c

. We note that T ∶ -1/2 (div , ) → -1/2 (div , ) is invertible if 2 is not an interior electric eigenvalue [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF].

So far we have assumed that is a smooth surface. his is a severe limitation since we could not even make a statement on suh "simple" problems as the scatering from a cube. Fortunately, the presented theory can be generalized to Lipshiz surfaces. his, however, renders the mathematical theory more complex given that is not deined on the edges and corners of a Lipshiz surfaces. A modiied version of -1/2 (div , ) must be introduced whih then allows to obtain a generalized Gårding inequality whih implies existence of a unique solution of (3.71) [START_REF] Bufa | Galerkin Boundary Element Methods for Electromagnetic Scatering[END_REF]. To obtain this result, extensive preliminary work was necessary [Cos88; McL00; Buf01; BC01b; BC01a; BCS02a; BCS02b; Buf03]. Open surfaces (i.e., Lipshiz screens) require a further extension of the theory, and so do T-junctions. In this thesis, however, we obtain theoretical results only for the case that is closed, and hence, we do not present the mathematical theory of these problems.

) Magnetic Field Integral Equation

As an alternative to the ansaz for the EFIE, we can leverage on (3.26), that is,

× s = -× i , resulting in × i = (I/2 + K) , (3.72)
and in the case of an excitation located in (interior problem), we ind the MFIE

× i = (I/2 -K) . (3.73)
As abbreviation, we introduce the exterior MFIE operator

M + = I/2 + K (3.74) Section 3.b
Scatering by or Radiation from a PEC Object and the interior MFIE operator

M -= I/2 -K . (3.75)
whih allows as to compactly write

× i = M ± . (3.76)
Since this thesis focuses on scatering problems, we only deal with the exterior problem and hence for the sake of simplicity we omit the plus symbol and use M for the exterior MFIE operator, though the theory presented in this thesis holds for the interior problem as well. We note that M ∶ -1/2 (div , ) → -1/2 (div , ) is invertible if 2 is not an interior electric eigenvalue and similarly to the EFIE the MFIE can be solved on Lipshiz surfaces [START_REF] Bufa | Galerkin Boundary Element Methods for Electromagnetic Scatering[END_REF].

) Combined Field Integral Equation

For interior electric eigenvalues 2 both the EFIE and the MFIE do not possess a unique solution due to interior resonances. hese interior resonances are unphysical (i.e., they cannot be excited by an exterior source). he classic approah in the engineering community is to form the CFIE [MH78]

C = -C × × i + (1 -C ) × i (3.77)
where

C = C × T + (1 -C )M (3.78)
is the CFIE operator with C ∈]0, 1[. In particular for Lipshiz surfaces, it has not been shown that the CFIE has a unique solution. To overcome this issue regularized CFIEs have been proposed in the past [BH04; ES07], though only in the case of [BH04] also the numerical solvability has been discussed. hese approahes, however, have never been adopted widely. In the case of the approah in [START_REF] Bufa | A Coercive Combined Field Integral Equation for Electromagnetic Scatering[END_REF], a reason might be that the edges of the geometry should be known to the solver. 20 Since no numerical evidence has ever been presented that would

Electromagnetic heory and Integral Equation Formulations

Chapter 3

imply that the CFIE admits non-unique solutions, we use the standard CFIE denoted in (3.77).

c) Electrostatics: Laplace's Equation and Integral Equation Formulations

In this thesis, we will frequently encounter integral operators known from the solution of Laplace's equation. his equation is obtained by considering that for = 0 the electric ield is conservative. his allows to express the electric ield as the gradient of a scalar potential, that is, =grad e . Combining =grad e together with (3.3) and (3.9) and assuming e = 0, we yield Laplace's equation on , where

Δ e = 0 , (
V ∶ -1/2 ( ) → 1/2 ( ) ∶= ∫ 0 (, ′ )( ′ )d( ′ ) , (3.84)
is the single layer (potential) operator, and we ind the electric potential e () = (S)(), where

S ∶ -1/2 ( ) → 1 loc� c � ∶= ∫ 0 (, ′ )( ′ )d( ′ ) , (3.85)
Section 3.c

Electrostatics: Laplace's Equation and Integral Equation Formulations is the single layer potential. 21For the Neumann problem, we assume that satisies the compatibility condition ∫ ()d() = 0 .

(3.86) By using the hypersingular operator

W ∶ 1/2 ( )/R → -1/2 ( ) ∶= ∫ ∂ 2 ∂ ∂ ′ 0 (, ′ )( ′ )d( ′ ) (3.87)
and solving

W = - (3.88)
the electric potential can be computed as e () = (D)(), where

D ∶ 1/2 ( )/R → 1 loc� c � ∶= ∫ ∂ ∂ ′ 0 (, ′ )( ′ )d( ′ ) (3.89)
is the double layer potential and 1/2 ( )/R is a quotient space22 , whih is necessary to guarantee a unique solution of (3.88) due to the null space of W. We note that the integral in (3.87) has to be understood in the Cauhy principal value sense. For practical implementations this is not relevant since one would solve the variational formulation [Ste10; Néd01]

(, W) 2 ( ) = (, ) 2 ( ) (3.90)
for all ∈ 1/2 , where we have

(, W) 2 ( ) = ∫ ∫ ( ′ ) ⋅ () 0 (, ′ )d( ′ )d() . (3.91)
he operators V and W satisfy the so-called Calderón identities

VW = I/4 -K 2 (3.92)
Electromagnetic heory and Integral Equation Formulations

Chapter 3

and

WV = I/4 -K ′2 (3.93)
where

K ∶ 1/2 ( ) → 1/2 ( ) ∶= ∫ ∂ ∂ ′ 0 (, ′ )( ′ )d( ′ ) (3.94)
is the double layer operator,

K ′ ∶ -1/2 ( ) → -1/2 ( ) ∶= ∫ ∂ ∂ ′ 0 (, ′ )( ′ )d( ′ ) (3.95)
is the adjoint double layer operator, and I is the identity operator.

Chapter 4

Discretization of Boundary Integral Operators and Equations N umerical formulations of the integral equations (3.71), (3.76), and (3.77) are necessary to ind solutions for non-canonical scatering problems. his hapter introduces the classical numerical formulations based on Galerkin theory. First, it discusses this theory and possible basis functions. hen discretized counterparts of (3.71), (3.76), and (3.77) are derived. Lastly, it analyzes the ill-conditioning of the system matrices resulting from these discretizations and shows partial remedies in form of quasi-Helmholz decompositions and projectors.

a) Petrov-Galerkin heory

Except for a few canonical objects (e.g., if is a sphere), analytic solutions are not known for (3.71), (3.76), and (3.77), so one must resort to a numerical approximation and discretize these equations. Diferent approahes are available suh as the Nyström or collocation method. In this thesis, we use the Petrov-Galerkin method, whih can guarantee that the numerical solution converges to the analytical solution.

For describing the Petrov-Galerkin method, we consider the equation

A = , (4.1)
where A ∶ → ′ is a bounded linear operator and and are Hilbert spaces. We can associate with A a bilinear form

∶ × → R that satisies [Ste10] � � (, ) � � ≤ ‖‖ � � � � (4.2)
� � =1 ⊂ with dim( ) = dim( ) = . he task is to ind ∈ suh that ( , ) = ⟨ , ⟩ ′ × (4.7)
for all ∈ . his variational formulation in (4.7) is equivalent to

Au = f (4.8)
where

� A � = � , A � 2 (4.9)
and

� f � = � , � 2 . (4.10)
Even though the inf-sup conditions for the analytic problem are satisied, this does not imply that we can expect to obtain a unique solution of (4.7). In addition, 23 Equations (4.5) and (4.6) can also be expressed as

ALBB ‖‖ ≤ sup 0≠∈ (, ) ‖‖ , ∀ ∈ (null A) ⟂ ⊂ . (4.4)
Equation (4.5) implies that must not be in the null space of A.

24 In literature, this is also known as the Ladyzhenskaya-Babuska-Brezzi (LBB) condition.

Section 4.b

Basis Functions a discrete inf-sup condition must be satisied, that is, we must have [Bab71; Bre74; XZ03]

sup ∈ � � (, ) � � > 0 , ∀ ≠ 0 , (4.11) inf ∈ sup ∈ (, ) ‖‖ ‖‖ ≥ DLBB > 0 , (4.12)
where DLBB is a positive constant. If this condition is satisied, then the solution is unique. If, in addition, the trial space satisies the approximation property

lim →∞ inf ∈ ‖ -‖ = 0 , ∀ ∈ , (4.13)
then we have → for → ∞, that is, the approximation converges to the analytical solution.

To correctly apply the Petrov-Galerkin method, it is decisive that the testing is always performed in the dual space of the range of A, where we notice that ′′ = when is a Hilbert space. As discussed in the next section, this requirement has been ignored in the case of the discretization of the MFIE in the past.

b) Basis Functions

For the numerical solution, it is assumed that is a Lipshiz polyhedral surface. If instead is a smooth surface, then we would need to approximate the surface by a polyhedral surface. We presume that this preprocessing step has happened at this point. he polyhedral surface is discretized with a mesh of triangular cells. For the discretization of the scalar operators V and W, we use piecewise constant functions25 ∈ ⊂ -1/2 ( ) and piecewise linear functions26 ∈ ⊂ 1/2 ( ). he piecewise constant functions are deined as where denotes the domain of the th cell of the mesh and its area. he piecewise linear functions are deined as

() = � 1/ , ∈ ,
() = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 for ∈ , 0 for ∈ ≠ , linear elsewhere, (4.15)
where ∈ is the th vertex of the mesh.

For the discretization of vector integral operators, we need RWG functions ∈ (that are equivalent to the zeroth order Raviart-homas functions on a manifold [RWG82; RT77]), whih we deine as

= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ -+ 2 + for ∈ + , - - 2 - for ∈ - (4.16)
following the convention depicted in Figure 4.1, where the vector denotes the th directed edge, + and - denote the domains of the cells on whih has its support, + and - denote the vertices on the edge , and + and - are the vertices opposite to the edge . In a slight abuse of notation, + and - denote the cell area of + and - , respectively. Diferent from [RWG82], we do not normalize by the edge length. We note that ⊂ -1/2 (div , ): the RWG functions are divergence conforming; the application of the divergence operator on RWG functions is well-deined in the sense that unphysical line harges cannot appear. For the rotated × ∈ × on the other hand, we have the property × ⊂ -1/2 (curl , ). Oten we need to discretize the identity operator I (or I, depending on the context). he resulting matrix is called Gram matrix. 27 In order to avoid deining the Gram matrix for diferent basis functions every time, we stik to the following convention: let ∈ and ∈ be functions of the function spaces and , respectively; the Gram matrix of these functions is deined and denoted as

� G � ∶= � , � 2 .
(4.17)

Naturally, this deinition extends to the case that functions are vector-valued. When the 2 -space is hosen as pivot space, the space -1/2 (curl , ) is the dual space of -1/2 (div , ) and vice versa. Unfortunately, and × are not dual to eah other in the sense that they do not satisfy inf-sup conditions. his led to the development of the BC basis functions ∈ . hey are divergence conforming just as the RWG functions, but they allow to obtain a well-conditioned mixed Gram matrix G × , , whih means that inf-sup conditions are satisied [START_REF] Bufa | A Dual Finite Element Complex on the Barycentric Reinement[END_REF]. 28

Discretization of Boundary Integral Operators and Equations Chapter 4 c) Discretization of the Field Integral Equations

In order to apply the Petrov-Galerkin method to the EFIE, we need to obtain a bilinear form associated with T . Suh a variational formulation is given by [Rum54]

� × , T � 2 ( ) ∶= i ∫ () ⋅ ∫ (, ′ )( ′ )d( ′ )d() + 1 i ∫ div () ∫ (, ′ ) div ′ ( ′ )d( ′ )d() , ∀ ∈ -1/2 (div , ), × ∈ -1/2 (curl , ) , (4.18)
where we used (2.43). hus we obtain as Petrov-Galerkin discretization of (3.71)

T j = -e , (4.19)
where

T = iT A + 1/(i)T Φ (4.20) with � T A � = � × , T A � 2 ( ) (4.21) = ∫ () ⋅ ∫ (, ′ ) ( ′ )d( ′ )d() , (4.22) � T Φ � = � × , T Φ � 2 ( ) (4.23) = ∫ div () ∫ (, ′ ) div ′ ( ′ )d( ′ )d() , (4.24
)

� e � = � × , × i � 2 ( ) , (4.25)
where the electric current density is approximated by

≈ ∑ =1 � j � . (4.26)
An important requirement of the Petrov-Galerkin theory is that both analytic and discrete inf-sup conditions must be satisied. hat this requirement is satisied Section 4.c

Discretization of the Field Integral Equations has been shown even for the case where is a Lipshiz surface [START_REF] Bufa | Galerkin Boundary Element Methods for Electromagnetic Scatering[END_REF]. For notational convenience, we will in general omit the superscript of T . he computation of the matrix elements of T A and T Φ is by no means trivial because of the singularity of the Green's function. If and ′ are suiciently far away from eah other, then Gaussian quadrature can be used for integration [START_REF] Gauss | Methodus nova integralium valores per approximationem inveniendi[END_REF][START_REF] Dunavant | High Degree Eicient Symmetrical Gaussian uadrature Rules for the Triangle[END_REF]. In the case that Gauss quadrature is not applicable, the singularity extraction method is employed in this thesis [START_REF] Graglia | On the Numerical Integration of the Linear Shape Functions Times the 3-D Green's Function or Its Gradient on a Plane Triangle[END_REF].

For the MFIE, the situation is slightly diferent. Classically, the bilinear form

(, M) 2 ( ) ∶= 1/2 ∫ () ⋅ ()d() -∫ () ⋅ × ∫ grad (, ′ ) × ( ′ )d( ′ )d() , ∀ ∈ -1/2 (div , ), ∈ -1/2 (div , ) (4.27)
has been used. Employing as expansion and testing functions, we obtain the system

M j ∶= � G /2 + K � j = h , (4.28)
where

� G � = � , � 2 ( ) = ∫ () ⋅ ()d() , (4.29) � K � = � , K � 2 ( ) (4.30) = -∫ () ⋅ × ∫ grad (, ′ ) × ( ′ )d( ′ )d() , (4.31) � h � = � , × i � . (4.32)
Suh a discretization approah is, however, not conforming with Petrov-Galerkin theory. Just as in the case of the EFIE, the testing functions should belong to Chapter 4 case, the bilinear form reads

� × , M � 2 ( ) ∶= 1/2 ∫ × () ⋅ ()d() -∫ () ⋅ ∫ grad (, ′ ) × ( ′ )d( ′ )d() , ∀ ∈ -1/2 (div , ), × ∈ -1/2 (curl , ) . (4.33)
Using as expansion and × as testing functions, we obtain the system

� M j ∶= � G × , /2 + � K � j = h , (4.34)
where

� G × , � = � × , � 2 ( ) = ∫ × () ⋅ ()d() , (4.35)
� � K � = � × , K � 2 ( ) (4.36) = -∫ () ⋅ ∫ grad (, ′ ) × ( ′ )d( ′ )d() , (4.37)
� h� = � × , × i � . (4.38)
We refer to � M as conformingly discretized MFIE. he classical discretization of the CFIE follows immediately as

Cj = -C e + (1 -C )h , (4.39)
where

C ∶= C T + (1 -C )M . (4.40)
For the conformingly discretized system, however, we have

� Cj = -C e + (1 -C )G G -1 × , h , (4.41)
where

� C = C T + (1 -C )G G -1 × , � M . (4.42) Section 4.d
On the Ill-Conditioning of System Matrices he Gram matrices are necessary to ensure that the CFIE is consistently tested [START_REF] Beghein | A Space-Time Mixed Galerkin Marhing-on-in-Time Sheme for the Time-Domain Combined Field Integral Equation[END_REF]. Lastly, we need to discretize the scalar operators V and W. Here, we obtain the system matrices

� V � = � , V � 2 ( ) (4.43)
and

� W � ∶= � , W � 2 ( ) (4.44)
where for the later we use the variational formulation of (3.91).

d) On the Ill-Conditioning of System Matrices

We have pointed out in Chapter 1 that equations suh as the discretized EFIE in (4.19) are typically solved with an iterative solver belonging to the family of Krylov subspace methods. he overall computational complexity in time then depends on the fast method used (a fast method suh as the MLFMM or ACA) and the number of iterations required by the solver to converge.

If we consider the system Ax = b where A is a symmetric, positive deinite matrix and if we apply the CG method, we have the upper bound for the number of iterations

iter ≤ ⌈ 1 2 √ cond A log 2 ⌉ (4.45)
with the condition number

cond A = ‖A‖ � � � A -1 � � � = max (A) min (A) , (4.46)
where max (A) is the largest singular value and min (A) the smallest singular value of A, and where is the relative residual error that serves as stopping criterion, that is, if‖Ax -b‖ ≤ ‖b‖, then the CG solver stops. Clearly, if cond A is large, then the CG solver will require many iterations to converge and one would say that A is ill-conditioned. here are many factors that can contribute to a large condition number, for example, a complicated geometry or a distorted mesh [START_REF] Steinbah | Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements[END_REF]. hese efects are, however, not on the focus of this thesis. Instead, the main atention is on the question how the condition number behaves if the mesh is reined, that is, the average edge length ℎ → 0.

Discretization of Boundary Integral Operators and Equations

Chapter 4

Why can a decreasing ℎ lead to an ill-conditioned system? For a symmetric, positive deinite matrix, eigenvalues and singular values as well as eigenvectors and singular vectors coincide. If an operator with eigenvalues growing to ininity or accumulating at zero is discretized, then it comes at no surprise that suh a behavior is observable in the corresponding system matrix: when the mesh is reined, the ininite spectrum of the respective operator is beter captured and thus also the system matrix has growing or decreasing eigenvalues, at least under the condition that an 2 -stable basis is used, that is, the Gram matrix of this basis is well-conditioned.

By introducing the weak and distributional derivative, the concept of derivatives was generalized. Closely linked to this generalization are pseudo-diferential operators. he Laplace-Beltrami operator is a pseudo-diferential operator of order +2, while the hypersingular operator W is of order +1, and the single layer operator V is of order -1. Operators of negative order have eigenvalues clustering around zero, while operators of positive order have eigenvalues growing to ininity. he rate at whih they grow or decrease corresponds to the order of the operator.

In general, if an 2 -stable basis is used for the discretization of an operator with non-zero order, then the resulting matrix is ill-conditioned. For integral operators, two of the most popular remedies are either the use of a basis, whih is itself 2 -unstable, these are the hierarhical basis preconditioners, or to use an operator of opposite order as preconditioner. For a hierarhical basis preconditioner, it is typical that these bases are constructed on classical function spaces suh as or . his allows to reuse the standard discretization and to apply the basis as a preconditioner in form of a transformation matrix. hat is, instead of solving the system

Ax = b (4.47)
we solve a system of the form

B T AB y = B T b (4.48)
so that in the best case

cond � B T AB � ≲ 1 (4.49)
would hold, where the expression ≲ means that there is a constant , independent of the average edge length ℎ, suh that ≤ holds. In addition, we will frequently use ≍ meaning that both inequalities ≲ and ≲ hold.

Section 4.d

On the Ill-Conditioning of System Matrices he approah to use an operator of the opposite order can be motivated by the Calderón identities, whih we introduced in (3.92) and (3.93). By applying V to W (or vice versa), an integral equation of the second kind is obtained, that is, the identity operator plus a compact operator. Since the eigenvalues of a compact operator cluster around zero, the eigenvalues I/4 -K 2 accumulate at 1/4. When an iterative solver suh as the CG method is used, the number of iterations to converge are bounded from above with a constant independent of the discretization density [SW98; Hip06; AK01]. In the case of the Calderón preconditioners, we solve a system of the form

P Ax = P b (4.50)
and have

cond(P A) ≲ 1 . (4.51)
he preconditioner in (4.48) is referred to as split preconditioner, and the one in (4.51) as let preconditioner. In order to ind new preconditioners, we will frequently use Rayleigh quotients. If A ∈ R × is symmetric, positive deinite, then we have For our analysis, we will use the notation

max (A) = max x x T Ax x T x (4.
min (A) x T x ≤ x T Ax ≤ max (A) x T x , ∀x ∈ R . (4.54)
If, for example, the inequality

x T Ax ≍ x T F -1 x , ∀x ∈ R . (4.55) holds, then cond(F A) ≲ 1 (4.56)
follows. 29 he EFIE is not directly applicable to suh an analysis since it is not a symmetric, positive deinite matrix. his is mainly due to the frequency 29 Consider that in this case also

x A 2 x ≍ x T F -2 x , ∀x ∈ R (4.57)
Chapter 4 dependency of the Green's function (, ′ ). We are not, however, able to consider the case → 0 due to the 1/ scaling of the scalar potential operator part of the EFIE. his observation, in fact, leads to another form of ill-conditioning particular to the EFIE: the low-frequency breakdown.

A close look at (4.20) reveals that T is ill-conditioned in , that is, the condition number of T grows as 1/ 2 for → 0: the scalar potential operator T Φ has a null space spanned by a solenoidal subspace of the RWG functions. Hence, the spectrum of T consists of two branhes, the spectrum of T Φ and of T A . Due to the scaling of the vector and the scalar potential operator with 1/ and , respectively, the two branhes are driven apart leading to the ill-conditioning.

he ℎ-dependency of the condition number T is more diicult to establish than the frequency dependency. In this case it is helpful to use a (quasi-)-Helmholz decomposition to make the transition to scalar quantities and to consider the contribution from T A separately from T Φ so that well-established inverse inequalities can be used (see, for example, [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF].) A relatively general framework for assessing the ill-conditioning of an operator was shown in [START_REF] Kirby | From Functional Analysis to Iterative Methods[END_REF]. Summarizing, one obtains for the EFIE

cond(T ) ≲ 1/(ℎ) 2 .
(4.60)

For M and � M , we obtain that the condition number can be bounded independently from or ℎ on simply connected geometries, but on topologically non-trivial structures, we obtain [START_REF] Bogaert | Low Frequency Stability of the Mixed Discretization of the MFIE[END_REF] cond( � M ) ≲ 1/ 2 . (4.61) he CFIE inherits these properties, and we have on simply connected geometries

cond � C = 1/ℎ (4.62)
and

cond � C = 1/( 2 ℎ) (4.63)
holds and by using the substitution y = F -1 x , we obtain

y F A 2 F y ≍ y T y , ∀y ∈ R . (4.58)
Given that F A 2 F = F T A T AF and the deinition of the singular value decomposition (SVD), we ind

cond(AF ) = � cond � F T A T AF � . (4.59)
Section 4.d

On the Ill-Conditioning of System Matrices on multiply connected. he reduction in the ill-conditioning from ℎ -2 to ℎ -1 results from the identity operator in � C: it introduces a lower bound of the spectrum so that the singular values of T A are shited and bounded away from zero.

his dissertation introduces new paradigms for curing the dense-discretization breakdown of the EFIE and the CFIE. A tool that will be used throughout the following hapters are quasi-Helmholz decompositions: they allow for a separate preconditioning of T A and T Φ , and they can be used to consider equivalent problems linked to scalar integral operators, whih makes it, for example, easier to devise a new basis.

) uasi-Helmholz Decompositions uasi-Helmholz decomposition have a long history, as pointed out in Chapter 1, since they were originally used to cure the low-frequency breakdown of the EFIE. Two families of basis functions, a solenoidal basis complemented with a non-solenoidal basis that can represent the harge, are employed. By rescaling these bases in , the low-frequency breakdown is prevented.

Examples for classical quasi-Helmholz decompositions are the loop-star and the loop-tree decomposition [WGK95; BK95; Vec99]. While for practical purposes the loop-tree decomposition works beter than the loop-star decomposition, we show in this section that the later is a beter foundation for the development of new preconditioning strategies. Diferent from a real Helmholz decomposition, where we would have a solenoidal and an irrotational basis, both the loop-star and the loop-tree decomposition provide a quasi-Helmholz decomposition: instead of forming an irrotational basis, the star and the tree functions are only non-solenoidal. In fact, there is no real Helmholz decomposition available for the RWG function space and thus we will frequently omit the word "quasi" and only use it if we want to stress the nature of the decomposition.

In more detail, let ∈ be loop functions, ∈ be global loops and ∈ be star functions [START_REF] Wu | A Study of Two Numerical Solution Procedures for the Electric Field Integral Equation[END_REF][START_REF] Vechi | Loop-Star Decomposition of Basis Functions in the Discretization of the EFIE[END_REF]. he global loops are the discrete counterpart of the quasi-harmonic Helmholz subspace. heir deinition, however, is subtle and cannot be treated here (see, for example, [START_REF] Wilton | On Improving the Stability of the Electric Field Integral Equation at Low Frequencies[END_REF][START_REF] Cools | Nullspaces of MFIE and Calderón Preconditioned EFIE Operators Applied to Toroidal Surfaces[END_REF]). As = ⊕ ⊕ , there are transformation matrices Λ ∈ R × V , H ∈ R × H , and Σ ∈ R × C that link the expansion coeicients of the current in the loop-star 57 Discretization of Boundary Integral Operators and Equations Chapter 4 basis to the expansion coeicients in the RWG basis, that is, we have

j = Λj + H j + Σ j (4.64)
and

∑ =1 � j � = V ∑ =1 � j � + H ∑ =1 � j � + C ∑ =1 � j � , (4.65)
where j , j , and j are the unknown vectors in the loop-star basis, V is the number of vertices (inner vertices, when is an open surface), C the number of cells, and H = 2, where is the genus of . Given the convention depicted in Figure 4.1, the loop transformation matrix is deined as

� Λ � = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 for = + , -1 for = - , 0 otherwise, (4.66)
where is the th vertex of the mesh. When the surface is open, are the inner vertices of the mesh and V is the number of inner vertices. Following the convention in Figure 4.1, the star transformation matrix is deined as

� Σ � = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 for = + , -1 for = - , 0 otherwise, (4.67)
where is the domain of the th cell of the mesh. In other words, the th column of Λ and Σ carries the coeicients with whih the th loop and star function, respectively, can be expressed as a linear combination of RWG functions. We deine the loop-star preconditioner as

Q = � Λ/ √ i H / √ i Σ √ i � . (4.68)
he discretization with the loop-star basis leads to the system matrix T Q , whih is related to T by

T Q = Q T T Q = � T ΛΛ T ΛΣ T ΣΛ T ΣΣ � (4.69) Section 4.d
On the Ill-Conditioning of System Matrices if is simply connected. 30 he loop-star decomposition has some interesting properties. First, we can relate the loop-loop and the star-star part of T Q to system matrices stemming from the discretization of scalar integral operators: we deine the functions

() = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 3/ for ∈ , -1/ for ∈
� T 0 ΛΛ � = ( , W ) 2 ( ) , (4.72)
and

� T 0 ΣΣ � = ( , V ) 2 ( ) . (4.73)
At this point, we should comment on the fact that the loop and the star functions are not independent if is closed; the all-one vectors

� 1 Λ� = 1 , = 1, … , V (4.74)
and

� 1 Σ� = 1 , = 1, … , C (4.75)
are in the null spaces of Λ and Σ , that is, Λ1 Λ = 0 and Σ 1 Σ = 0. If is open, the loop functions are linearly independent, but since the overall harge is zero, the star functions remain linearly dependent. he consequence for the classical loop-star preconditioner is that if is closed, then a loop and a star function must be eliminated, and if is open, then only a star function must be eliminated resulting in transformation matrices Λ ∈ R × Λ and Σ ∈ R × Σ , where Λ and Σ are the dimensions suh that the loop and star functions are linearly independent and we have

= Λ + H + Σ [Vec99].
30 In the multiply connected case, it reads

T Q = Q T T Q = ⎡ ⎢ ⎢ ⎣ T ΛΛ T ΛH T ΛΣ T HΛ T HH T HΣ T ΣΛ T ΣH T ΣΣ ⎤ ⎥ ⎥ ⎦ (4.70)
.

For the preconditioning shemes we are developing in this thesis, this is not a customary hoice. For example, the equality in (4.72) only holds if we do not eliminate a loop function since W has a null space spanned by 1 Λ (and in the end, we are developing a preconditioner for W , whih we can then only apply to T A if this equality holds). Another issue by eliminating a function is that the matrices Λ and Σ become even more ill-conditioned [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF]. For the (quasi-)Helmholz projectors, we are introducing in Section 4.d., it is necessary to solve the systems

Λ T Λx = b and Σ T Σ x = b.
It turns out that if the functions are not eliminated, then it is easier to precondition these systems. Depending on the preconditioner used, however, a delection of the null spaces might be necessary (see Section 4.d.). he overall Helmholz projectors are, however, identical, regardless whether we eliminate a function or not. We leverage this fact in Chapter 6 in order to keep the notation concise.

) uasi-Helmholz Projectors

We have seen that the loop-star basis allows to link T A to W and T Φ to V . Another important property of the loop-star decomposition is the orthogonality between the transformation matrices: we have Λ T Σ = 0, Λ T H = 0, and H T Σ = 0. he orthogonality can be used to recover the loop or the star components of j . he right-inverses of Λ and Σ are given by (Λ T Λ) + Λ T and (Σ T Σ ) + Σ T . If, for example, (Λ T Λ) + Λ T is applied to (4.64), then we obtain the loop expansion coeicients, that is, = (Λ T Λ) + Λ T j . he symbol "+" denotes the Moore-Penrose pseudo-inverse, whih is necessary due to the linear dependency of the loop and the star functions.

his motivates the deinition of quasi-Helmholz projectors

P Λ ∶= Λ � Λ T Λ � + Λ T (4.76)
and

P Σ ∶= Σ � Σ T Σ � + Σ T (4.77)
where P Λ projects to the solenoidal and P Σ to the non-solenoidal Helmholz subspace (for a detailed discussion and derivation of the Helmholz projectors, see [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF]And+13]). When the geometry is multiply connected, there is the projector P H that projects to the harmonic subspace. We can implicitly obtain this projector as

P H = I -P Λ -P Σ . (4.78) Section 4.d
On the Ill-Conditioning of System Matrices he matrices Σ T Σ and Λ T Λ are graph Laplacians and thus ill-conditioned. Standard tehniques, suh as algebraic multigrid preconditioners, allow an eicient inversion [LB12; NN12; Not]. In practice, the null spaces of Σ T Σ and Λ T Λ can prevent convergence when the residual error is low. To stabilize the iterative solver, we delect the null space as follows. For a single body problem, we can invert the matrix Σ T Σ + 1Σ 1T Σ , where 1Σ ∈ R C is the normalized all-one vector (i.e., 1Σ = 1 Σ /‖1 Σ ‖ 2 ) spanning the null space of Σ T Σ . We note that

� Σ T Σ + 1Σ 1T Σ � -1 = � Σ T Σ � + + 1Σ 1T Σ (4.79)
holds (see [And+13]). Other normalizations may be used as well. he best performance of the iterative solver can be expected when the singular value associated with 1Σ 1T Σ is shited into the spectrum of Σ T Σ . For a multibody problem, the dimensionality of the null space is the number of bodies B . For the th body, the th vector 1Σ, is deined as

� 1Σ, � = � 1/ � C, , when ∈ 0 , else, (4.80)
where C, is the number of cells and is the surface of the th body, respectively, and denotes the th cell of the entire surface = ∪ B =1 . We deine the matrix

̂1 Σ ∈ R C × B as � ̂1 Σ� = � 1Σ, � (4.81)
and then the matrix Σ T Σ + ̂1 Σ ̂1 T Σ is invertible. Similarly, we can deal with Λ T Λ. Diferent from Σ T Σ though, the graph Laplacian Λ T Λ has a null space only when the surface is closed. For stabilizing the inversion of Λ T Λ in the presence of a null space, we can invert the matrix Λ T Λ+ 1Λ 1T Λ (i.e., using the same strategy, as we did in the case of Σ T Σ ), where 1Λ is the normalized all-one vector (i.e., 1Λ = 1 Λ /‖1 Λ ‖ 2 ) spanning the null space of Λ T Λ. For a multibody problem, the dimensionality of the null space is the number of closed bodies B C . For the th closed body, we have the vector 1Λ, with entries

� 1Λ, � = � 1/ � V, , when ∈ 0 , else, (4.82) Chapter 4
where is the surface of the th closed body, V, is the number of vertices on this surface, and denotes the th vertex of the entire surface . We deine the matrix

O Λ ∈ R Λ × B C as � ̂1 Λ� = � 1Λ, � (4.83)
and then the matrix Λ T Λ + ̂1 Λ ̂1 T Λ is invertible. As demonstrated in [And+13], the projectors are an ideal means to cure the low-frequency breakdown. We deine P ΛH = P Λ + P H . Applying the projectors to the system matrix, we ind

(P ΛH + P Σ )T (P ΛH + P Σ ) = (P ΛH + P Σ ) � iT A + 1/(i)T Φ� (P ΛH + P Σ ) = (i)P ΛH T A P ΛH + (i)P ΛH T A P Σ + (i)P Σ T A P ΛH + (i)P Σ T A P Σ + 1/(i)P Σ T Φ P Σ . (4.84)
To prevent the low-frequency breakdown, we rescale the projectors with the square root of the wavenumber resulting in the preconditioner is free from the low-frequency breakdown.he imaginary unit +i is used to prevent the numerical cancellation due to the diferent scaling of the solenoidal and non-solenoidal components in the right-hand side and the current vector (for an exhaustive discussion, we refer the reader to [And+13]). While (4.86) is free from the low-frequency breakdown, its condition number still grows with 1/ℎ 2 . In the following parts, new paradigms in preconditioning are introduced to handle the dense-discretization breakdown.

P = � 1/ √ � P ΛH + i √ P Σ . ( 4
Part II.

Hierarhical Bases on Structured and Unstructured Meshes

Chapter 5

Primal and Dual Haar Bases on Unstructured Meshes for the EFIE A new hierarchical basis preconditioner for the EFIE operator is introduced. In contrast to other hierarchical basis preconditioners, it works on arbitrary meshes and preconditions both the vector and the scalar potential within the EFIE operator. his is achieved by taking into account that the vector and the scalar potential operator discretized with loop-star basis functions are related to the hypersingular and the single layer operator (i.e., the well known integral operators from electrostatics). he strategy proposed in this chapter for preconditioning the EFIE is the transformation of the scalar and the vector potential operator into operators equivalent to the single layer operator and to its inverse. Further mathematical considerations show that this allows to use generalized primal and dual Haar functions as preconditioner. It turns out though that in the case of the dual Haar wavelets, the inverse transformation matrix must be used. he numerical results show the efectiveness of the proposed preconditioner and the practical impact of theoretical developments in real case scenarios. his chapter is based on [START_REF] Adrian | A Hierarhical Preconditioner for the Electric Field Integral Equation on Unstructured Meshes Based on Primal and Dual Haar Bases[END_REF].

Q uasi-Helmholtz decompositions-as we have seen in Chapter 4-are commonly used to cure the low-frequency breakdown. Typical examples are the loop-star and the loop-tree basis preconditioners, where the loop functions form a set of solenoidal functions and the star/tree functions form a set of non-solenoidal functions. With suh a Helmholz decomposed basis, the solenoidal null space of the scalar potential is exploited to separate the vector from the scalar potential operator allowing to rescale them in frequency [BK95; WGK95].

While loop-star and loop-tree preconditioners takle the low-frequency breakdown, they do not cure the dense-discretization breakdown. he dense-discretization breakdown is due to the integrative and derivative strength of the vector and of the scalar potential operator giving rise to a condition number of the EFIE system matrix whih scales with 1/ℎ 2 , where ℎ is the average edge length of the mesh. Hierarhical basis preconditioners cure the dense-discretization breakdown when the basis is constructed suh that it relects the Sobolev norm induced by the EFIE operator [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF].

hey are, however, not a panacea. In fact, the construction of a hierarhical basis for the discretization of vector operators suh as the EFIE is far from being trivial, in particular, when three constraints are applied: (i) it should be possible to apply the hierarhical basis multiplicatively to an EFIE that is discretized with RWG functions (so that the hierarhical preconditioner can easily be integrated into existing codes), (ii) the construction and application of the hierarhical basis should have at most a quasilinear complexity (i.e., linear up to logarithmically growing multiplicative terms) so that the advantage of using fast matrix-vector multiplication algorithms is not jeopardized), and (iii) the hierarhical basis should be applicable to unstructured meshes. An unstructured mesh is the output of a mesher where, given a spline geometry, an optimized triangulation subject to a certain average edge length ℎ is generated. A structured mesh, on the other hand, is generated from a (usually coarse) unstructured mesh by sequentially reining it, for example, dyadically (i.e., the midpoints of the three edges of a cell are connected resulting in four smaller, new cells) until the desired average edge length ℎ is obtained.

While the irst and second condition are met by virtually all hierarhical basis preconditioners [VPV05; ATV07; And12b; ATV10; Che+09; HM12], there have been only few hierarhical bases reported for the EFIE that work on unstructured meshes as well [START_REF] Andriulli | Hierarhical Bases for Nonhierarhic 3-D Triangular Meshes[END_REF]. he satisfaction of the third criterion, however, is decisive for the practicality of the hierarhical basis preconditioner. Structured meshes are, in industrial applications, rarely used for two reasons. Usually one cannot start from a coarse mesh and perform a structured reinement. Most of the time, one has to start with a mesh that is relatively ine since only then the details of the spline geometry can be captured. In addition, the hierarhical basis only reduces the condition number to one obtained by a discretization of the EFIE on coarsest mesh. his condition number can still be prohibitively large so that the efectiveness of the hierarhical basis preconditioner is insuicient.

Currently available hierarhical basis preconditioners for unstructured meshes only precondition the scalar potential operator [AVV08; DJC11]. As part of the underlying quasi-Helmholz decomposition, loop functions are used for the solenoidal basis whih, however, do not cure the dense-discretization breakdown of the vector potential operator [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF]. hus the condition number of the entire system matrix still grows with 1/ℎ and we face the open task to ind a hierarhical basis that preconditions the vector potential operator on unstructured meshes.

he reason why there is a hierarhical basis preconditioner available for the scalar potential operator but not for the vector potential operator can be explained as follows. When the EFIE operator is discretized with RWG functions as expansion and testing functions, the scalar potential operator is a single layer operator discretized with piecewise constant basis functions related to the two divergence terms of the current basis functions. his allows, once these divergence terms are resolved, to reuse a hierarhical basis deined on the space of piecewise constant functions [START_REF] Andriulli | Hierarhical Bases for Nonhierarhic 3-D Triangular Meshes[END_REF].

he vector potential operator, on the other hand, is a single layer operator on the vector function space of the electric surface current density . Unlike the scalar potential operator, we cannot turn it directly into a scalar single layer operator. Yet, there is an important property of the vector potential operator: we have seen in Chapter 4 that when loop functions are used, the discretized vector potential operator is equivalent to the hypersingular operator W discretized with scalar piecewise linear functions. his simpliies our task of preconditioning the vector potential operator since we do not have to deal with a vector but a scalar operator, that is, we "only" have to ind a hierarhical basis for a scalar function space.

In this hapter, a provable hierarhical preconditioning strategy for the vector potential operator is reported. he three criteria (i)-(iii) for a good hierarhical basis preconditioner stated before are satisied. Both the scalar and the vector potential operator are preconditioned in this work by generalized primal and dual Haar bases (i.e. generalized Haar bases deined on the standard and dual mesh respectively). In order to precondition the scalar potential operator with the primal Haar basis, we use graph Laplacians to transform it into the single layer operator V. For the vector potential operator instead, whih relates to the hypersingular operator W, we exploit the spectral equivalency of the later to the inverse single layer operator to precondition it with a linear-in-complexity, Primal and Dual Haar Bases on Unstructured Meshes for the EFIE Chapter 5 closed-form inverse of the dual Haar basis. Since the generalized Haar basis is easy to construct and the same code can be used for the generation of the primal and the dual Haar bases, the implementational efort of using the preconditioner is minimal.

he hapter is structured as follows: in Section 5.a, we comment on the scattering scenario. Section 5.b introduces the primal and the dual Haar basis. In Section 5.c, we show how the vector and the scalar potential operator can be transformed into operators spectrally equivalent to the single layer operator and its inverse and how they can be preconditioned with a generalized Haar bases in these representations. In Section 5.d, we present numerical results that demonstrate the efectiveness of the proposed approah.

a) Bakground

As required in Chapter 4, the scaterer ⊂ R 3 is Lipshiz polyhedral, we constrain ourselves, however, to the discussion of simply connected geometries to keep the notation concise. For multiply connected geometries, the hierarhical basis can be complemented with global loops as discussed in Section 4.d.. If the number of global functions is large, or even scales with , this is not eicient anymore. Chapter 6 discusses how hierarhical basis preconditioners can be combined with a Helmholz projector approah, where the quasi-harmonic Helmholz subspace is only recovered implicitly. Whenever relevant for the implementation of the hierarhical basis preconditioner, we will point out to the reader the necessary modiications to apply the tehnique to multi-body problems, both closed and open.

b) Construction of the Generalized Haar Basis

Generalized Haar bases have been constructed before, also for the case of unstructured meshes (see, for example, [START_REF] Harbreht | Wavelet Galerkin BEM on Unstructured Meshes[END_REF], where an octree was used for the partitioning of the mesh). Classically, a Haar basis would be constructed on the space of piecewise constant functions . As it turns out in Section 5.c, we also need a Haar basis for the piecewise constant functions deined on the dual mesh, whih we denote as ∈ . Diferent from the standard piecewise constant functions, whih are deined on the cells of the mesh, these functions are associated with the vertices of the mesh. We can deine the dual piecewise constant functions by using a barycentric mesh reinement. he support of the dual piecewise constant function is given by the cells on the barycentrically reined mesh that are atahed to the th vertex (Figure 5.1 shows the support of a dual piecewise constant function). When is in the support of , the function value () = 1/ � , where � is the area of the support of , and is zero when is not in the support of . Let bar be the standard piecewise constant functions deined on the barycentrically reined mesh. For the example given in Figure 5 We start with the construction of ĤΣ . he construction algorithm requires as input the cell-based graph Laplacian Σ T Σ and the areas of the cells of the mesh. he construction of the Haar basis begins with deining the entire mesh as a macro cell (see Figure 5.2a). On this macro cell, the irst generalized Haar function is deined as the constant function

H,0 = 1/ , ∈ , (5.2)
where is the area of the surface . Eventually, we seek to obtain a multiplicative preconditioner in the form of a matrix ĤΣ that maps from the generalized Haar functions to the piecewise constant functions ∈ . We have the relationship

H,0 = 1/ C ∑ =1 (5.3)
between the irst generalized Haar function and the functions ∈ . We insert the expansion coeicients for H,0 , whih are

� ĤΣ � 1 = /
(5.4) with = 1, … , C , in the irst column of ĤΣ . he next Haar functions are obtained by dividing the mesh into m = 4 macro cells. Although alternative strategies are possible [START_REF] Mitharwal | On the Multiplicative Regularization of Graph Laplacians on Closed and Open Structures With Applications to Spectral Partitioning[END_REF], in our implementation, we have used the graph partitioning algorithms made available by the METIS library [START_REF] Karypis | A Fast and High uality Multilevel Sheme for Partitioning Irregular Graphs[END_REF]. Other graph partitioning algorithms could be used as well as long as they can ensure that the macro cells have (approximately) the same area and that the cells in a macro cell are all connected by their edges. An example for the division is shown in Figure 5.2.

We denote the domains of the thereby generated macro cells by m, , = 1, … , m and deine macro piecewise constant functions where ( m, ) is the area of the macro cell m, . Notice, that these functions are normalized suh that � m, d() = 1. he functions m, are given in terms of

m, = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 ( m, ) ∈ m, , 0 ∉ m, , (5.5)
as m, = 1/( m, ) ∑ ∈ � ∈ | ⊂ m, � .
(5.6)

With this relationship, we obtain the vectors

� b � = � /( m, ) when ⊂ m, , 0 otherwise, (5.7)
and we can combine them columnwise in the matrix

B = � b 1 … b m � .
he functions m, with = 1, … , m and H,0 are linearly dependent, and they are not orthogonal to H,0 . herefore, we need to form m -1 linear combinations of macro piecewise constant functions H, suh that � H,0 , H, � 2 = 0 for = 1, … , m -1. here is not a unique solution, and so we strive to orthogonalize the Haar functions of the same group as well, whih we obtain in the case of uniform meshes by considering the unweighted cell-based graph Laplacian L deined by the m macro cells m, and apply the SVD, that is, we have U SU T = L. hen we deine the matrix R = U(1 ∶ end, 1 ∶ end -1) (Matlab notation). Using this matrix, we deine the Haar functions as

H, = m ∑ =1 m, � R � , = 1, … , m -1 (5.8)
noting that ∑ R(, ) = 0 whih ensures the orthogonality of the Haar wavelets to H,0 . Given these deinitions, the matrix B R is the transformation matrix that maps from the Haar functions to the piecewise constant functions ∈ . he matrix B R is added to the matrix ĤΣ , whih at this point consists of a single column associated with H,0 . he remaining generalized Haar functions are obtained recursively: eah macro cell, when it contains more than one cell, is subject to a further division into m cells (as it is depicted in Figure 5 Since the primal generalized Haar basis is used for representing the electric harge and since the total harge is zero, we discard the function H,0 and denote the resulting matrix as

H Σ ∈ R C × Σ .
Analogously, the dual generalized Haar basis ĤΛ can be constructed. As input, we now need the vertex-based graph Laplacian matrix Λ T Λ and the areas � of the dual cells , where denotes the support of the th dual piecewise constant function. he deinition of H,0 stays the same as in (5.2), and we represent it as a linear combination of dual piecewise constant functions

H,0 = 1/ V ∑ =1 � .
(5.9) herefore, the irst column of ĤΛ is given by

� ĤΛ � 1 = � / (5.10) with = 1, … , V .
On eah of these dual macro cells we deine a dual macro piecewise constant function

m, = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 � ( m, ) ∈ m, , 0 ∉ m, , (5.11)
where � ( m, ) is the area of the dual macro cell m, . Notice, that these functions are normalized suh that � m, d() = 1. We deine the vectors

� b� = � � / � ( m, ) when ⊂ m, , 0 otherwise, (5.12)
and we combine them columnwise in the matrix � B = � b1 … b m � . he dual macro functions are combined to the dual generalized Haar functions

H, = m ∑ =1 m,� � R � , = 1, … , m -1 (5.13)
suh that � H,0 , H, � 2 = 0 and where � R is obtained analogously to R by using the SVD applied to the cell-based graph Laplacian of the m dual macro cells m, .

Chapter 5 he matrix � B � R is the transformation matrix that maps from the Haar functions to the dual piecewise constant functions ∈ . he matrix � B � R is added to ĤΛ , whih at this point consists of a single column whih is associated with H,0 . hen the algorithm continues recursively by further dividing the dual macro cells and generating for eah new division step the matrices � B � R. When the surface is closed, the function H,0 and the associated column in ĤΛ are discarded. We denote the resulting matrix as H Λ . Diferent from the generation of H Σ , special care must be taken when the surface is open. First of all, only the dual piecewise constant functions associated with the inner vertices of the mesh are considered. Secondly, the function H,0 must not be discarded (i.e., H Λ ∶= ĤΛ ) as in this case the loop functions are a set of linearly independent functions and thus Λ T T A Λ has full rank.

c) New Hierarhical Basis

In the following, we transform the vector and the scalar potential operator suh that we can apply a generalized Haar basis for preconditioning.

) Scalar Potential Operator

It is known that V can be preconditioned by applying the Haar basis transformation matrix ĤΣ -at least on structured meshes-followed by the diagonal rescaling matrix [Osw94; Osw98; HKS05]

� DΣ � = 2 -Σ ()/2 , (5.14)
where the function Σ (), ∈ {1, … , C }, returns the level on whih the associated Haar function is deined, so that the condition number bound

cond � DΣ Ĥ T Σ V ĤΣ DΣ � ≲ log 2 (1/ℎ) (5.15)
holds [START_REF] Oswald | Multilevel Norms for -1/2[END_REF]. he numerical results show that suh a basis is an efective preconditioner even for unstructured meshes.

Section 5.c

New Hierarhical Basis he divergence of is given by [RWG82]

div = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1/ + , ∈ + , -1/ - , ∈ - , 0 ,
otherwise.

(5.16)

Since the functions div are piecewise constant, they can be deined as a linear combination of functions, thereby giving rise to a transformation matrix X . hen we have

� T Φ� ∶= -� × , T Φ � = X T V X , (5.17)
where the matrix X ∈ R C × is deined as

� X � = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 for = + , -1 for = - , 0
otherwise.

(5.18)

Comparing (5.18) with (4.67), we see that X = Σ T , and another application of Σ yields

T 0 ΣΣ = Σ T ΣV Σ T Σ . (5.19)
To apply the generalized Haar basis to the scalar potential operator, we must remove the star transformation matrices Σ T Σ in (5.19). We apply from let and right (Σ T Σ ) + , where the "+"-symbol denotes the Moore-Penrose pseudo-inverse, and ind

� Σ T Σ � + Σ T ΣV Σ T Σ � Σ T Σ � + = � Σ T Σ � + T 0 ΣΣ � Σ T Σ � + .
(5.20) he pseudo-inverse is necessary since there are only Σ = C -1 linearly independent star functions as the total harge is always zero [START_REF] Vechi | Loop-Star Decomposition of Basis Functions in the Discretization of the EFIE[END_REF].

We take this later fact into account by using H Σ ∈ R C × Σ , whih is obtained from ĤΣ by discarding the constant Haar function (see Section 5.b). In order to apply H Σ D Σ to (5.20), we must have

D Σ H T Σ � Σ T Σ � + Σ T ΣV Σ T Σ � Σ T Σ � + H Σ D Σ = D Σ H T Σ V H Σ D Σ , (5.21)
where � D Σ� = 2 -Σ ()/2 with ∈ {1, … , Σ } and Σ () = Σ ( + 1). hat (5.21) holds can be seen as follows: the system Σ T Σ x = y is solvable only when y is in the range of Σ T Σ . It is easy to see that

� Σ T Σ � = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ deg if = , -1 if ≠ and is adjacent to , 0 otherwise, (5.22)
where deg is the degree of (i.e., the number of cells atahed to th cell). In other words Σ T Σ is the graph Laplacian matrix associated with the cell-based graph. From this deinition it follows that the let null space of Σ T Σ is spanned by the all-one vector. Only vectors whose row sum is zero are orthogonal to this vector. Hence, the row sum of y = H Σ x must be zero for all x , as y must be in the range of Σ T Σ . hat the row sum is zero for all x , can be seen from the deinition of H Σ (see (5.7) and (5.8) in Section 5.b). We deine

� Σ ∶= � Σ � Σ T Σ � + H Σ � , (5.23)
where the wide hat symbol " � " signiies that we deal with a hierarhical basis. hen we have

cond � D Σ � Σ T T 0 Φ � Σ D Σ � ≲ log 2 (1/ℎ) (5.24) since D Σ � Σ T T 0 Φ � Σ D Σ = D Σ H T Σ � Σ T Σ � + Σ T T 0 ΣΣ Σ � Σ T Σ � + H Σ D Σ (5.20) = D Σ H T Σ � Σ T Σ � + Σ T ΣV Σ T Σ � Σ T Σ � + H Σ D Σ (5.21) = D Σ H T Σ V H Σ D Σ (5.25) and cond � D Σ H T Σ V H Σ D Σ � ≲ cond � DΣ Ĥ T Σ V ĤΣ DΣ � (5.15) ≲ log 2 (1/ℎ) , (5.26)
where we used that for any symmetric, positive deinite (SPD) matrix the symmetric elimination of a row and column leads to a smaller or at most equal condition number than the original matrix.

Section 5.c

New Hierarhical Basis

) Vector Potential Operator

According to (4.72), the loop-loop blok matrix T 0 ΛΛ is identical to the discretization of the hypersingular operator with piecewise linear functions, that is,

� W � = � T 0 ΛΛ � . (5.27)
Since a set of hierarhical piecewise linear functions for unstructured meshes is not available, we must resort to a strategy that does not explicitly require suh a set of hierarhical functions.

To simplify the following analysis, we consider a modiied hypersingular operator. Without the modiication, the operator W is not 1/2 ( )-elliptic, and its discretization possesses a null space spanned by the constant functions. Following a standard approah [START_REF] Steinbah | Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements[END_REF], we introduce the delected operator

Ŵ ∶ 1/2 ( ) → -1/2 ( ) (5.28)
deined by the bilinear form

� , Ŵ � 2 ( ) ∶= (, W) 2 ( ) + (1, ) 2 ( ) (1, ) 2 ( ) (5.29)
for all , ∈ 1/2 ( ). We note that the unique solution of Ŵ = is also a solution of W = when satisies the solvability condition � d( ′ ) = 0. his can be seen when = 1 in (5.29), whih reduces to (, 1) 2 ( ) (1, 1) 2 ( ) = 0 whih implies (, 1) 2 ( ) = 0. he discretization of Ŵ is given by

� Ŵ � = � , Ŵ � 2 ( ) .
(5.30)

With the dual piecewise constant functions we can discretize the single layer resulting in the matrix

� � V � = � , V � 2 ( ) .
It should be noticed that by using dual piecewise constant functions, we obtain system matrices �

V and W with equal dimensionality. Proposition 5.1. Let H Λ be the transformation matrix from the dual generalized Haar basis to the dual piecewise constant functions as deined in Section 5.b, and let

� D Λ� = 2 -Λ ()/2 ,
(5.31)

where the function Λ (), ∈ {1, … , Λ } returns the level on which the associated Haar function is deined with Λ = V -1 the number of linearly independent loop functions. hen we have

cond � D -1 Λ H ‡ Λ G -1 W G -T � H ‡ Λ� T D -1 Λ � ≲ log 2 (1/ℎ) , (5.32)
where � G � = � , � 2 ( ) and H ‡ Λ is an 2 -generalized inverse matrix deined as

H ‡ Λ = � H T Λ G H Λ � -1 H T Λ G .
(5.33)

Proof. From (5.15), we have

cond � DΛ Ĥ T Λ � V ĤΛ DΛ � ≲ log 2 (1/ℎ) , (5.34)
where ĤΛ is deined in Section 5.b and � DΛ � = 2 -Λ ()/2 with ∈ {0 … , V }. Based on the scalar Calderón identities (see Section 3.c and [SW98; Hip06]), we have the spectral equivalence

x T � V x ≍ x T G T Ŵ -1 G x , ∀x ∈ R V (5.35)
and therefore obtain

cond � DΛ Ĥ T Λ G T Ŵ -1 G ĤΛ DΛ � ≲ log 2 (1/ℎ) (5.36) from whih cond � D-1 Λ Ĥ -1 Λ G -1 Ŵ G -T Ĥ -T Λ D-1 Λ � ≲ log 2 (1/ℎ) (5.37)
holds, since all matrices appearing are invertible. We should now make the transition from Ŵ to W . It follows from (5.30), that Ŵ x can be replaced by W x when the associated function x ∶= ∑ � x � satisies ( x , 1) 2 ( ) = 0. In matrix form, this can be expressed as ãT G x = 0 , (5.38)

where � ã� = � . Any preconditioner X ∈ R V × Λ applicable to W suh that X T W X is well-conditioned must ensure for any y ∈ R Λ that the expansion coeicients x = X y satisfy the discrete solvability condition (5.38).

Section 5.c

New Hierarhical Basis First, we note that

Ĥ -1 Λ = ( Ĥ T Λ G ĤΛ ) -1 Ĥ T Λ G
. Since H Λ is obtained by eliminating the irst column of ĤΛ , we use as let inverse of H Λ the matrix H ‡ Λ , whih is obtained from Ĥ -1 Λ by eliminating the irst row, and whih we can write explicitly as

Ĥ ‡ Λ = (H T Λ G H Λ ) -1 H T Λ G . hen we have cond � D -1 Λ H ‡ Λ G -1 Ŵ G -T � Ĥ ‡ Λ� T D -1 Λ � ≲ log 2 (1/ℎ) , (5.39)
since the matrix in (5.39) is symmetric, positive deinite and for any suh matrix the elimination of the th row and column yields a condition number bounded by the condition number of the original matrix. In addition, by the deinition of H ‡ , we satisfy (5.38): for any y ∈ R Λ , we ind a z ∈ R Λ suh that

y = (H T Λ G H Λ ) -1 D -1 Λ z .
(5.40)

Deining x = G -T G H Λ y and inserting in (5.38), we obtain

ãG x = ãG G -T G H Λ y = ãG H Λ y .
(5.41)

Since � G � = 1/ � , eq. (5.41) can be writen as

1 T Λ H Λ y = 0 , (5.42)
where 1 Λ ∈ R V is the all-one vector and the equality results from the deinition of H Λ , where we notice that the column vectors of H Λ have a zero mean value (see Section 5.b).

Summarizing, we deine the transformation matrix for the new solenoidal basis as

� Λ ∶= ΛG -T G H Λ � H T Λ G H Λ � -1
.

(5.43)

he Gram matrix G can be computed analytically. We have where () is the set of vertices adjoint to vertex , + and - are the areas of the cells atahed to the edge that connect the vertices and , and � is the area of the dual cell atahed to vertex (see Figure 5.1 for an example of a dual cell). Since this Gram matrix is well-conditioned and sparse, iterative solvers can be used for computing the application of G -1 to a vector. We note that H T Λ G H Λ is the Gram matrix of the Haar functions, that is,

� G � = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
� H T Λ G H Λ � = � , , , � 2 (5.45)
(for an explicit deinition of the Haar functions , , see Section 5.b). he inverse of the matrix H T Λ G H Λ can be computed fast (i.e., in O( V ) complexity). his can be seen by considering the structure of this Gram matrix: the only Haar functions whih are not orthogonal to a Haar function H, in the 2 -sense are the Haar functions deined on the same macro cell as H, ; by construction of the Haar functions, we have ensured that the functions from diferent levels are 2 ( )-orthogonal. Haar functions on the same level but deined on diferent macro cells have orthogonal supports and thus are orthogonal in the 2 ( )-sense. As outlined in Section 5.b, every macro cell is split in (at most) four smaller macro cells, and hence, there are at most three Haar functions deined on a macro cell. hen for eah Haar function there are at most only two other non-orthogonal Haar functions. hus H T Λ G H Λ is a blok diagonal matrix with bloks of at most size 3 × 3 (i.e., to be precisely, it is blok diagonal when the non-orthogonal Haar functions are grouped together). Clearly, this matrix is sparse with the number of elements scaling as O( V ) and its inverse is given by the inversion of eah of the matrix bloks. Since the costs for inverting a single blok is independent from V and the number of bloks are scaling linearly in V , the complexity for inding the inverse is O( V ).

) Proposed Preconditioner for the EFIE Operator

We deine the overall basis transformation matrix as

� Q = � � Λ � Σ � , (5.46)
Section 5.c

New Hierarhical Basis where � Λ was deined in (5.43) and � Σ in (5.23), and the rescaling matrix as

D = � D -1 Λ / √ i D Σ √ i � , (5.47)
where D Λ was deined in (5.31) and D Σ in (5.14). We observe for the static limit

lim →0 D � Q T T � Q D = � D -1 Λ � Λ T T 0 A � ΛD -1 Λ D Σ � Σ T T 0 Φ � Σ D Σ� (5.48)
and because of (5.24) and (5.32), we have

cond � lim →0 D � Q T T � Q D � ≲ log 2 (1/ℎ) .
(5.49)

Since the dynamic kernel only introduces a compact perturbation, we conclude that D � Q T T � Q D is well-conditioned up to a logarithmic perturbation. As a side note, we also consider multiply connected geometries. Similar to loop-star or loop-tree quasi-Helmholz preconditioners (see also Section 4.d.), we add the global loops transformation matrix H to � Q resulting in

� Q = � � Λ H � Σ � (5.50)
and use

D = ⎡ ⎢ ⎢ ⎢ ⎣ D -1 Λ / √ i D H / √ D Σ √ i ⎤ ⎥ ⎥ ⎥ ⎦ , (5.51)
where

� D H� = 1/ � � H T T 0 A H � .
(5.52) hen we have

lim →0 D � Q T T � Q D = ⎡ ⎢ ⎢ ⎢ ⎣ D -1 Λ � Λ T T 0 A � ΛD -1 Λ D -1 Λ � Λ T T 0 A H D H D H H T T 0 A � ΛD -1 Λ D H H T T 0 A H D H D Σ � Σ T T 0 Φ � Σ D Σ ⎤ ⎥ ⎥ ⎥ ⎦ . (5.53)
Primal and Dual Haar Bases on Unstructured Meshes for the EFIE Chapter 5 he analysis of the conditioning is more complicated than for (5.48) due to the of-diagonal blok matrices. First, we note that D H H T T 0 A H D H is well-conditioned since mesh reinements do not hange the global loops. hus all the blok matrices on the main diagonal are well-conditioned. he transformation matrix H has no preconditioning strength, whih would indicate that the matrices

D -1 Λ � Λ T T 0 A H D H and D H H T T 0 A � ΛD -1
Λ are only "half preconditioned" so that we have for their singular values an upper bound. We note that by adding the global loops, we cannot exclude the case that some eigenvalues of lim →0 D � Q T T � Q D are shited closer to zero, but since the basis transformation matrix � Q has full rank, we can exclude that a real null space is introduced. Numerical evidence suggests that even a shiting of eigenvalues closer to zero does not occur (see Chapter 6). hus it seems that we can safely assume that the logarithmic bound persists in the presence of global loops.

Obtaining H explicitly is, however, not a trivial task and using global loops explicitly can be a costly operation. herefore, we present a sheme in the next hapter that allows to use hierarhical basis preconditioners for the EFIE suh that a searh for the global loops is not required.

Returning to the simply connected case, beter results are observed in practice when a normalization is used for curing the low-frequency breakdown, that is, we use

D = � D -1 Λ / √ Λ D Σ / √ Σ � , (5.54)
where Λ and Σ are the largest singular values of D -1 Λ � Λ T T � ΛD -1 Λ and D Σ � Σ T T � Σ D Σ , respectively. he diference in the norm of these two matrices is properly taken into account in (5.54) by the rescaling of D Λ and D Σ . Note that Λ and Σ can be obtained in linear complexity by using, for example, a power iteration method [START_REF] Golub | Matrix Computations. 4th ed. Johns Hopkins Studies in the Mathematical Sciences[END_REF].

he matrices D Λ and D Σ where deined by simply taking into account the level on whih the respective Haar function is deined (see (5.14) and (5.31)). his works well when partitions on a level have all the same area. Since graph partitioning shemes are usually heuristic this is only approximately ensured. For this reason it is typically proposed to compute where T is usually replaced by the near-interaction part of the system matrix in order to compute the matrix-matrix products rapidly. In our sheme, it is not possible to compute D in this manner due to the inverse matrices appearing in the deinition of � Λ and � Σ . However, we have an alternative by leveraging on the fact the Haar functions we are using are not 2 -stable, so that the Gram matrices of the primal and the dual Haar bases scale as

� D � = 1/ � � � Q T T � Q � , ( 5
� H T Λ G H Λ� ≍ 2 2 Λ () , (5.56)
and

� H T Σ G H Σ� ≍ 2 2 Σ () .
(5.57)

Hence, we can use the diagonal of their Gram matrices to obtain D Λ and D Σ , that is, we deine

� D Λ� = � � H T Λ G H Λ � � -0.25 , (5.58) � D Σ� = � � H T Σ G H Σ� � -0.25
.

(5.59)

d) Numerical Results

We begin with studying the spectral properties of the new formulation by using a cube with side length 1 m. We have generated a regular sequence of meshes, for diferent values of the average edge length ℎ, resulting in numbers of unknowns ranging from 18 to 18 432. At irst, we analyze how well the dense-discretization breakdown is prevented by the proposed preconditioner. his is done by computing the condition number for diferent values of the spectral index 1/ℎ. We begin our analysis by comparing the preconditioned scalar potential operator in (5.24) with a star, a tree, and a standard hierarhical non-solenoidal basis preconditioner (like the non-solenoidal part of the method used in [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF]). Figure 5.3a shows that both hierarhical preconditioners have the same qualitative behavior, whereas the star and tree preconditioner result in ill-conditioned system matrices.

Next, we analyze the efectiveness of our proposed preconditioning strategy of the vector potential operator in (5.32). We compare it with a vector potential operator that is discretized with loop functions, the modiied hierarhical loop functions proposed in [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF], and with hierarhical loop functions. he later are constructed by adding the loop functions deined on all the new vertices of eah reinement step. his construction does not enforce an orthogonality between the levels, and thus we cannot expect it to show any sign of preconditioning efectiveness. Figure 5.3b) displays the condition number as a function of the spectral index 1/ℎ. he modiied hierarhical loops compare well with our proposed method. his can be expected from the theory presented in [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF]. he modiied hierarhical loops lead to a discretization where the condition number of the matrix is bounded by a constant, whereas for the proposed preconditioner the condition number is still growing logarithmically. Diferent from the proposed preconditioner, the modiied hierarhical loops are limited to structured meshes and lak, therefore, the necessary lexibility for practical applications.

Figure 5.3c shows the results when the entire EFIE operator is preconditioned with our new method, with the preconditioner presented in [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF], and with a standard hierarhical basis preconditioner. he standard hierarhical basis preconditioner consists of loop functions and a hierarhical non-solenoidal basis. his preconditioner is representative of the state of the art, since it is applicable to structured and unstructured meshes. In Figure 5.3c, the condition number is shown as a function of the spectral index 1/ℎ. hese results show that the new preconditioner presented in this work compares favorably with previously proposed tehniques. In fact, it has a performance whih is comparable with the preconditioner in [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF], whih however is applicable only to structured meshes. Since the standard hierarhical basis preconditioner employs loop functions, the vector potential operator is not efectively preconditioned; it is for this reason that in Figure 5.3c we observe the growth in the number of iterations for this preconditioner. In Figure 5.3d, the condition number of the system matrix is shown as a function of the frequency . he frequency is related to the wavenumber by = 2π /, where is the speed of light. he results show that the new preconditioner efectively cures the low-frequency breakdown.

Finally, we have tested the new preconditioner on more realistic geometries. he proposed preconditioner requires the inversion of the mixed Gram matrix G and the graph Laplacian Σ T Σ . In general, we are going to use the conjugate gradient squared (CGS) method as iterative solver with the exception of our treatment of Σ T Σ (other Krylov subspace methods could be used as well). he matrix G is sparse and well-conditioned, hence one can easily use a solver Section 5.e Conclusion tolerance close to mahine precision. As Σ T Σ is a graph Laplacian, the matrix is ill-conditioned and a preconditioner necessary. We are using the aggregationbased algebraic multigrid preconditioner AGMG presented in [START_REF] Napov | An Algebraic Multigrid Method with Guaranteed Convergence Rate[END_REF]Not]. As solver tolerances, we use 1 ⋅ 10 -14 for the AGMG solver and 1 ⋅ 10 -16 for the CGS for G , thereby showing that even for small tolerances our method remains practical.

As closed realistic structure, we use a model of a Space Shutle discretized with 1 148 400 basis functions. he length of the Space Shutle is /20. We compared the preconditioner proposed in this work with a classical loop-tree preconditioner [START_REF] Wu | A Study of Two Numerical Solution Procedures for the Electric Field Integral Equation[END_REF] and the case that a hierarhical non-solenoidal basis is complemented by loop functions (denoted as "standard hierarhical preconditioner"). For obtaining the numerical results, we used the CGS solver with tolerance 1 ⋅ 10 -4 . To accelerate the computation, we employed the ACA with precision 1 ⋅ 10 -4 . As excitation, we used a plane wave and a voltage gap, respectively. he results, summarized in Table 5.1, show clearly the improvements that the preconditioner proposed in this work presents over the state of the art. Figure 5.4 shows a good agreement of the bistatic scatering cross sections. Like other hierarhical preconditioner shemes or Calderón preconditioners, the new preconditioner was derived under the assumption that the structures are closed. In practice, we ind that the new preconditioner can be used on open structures as well. We simulated the scatering problem for the model of a MiG-15 with 306 036 unknowns. We used the same ACA precision and solver tolerance as for the Space Shutle, and we studied the efect of a plane wave and a voltage gap excitation. Table 5.2 shows the number of iterations and the timings, and Figure 5.5 shows the bistatic scatering cross section. Clearly, the new preconditioner remains efective for open structures.

e) Conclusion

We presented a hierarhical basis preconditioner, whih works on structured and on unstructured meshes. Diferent from other hierarhical basis preconditioners with comparable applicability, it preconditions the electric ield integral equation operator completely. he numerical results do not only conirm the theoretical predictions but also show that the presented preconditioner outperforms standard methods for real case scenarios. Chapter 6

Hierarhical Bases on Multiply Connected Objects for the EFIE his chapter extends hierarchical basis preconditioners applicable to the EFIE such that on multiply connected geometries no search for global loops is required. Currently available hierarchical basis preconditioners need an explicit representation of global loops. Finding these requires a computational complexity exceeding the linearithmic complexity of fast matrix-vector multiplication methods. Instead of using an explicit representation of global loops, quasi-Helmholz projectors are utilized to precondition the EFIE separately on the solenoidal, non-solenoidal, and quasi-harmonic Helmholz subspace. hereby, we avoid the explicit recovery of the global loops and maintain the leading complexity of fast multiplication methods. Numerical results prove the efectiveness of the proposed approach. his chapter is based on [AAE14b].

H ierarchical basis preconditioners, as discussed in Chapter 1 and in Chapter 5, can cure the ill-conditioning of the EFIE up to a logarithmic perturbation. Various bases have been presented for the EFIE operator and their efectiveness has been demonstrated in the past [VPV05; ATV07; Che+09; ATV10; AVV08; HM12; AAE17]. A hierarhical basis for the EFIE operator must precondition both the vector and the scalar potential operator. he separate preconditioning of the vector and the scalar potential operator is usually ahieved by using a set of hierarhical solenoidal and non-solenoidal basis functions. his enforces a quasi-Helmholz decomposition that exploits the solenoidal null space of the scalar potential operator in order to split it of from the vector potential operator.

On multiply connected geometries, however, the hierarhical bases share the same issue as the classical loop-star/tree quasi-Helmholz decompositions: they require an explicit representation of the quasi-harmonic subspace. his subspace is spanned by the so-called global loops, whih are associated with the handles and holes of the geometry [START_REF] Cessenat | Mathematical Methods in Electromagnetism: Linear heory and Applications[END_REF][START_REF] Vechi | Loop-Star Decomposition of Basis Functions in the Discretization of the EFIE[END_REF]. he explicit representation of the quasi-harmonic subspace requires to ind the global loops of the geometry. Existing loop-inding algorithms have at least a complexity of O( ), where is the genus of the geometry [VY90; Laz+01]. When ∝ , the loop-inding algorithms cannot have a complexity less than quadratic [START_REF] Erikson | Optimally Cuting a Surface into a Disk[END_REF].

For a convenient application of the hierarhical preconditioner, it is desirable that the hierarhical basis can be applied multiplicatively. In standard codes the EFIE is discretized with RWG basis functions and the preconditioner is a sparse basis transformation matrix applied from let and right to the system matrix. Standard loop-inding algorithms do not generally ind the shortest loops, so that the transformation matrix mapping the global loops to RWG functions is dense. Algorithms that do ensure to ind the shortest loops (and thereby avoid a dense transformation matrix) have at least a cubic complexity [START_REF] Erikson | Optimally Cuting a Surface into a Disk[END_REF][START_REF] Dey | An Eicient Computation of Handle and Tunnel Loops via Reeb Graphs[END_REF].

As the leading complexity set by the fast multiplication methods is at most linearithmic [CRW93; ZVL05], the use of loop-inding algorithms is unatractive. Considering that the implementation of suh an algorithm is cumbersome and their application is only satisfactory for certain classes of geometries, it is desirable to ind a general purpose formulation where the searh for global loops is unnecessary, that can be integrated easily into existing codes, and that does not jeopardize the efectiveness of the fast multiplication method.

In this hapter, we present suh a formulation applicable to hierarhical bases for the EFIE operator meeting these criteria. To this end, we leverage quasi-Helmholz projectors allowing for an implicit representation of the quasi-harmonic Helmholz subspace. Helmholz projectors were irst introduced in [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF], where the projectors were used for stabilizing the Calderón preconditioned EFIE in the static limit. We show that we cannot directly integrate these Helmholz projectors in a hierarhical preconditioning sheme to bypass the searh for the global loops. Instead, it turns out that the standard split preconditioning formulation must be abandoned in favor of a let preconditioning formulation. he numerical results corroborate the efectiveness of the presented theory. In this hapter, we allow to be a multiply connected Lipshiz polyhedral domain. Available hierarhical bases rest upon a quasi-Helmholz decomposition [VPV05; ATV07; Che+09; ATV10; AVV08; AAE17] as the scalar potential operator must be preconditioned separately from the vector potential operator. For any hierarhical basis that shall be applicable to the formulation presented in this work, we require that the space of hierarhical functions is equal to the space of RWG functions:

= � ⊕ ⊕ �
, where is the space of RWG basis functions, � the space of hierarhical solenoidal functions, is the space of quasi-harmonic global loop functions, and � the space of hierarhical non-solenoidal basis functions. his allows us to deine linear mappings � Λ ∈ R × Λ , H ∈ R × H , and � Σ ∈ R × Σ from the hierarhical solenoidal and non-solenoidal functions to the RWG basis functions. We do not give an explicit representation of � Λ and � Σ , as our sheme is not limited to a speciic hierarhical basis.

In other words, we can express the system matrix T � Q whih is obtained by using the hierarhical basis in terms of a linear mapping

� Q = � � Λ H � Σ � (6.1) applied to T as T � Q = � Q T T � Q .
In general for the hierarhical basis preconditioner to be efective, the matrix has to be rescaled by using the diagonal preconditioner

� D � = 1/ � � T � Q � . (6.2)
his diagonal preconditioner is necessary for both curing the low-frequency and the dense-discretization breakdown. Summarizing, the preconditioned system matrix is

DT � Q D . (6.3)
Given the discussion in the previous hapter on the conditioning of the matrix (5.53), we assume that we have condition number bound It remains to link the projectors to the hierarhical basis preconditioner. Inspired by (4.85), a irst idea would be to use

cond � DT � Q D � ≲ log 2 � 1/ℎ 2 � . ( 6
� P = � ΛD � Λ X � Λ + P H √ + � Σ D � Σ X � Σ , (6.5)
requiring that

� P T T � P (6.6) is well-conditioned. he matrices D � Λ and D � Σ are the diagonal preconditioners of � Λ T T � Λ and � Σ T T � Σ , respectively. he matrices X � Λ ∈ C Λ × and X � Σ ∈ C Σ × are necessary to math the column dimensionality of P H since D � Λ ∈ C Λ × Λ and D � Σ ∈ C Σ × Σ but P H ∈ R × .
he parameter (instead of 1/ √ as in (6.5)) is necessary, because the diagonal preconditioning of D � Λ might have an efect diferent from a mere rescaling with 1/ √ (the hoice of is discussed in Section 6.c). All in all, the projector-based preconditioning of the quasi-harmonic subspace becomes more complicated than in (6.5) or in [And+13].

It might seem that Λ T and Σ T are natural hoices for X � Λ and X � Σ . But these matrices are ill-conditioned, as they have a derivative strength [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF]. Applying an ill-conditioned matrix from let and right to a well-conditioned matrix results, in general, in an ill-conditioned matrix and so (6.6) is not well-conditioned. 31 31 Alternatively, we could resort to the hierarhical basis transformation matrices � Λ T and � Σ T and rescale them suh that they become2 -stable transformations, that is that the discretization of the identity operator � Λ T G � Λ and � Σ T G � Σ is well-conditioned. However, both the solenoidal and the non-solenoidal hierarhical basis should be 2 ( )-stable, whih they not necessarily are (e.g., Section 6.b New Formulation Without the Searh for Global Loops Instead of hoosing Λ and Σ for X T � Λ and X T � Σ , one might consider using the transformation matrices Γ and Θ of cotree and tree functions, respectively, since Γ and Θ do not possess a derivative (or integrative) strength. For the deinition of tree/cotree functions see [START_REF] Sun | Spurious Modes in Finite-Element Methods[END_REF] (what we call tree functions in this hapter, is referred to as cotree function in [START_REF] Sun | Spurious Modes in Finite-Element Methods[END_REF] and vice versa; they can be obtained in linear complexity by a depth-irst searh). he dimensionality of the tree functions Σ (as they must span the harge space), and thus, the dimensionality of the cotree space is -Σ = Λ + H , where H is the dimensionality of the quasi-harmonic subspace. To math the dimensionality of � , we must eliminate H cotree functions resulting in the reduced cotree transformation matrix Γ . Despite their well-conditionedness, Γ and Θ may not be used, because their null spaces are not orthogonal to the null space of P H . he non-orthogonality of the null spaces induces a new form of ill-conditioning. To illustrate this efect, we used a cube with a hole on eah side so that the cube has ive global loops (this cube is shown in Figure 6.1; there are ive and not six global loops, because one hole makes the cube an open but still simply connected structure). A plane wave with frequency 1 MHz impinges on this cube with side length /150, where is the wavelength. Figure 6.4 shows the spectrum of the system matrix. When we use the cotree/tree formulation, ive isolated singular values appear. hese are the singular values associated with the global loops. hese singular values are not clustered meaning that the quasi-harmonic subspace has become ill-conditioned.

Hence, we need matrices that have null spaces that are orthogonal to the null space of P H , but whih are, in contrast to Λ T and Σ T , well-conditioned. he following lemma provides us with matrices having the desired properties. Lemma 6.1. he irst Λ ( Σ ) singular values of (Λ T Λ) -1/2 Λ T ((Σ T Σ ) -1/2 Σ T ) are equal to one, the rest of the singular values are zero.

Proof. We proof this for (Λ T Λ) -1/2 Λ T . he proof for (Σ T Σ ) -1/2 Σ T follows analogously. Let for = 1, … , Λ and Π ∈ R Λ × Λ . he inverse square root of Λ T Λ is given by the decomposition RΠ -1/2 R T . We then ind

� Λ T Λ � -1/2 Λ T = RΠ -1/2 R T R ⏟⏞⏞⏞ �⏞⏞⏞ � I ΞU T = RΨU T , (6.8)
where

Ψ = Π -1/2 Ξ, Ψ ∈ R Λ × with � Ψ � = 1, = 1, … , Λ and zero otherwise.
It follows from (6.7) and (6.8) that the range and null spaces of Λ T and (Λ T Λ) -1/2 Λ T are identical.

Lemma 6.1 infers that the matrices (Λ T Λ) -1/2 Λ and (Σ T Σ ) -1/2 Σ are well-conditioned up to the null space of Λ and Σ , respectively, and that they, because of their range and null space mathing those of P H , can be used instead of Λ T and Σ T . Section 6.b New Formulation Without the Searh for Global Loops Modifying the preconditioner proposed in (6.5) to

� P = � ΛD � Λ� Λ T Λ � -1/2 Λ T + P H √ + � Σ D � Σ � Σ T Σ � -1/2
Σ T , (6.9)

we obtain a hierarhical preconditioner, whih takes implicitly care of the quasi-harmonic Helmholz subspace. Figure 6.4 demonstrates the correctness of this square root based approah. We see that all the singular values are clustered for a suitably hosen as described in Section 6.c. he downside of this approah is obvious: we have to compute the inverse square root of a matrix. To avoid this, we propose a let preconditioner instead of this split preconditioner: instead of � P T T � P (6.10) we suggest to use � P � P T T , (6.11)

where we note that � P � P T does not contain inverse square roots anymore, as we have

� P � P T = � ΛD 2 � Λ � Λ T + P H + � Σ D 2 � Σ � Σ T .
(6.12)

We note that the matrices � P � P T T and � P T T � P are similar, since

� P -1 � � P � P T T � � P = � P T T � P . (6.13)
and thus they have the same eigenvalue spectrum, whih typically results in a convergence behavior of the iterative solver comparable to the split preconditioned case.

For extremely low frequencies this approah needs a slight modiication: since the solenoidal/quasi-harmonic and the non-solenoidal part of the current, P ΛH j and P Σ j , scale diferently in frequency, numerical cancellation appears in j , whih would render it impossible to compute the electric and magnetic ield (see [And+13] for more details). In this case, one should apply the low-frequency preconditioner from Section 4.d. and replace T by -iP T T P so that P does not have to take care of the low-frequency breakdown anymore. In more detail, the preconditioned matrix would read

� P � P T � -iP T T P � (6.14)
where the diagonal preconditioners should be obtained from the frequency independent system matrices

� D � Λ � = 1/ � � � Λ T T 0 A � Λ � (6.15)
and

� D � Σ � = 1/ � � � Σ T T 0 Φ � Σ � (6.16)
since -iP T T P is well-conditioned in frequency. If the basis from the previous hapter is used, then (5.58) and (5.59) can be used to determine D � Λ and D � Σ . We note that -iP T T P and � P T (-iP T T P ) � P become SPD for → 0. his allows to obtain an upper bound for the condition number of � P � P T (-iP T T P ) in the static limit. Lemma 6.2. If we have a hierarchical basis such that

x T x ≲ lim →0 x T � P T � -iP T T P � � P ≲ log 2 (1/ℎ)x T x , ∀x ∈ R ≲ log 2 (1/ℎ) (6.17) is satisied, then cond � lim →0 � P � P T � -iP T T P � � ≲ log 2 (1/ℎ) (6.18)
holds.

Proof. In the following, we assume → 0 and suppress the limit. Using Rayleigh quotients, we have

x T x ≲ x T � P T � -iP T T P � � P x ≲ log 2 (1/ℎ)x T x , ∀x ∈ R . (6.19)
Using the substitution � P x = y , we have

y T � P -T � P -1 y ≲ y T -iP T T P y ≲ log 2 (1/ℎ)y T � P -T � P -1 y , ∀y ∈ R (6.20)
and thus

y T � � P -T � P -1 � 2 y ≲ y T � -iP T T P � 2 y ≲ log 4 (1/ℎ)y T � � P -T � P -1 � 2 y , ∀y ∈ R . (6.21) Section 6.c
Implementational Issues

Using the substitution � P -T � P -1 y = x , we obtain

x T x ≲ x T � P � P T � -iP T T P � 2 � P � P T x ≲ log 4 (1/ℎ)x T x , ∀y ∈ R . (6.22) his implies cond � � P � P T � -iP T T P � 2 � P � P T � ≲ log 4 (1/ℎ) (6.23)
and hence

cond � � P � P T � -iP T T P � � ≲ log 2 (1/ℎ) . (6.24)
Considering that the efect of the non-dynamic kernel is only a compact perturbation, we do not expect a signiicant impact on the employed method. he numerical results shown in Section 6.d conirm our expectations and demonstrate the legitimacy of (6.11).

c) Implementational Issues

Since we were not interested in extremely low-frequencies, we used (6.11) to demonstrate the applicability of our approah. So far it has not been discussed, how to select the scaling parameter . he spectrum of the global loops scales linearly in frequency. For that reason, the preconditioner in (4.85) rescales both the solenoidal and quasi-harmonic subspace with the same factor 1/. When we use the hierarhical basis and a diagonal preconditioner is obtained by using (6.2), one could simply use the average factor by whih the spectrum is shited, that is, Albeit (6.11) is a signiicant improvement compared with (6.10), further enhancements are possible. If a hierarhical basis is used, where it is advisable to use the diagonal preconditioner to obtain D , then numerical results show that it suices to compute D � Λ and D � Σ as the diagonal preconditioners of � Λ T T near � Λ and � Σ T T near � Σ , where T near is the near-interaction part of the system matrix as deined in the ACA sheme. his matrix is sparse and hence the products can be formed explicitly.

= Λ ∑ Λ =1� � Λ T (iT A ) � Λ � . ( 6
As D = � D � Λ D � Σ � , D � Λ and D �
Σ can be obtained without modiication of an existing hierarhical basis preconditioner code.

d) Numerical Results

To prove the general efectiveness of (6.11), we tested it with both kinds of hierarhical bases: a hierarhical basis for unstructured meshes (HB-U) [START_REF] Andriulli | Hierarhical EM Preconditioners with Spectral Domain Partitioning[END_REF] (whih in this case preconditions only the scalar potential operator) and a hierarhical basis for structured meshes (HB-S) [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF] (whih preconditions in this case the vector and the scalar potential operator). Section 6.d

Numerical Results

In the irst test, we veriied the applicability of the new formulation with a simply connected geometry. A plane wave with frequency 1 MHz impinges on a cube with side length /300. In order to use the HB-S, we used a dyadic mesh reinement. To accelerate the computation, we compressed the system matrix, compression precision 1 ⋅ 10 -7 , with the adaptive cross approximation (ACA), whih reduces the cost of a single matrix-vector product to O( log ) [START_REF] Zhao | he Adaptive Cross Approximation Algorithm for Accelerated Method of Moments Computations of EMC Problems[END_REF]. We compared the new formulation from (6.11) with the loop-tree preconditioner and with the standard formulation of eah hierarhical basis preconditioner as it is denoted in (6.3). From Figure 6.2, we can see that the proposed formulation has a comparable eicacy as the standard formulation of (6.3) in terms of number of iterations. In fact, for some cases the let preconditioning formulation outperforms the standard split preconditioning formulation in terms of number of iterations. he beter performance of the HB-S compared with the HB-U stems from the fact that the used HB-U only preconditions the scalar potential operator.

As a second example, we veriied the low-frequency stability. We used the multiply connected cube with six holes (see Figure 6.1). For the generation of the results we used the HB-U. We compared the proposed method with the case where = 1 independent from the frequency. One can see in Figure 6.3 that the condition number remains constant, while when = 1 the condition number hanges with the frequency. Figure 6.4 shows the spectrum for the case when = 1 kHz. One can see that the part of the spectrum associated with the global loops is well-contained in the rest of the spectrum, while for = 1 the singular values associated with the global loops are separated from the rest of the spectrum. In comparison with Figure 6.4, we can see the integrity of the preconditioner: the singular values associated with the global loops are clustered.

As a third example, we tested the HB-U on a plate with an increasing amount of holes (as an example, a plate with 64 holes is shown in Figure 6.6). A plane wave with frequency 1 MHz impinges on the plate. We compared the HB-U in the new formulation with a classical loop-tree preconditioner that does not take into account the global loops. Figure 6.5 shows the relative error. he CGS precision was 1 ⋅ 10 -6 . he large and with the number of holes growing error shows the importance of including the global loops-explicitly or implicitly.

Lastly, we were interested in the performance when we deal with more complex structures. To evaluate the performance, we used the model of a MiG-15 as shown in Figure 6.7. his model is an open geometry with one hole. We tested the proposed preconditioner with the HB-U. he precision of the iterative solver is 1 ⋅ 10 -6 and the precision of the ACA is 1 ⋅ 10 -7 . Figure 6.9 shows the number of iterations. We observe that the number of iterations needed by the new formulation is in the same order as the number of iterations needed by the standard formulation of hierarhical basis preconditioners. his demonstrates that the optimality efect of the hierarhical basis-that the number of iterations grows at most logarithmically-is maintained.

e) Conclusion

We presented a formulation that allows to apply hierarhical bases for preconditioning the EFIE on multiply connected geometries without searhing for global loops-a searh whih has, in general, a quadratic cost. Instead, the quasi-harmonic subspace is preserved by implicitly recovering the global loops using a let preconditioning formulation instead of the standard split preconditioning formulation of the hierarhical bases preconditioner. hereby, the leading complexity O( log ) set by the fast multiplication method is maintained regardless of the 10 -1 10 0 10 1 Fig. 6.4.: Cube with six holes: the singular value spectrum using cotree/tree functions (formulation of (6.5) with X � Λ = Γ T and X � Σ = Θ T ) and preconditioned loop/star functions (formulation of (6.8)), and the singular value spectrum of the proposed method (formulation of (6.11) with hosen as in (6.25) and with = 1). he frequency is 1 kHz. he plot of the square root formulation is hidden behind the plot of "his work". Fig. 6.9.: MiG-15: number of iterations for a CGS solver with precision 1 ⋅ 10 -6 and ACA accuracy ⋅ 10 -7 .

Section 6.e Conclusion number of global loops. he proposed formulation was evaluated with diferent hierarhical bases and for all bases the results demonstrate a performance comparable with the results obtained by the standard formulation of the hierarhical bases preconditioners, where a loop-inding algorithm had been employed to construct the quasi-harmonic subspace explicitly.

Chapter 7

On the Hierarhical Preconditioning of the CFIE his chapter analyzes how hierarchical bases preconditioners constructed for the EFIE can be efectively applied to the CFIE. In addition, a new scheme is proposed: the CFIE is implicitly preconditioned on the solenoidal Helmholz subspace by using a Helmholz projector, while a hierarchical non-solenoidal basis is used for the non-solenoidal Helmholz subspace. Numerical results corroborate the efectiveness of the new formulation. his chapter is based on [AAE16b].

A t resonance freqencies, the EFIE and the MFIE are not uniquely solv- able (see the discussion in Section 4.c), an issue that has been overcome by solving the CFIE [START_REF] Mauz | H-Field, E-Field, and Combined-Field Solutions for Conducting Bodies of Revolution[END_REF]. Since, however, the CFIE is a combination of the EFIE and of the MFIE, it is not free from the low-frequency and the dense-discretization breakdown. For this reason, eforts have been made to extend the hierarhical basis preconditioners to the CFIE [START_REF] Francavilla | Applications of a Hierarhical Multiresolution Preconditioner to the CFIE[END_REF].

he conference contribution [START_REF] Francavilla | Applications of a Hierarhical Multiresolution Preconditioner to the CFIE[END_REF] reported on the fact that the application of a hierarhical loop/hierarhical non-solenoidal basis preconditioner to the CFIE resulted in a well-conditioned equation. Given that the hierarhical loop preconditioner is not capable of preconditioning the EFIE [ATV10; LO98], we found the result surprising and decided to perform a theoretical investigation of the problem.

his leads to this hapter whose novelty content is twofold: (i) we show on a theoretically sound basis that a direct application of the hierarhical basis preconditioner to the CFIE is possible when the hierarhical loop [START_REF] Vipiana | A Multiresolution Method of Moments for Triangular Meshes[END_REF] or the three-point hierarhical loop functions [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF] are used as solenoidal basis. Beyond the discussion in [AAE16b], we discuss the applicability of the new dual On the Hierarhical Preconditioning of the CFIE Chapter 7

Haar basis developed in Chapter 5. (ii) We propose a new hierarhical let preconditioning sheme that diferent from any other hierarhical approah for the preconditioning of the CFIE (including those communicated in [START_REF] Francavilla | Applications of a Hierarhical Multiresolution Preconditioner to the CFIE[END_REF] and detailed in [START_REF] Righero | Hierarhical Bases Preconditioner to Enhance Convergence of CFIE With Multiscale Meshes[END_REF]) can provide optimal (up to a logarithmic term) preconditioning both on structured and unstructured meshes. Numerical results will show the efectiveness of the new sheme for canonical and realistic examples.

a) Spectral Analysis of the EFIE

We assume in this analysis that is simply connected. he operators W and V are known to sufer from the dense-discretization breakdown [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF]. If we precondition these operators successfully, we can precondition the EFIE (see also Chapter 5). Diferent hierarhical bases have been introduced in the past. For preconditioning T Φ , the bases developed are all equivalent, and we hoose as a representative the basis from [START_REF] Andriulli | Hierarhical Bases for Nonhierarhic 3-D Triangular Meshes[END_REF].

For T A , two hierarhical bases have been commonly used until the publication of [START_REF] Adrian | A Hierarhical Preconditioner for the Electric Field Integral Equation on Unstructured Meshes Based on Primal and Dual Haar Bases[END_REF]: the hierarhical loops � H [VPV05] (the subscript "H" denotes that we deal with hierarhical loops. It has nothing to do with global loops, whih are denoted as in this thesis) and the three-point hierarhical loops � T [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF]. 33 Only the later can cure, however, the dense-discretization breakdown of the EFIE [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF]. Both � H and � T require a structured mesh: a mesh that is obtained by iteratively reining, for example dyadically, times a coarse mesh. Accordingly, � H and � T are deined on hierarhical levels (for details, see [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF]). hese functions are related to scalar functions by

� H = ∇ × � H (7.1) and � T = ∇ × � T , (7.2)
where � H are the hierarhical nodal functions presented by Yserentant [Yse86] and �

T are the three-point hierarhical nodal functions by Stevenson [START_REF] Stevenson | Stable hree-Point Wavelet Bases on General Meshes[END_REF]. As 33 Due to the relationship of loop functions with the piecewise linear functions, as will be outlined in this section, it should be possible to generate further hierarhical solenoidal bases by using, for example, a basis as presented in [START_REF] Dahmen | Element-by-Element Construction of Wavelets Satisfying Stability and Moment Conditions[END_REF]. his would, however, not impact the discussion in this hapter since � H are -stable, 0 ≤ ≤ 1, that is, they are optimal for preconditioning in the relevant range of (and for any other hierarhical basis the analysis could be adapted).

Section 7.a Spectral Analysis of the EFIE we did for the loop basis, we can deine transformation matrices � Λ H and � Λ T for the hierarhical loop and three-point hierarhical loop functions, respectively. In this hapter we assume, following the discussion in Section 4.d., that the bases are all linearly independent, that is, a loop function has been eliminated so that

Λ, � Λ H , � Λ T ∈ × Λ .
In addition, we consider the new solenoidal hierarhical basis introduced in Chapter 5, where we denote the transformation matrix as � Λ. For notational convenience, we assume that these functions have been rescaled as explained in Chapter 5 suh that cond( � Λ T T A � Λ) ≲ log 2 � 1/ℎ 2 � , thus there is no need for an additional diagonal preconditioner matrix in order to precondition the EFIE.

he operator W maps from the Sobolev space 1/2 ( ) to -1/2 ( ) and induces an inner product on 1/2 /R [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF]. he three-point hierarhical nodal functions T are 1/2 -stable, that is, when rescaled appropriately, the three-point hierarhical loop functions can precondition the vector potential operator [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF] resulting in

cond � D 1 Λ � Λ T T T A � Λ T D 1 Λ � ≲ 1 , (7.3)
with

� D Λ � = 2 (1-) Λ () , (7.4)
where Λ (), ∈ {1, … , Λ }, is the number of the level of the th three-point hierarhical loop function. We note that D 1 Λ makes the three-point hierarhical nodal functions 1/2 -stable, while D 0 Λ results in 2 -stable and D 2 Λ in 1 -stable three-point hierarhical nodal functions. 34 he hierarhical nodal functions, on the other hand, are not 1/2 -stable [START_REF] Lorenz | Multilevel Finite Element Riesz Bases in Sobolev Spaces[END_REF], and thus, the hierarhical loop functions fail to precondition T A . Likewise, the nodal functions are not 1/2 -stable resulting in [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF] 35

cond � Λ T T A Λ � ≲ 1/ℎ . (7.5)
We note that an application of a Jacobi preconditioner can improve the behavior quantitatively but not qualitatively (see Fig. 7.1a).

34 his means that for D 1 Λ the three-point hierarhical loop functions are (div 0, )-stable and for D 2 Λ they are 2 ( )-stable, where (div 0, ) = � ∈ (div , ) ∶ div = 0 � . he deinition of D Λ always depends on the deinition of the hierarhical basis. Naturally, one could have tailored the hierarhical basis suh that it is 2 -stable without the application of a diagonal preconditioner.

On the Hierarhical Preconditioning of the CFIE Chapter 7

Let � be the hierarhical non-solenoidal functions for whih we deine the transformation matrix � Σ . Since V is inducing an -1/2 inner product, we need a rescaling deined by � D Σ� = 2 + Σ ()/2 to make div � stable in -1/2 , where Σ (), ∈ {1 … , Σ }, is the number of the level of the th hierarhical non-solenoidal function. hen it can be proved that (see [START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF] and references therein)

cond � D Σ � Σ T T Φ � Σ D Σ � ≲ log 2 � 1/ℎ 2 � . (7.6)
We deine the overall transformation matrix

Q X /, � Σ ∶= � X D Λ / √ � Σ D Σ √ � , (7.7)
where X can be Λ, � Λ H , or � Λ T . Summarizing, we ind for the preconditioned EFIE that the loop/hierarhical non-solenoidal basis yields

cond � � Q 0 Λ/, � Σ � T T Q 0 Λ/, � Σ � ≲ 1/ℎ (7.8)
and that the three-point hierarhical loops/hierarhical non-solenoidal basis yields

cond � � Q 1 � Λ T /, � Σ � T T Q 1 � Λ T /, � Σ � ≲ log 2 � 1/ℎ 2 � . (7.9)
b) Spectral Analysis and Preconditioning of the CFIE Following these considerations, we study the CFIE. For the sake of brevity, the analysis is carried out only for the conformingly discretized CFIE. he indings are, however, the same as for the standard CFIE. Because of the identity operator of the MFIE, the conditioning of the CFIE is beter than that of the EFIE with

cond � � C � ≲ 1/ℎ . (7.10)
While the largest singular value of T Φ still grows to ininity, the singular values of T A are shited from 0 to 1/2 by I/2. Next, we have to study a Helmholzdecomposed CFIE. Following the argumentation in [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF], the compact operator K (i.e., K is compact on smooth surfaces; the numerical results imply that Section 7.b Spectral Analysis and Preconditioning of the CFIE the spectral properties hold for non-smooth surfaces as well) can be neglected and it suices to analyze how G hanges the EFIE behavior. We notice that when I is discretized with loop, hierarhical loop, and three-point hierarhical loop functions, the resulting matrix is equivalent to the discretization of the Laplace operator in its weak formulation with nodal, hierarhical nodal, and three-point hierarhical nodal functions since [And12a]

� ∇ × , ∇ × � 2 = � ∇ , ∇ � 2 .
(7.11)

he Laplace operator induces an inner product on 1 /R, and hence, we need 1 -stable basis functions.

It is well-known that the nodal functions are not 1 -stable [START_REF] Lorenz | Multilevel Finite Element Riesz Bases in Sobolev Spaces[END_REF]; the condition number grows with (1/ℎ 2 ), and given that the Laplace operator Δ is a pseudo-diferential operator of order +2 and W is a pseudo-diferential operator of order +1, the total order of W + Δ is +2, and hence, we have

cond � Λ T � CΛ � ≲ 1/ℎ 2 . (7.12)
In other words, loop functions applied to the CFIE result in a conditioning worse than when they are applied to the EFIE. Similar to [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF], we ignore K for the condition number analysis since its eigenvalues cluster around zero. Hence, we only consider I and T . We note that on multiply connected geometries, we cannot neglect K [START_REF] Bogaert | Low Frequency Stability of the Mixed Discretization of the MFIE[END_REF].

he hierarhical nodal functions are not 1 -stable. Yet, for 1 and when rescaled appropriately, they result in a condition number that grows with [Yse86]

O � log 2 � 1/ℎ 2 � � , (7.13)
and thus, we have

cond � � Λ T H � C � Λ H � ≲ 1/ℎ 2 . (7.14)
he three-point hierarhical nodal functions are 1 -stable [START_REF] Stevenson | Stable hree-Point Wavelet Bases on General Meshes[END_REF], and thus, we have

cond � D 2 Λ � Λ T T � C � Λ T D 2 Λ � ≍ 1 . (7.15)
On the Hierarhical Preconditioning of the CFIE Chapter 7

Notice that the matrix

D 2 Λ � Λ T T T � Λ T D 2
Λ is ill-conditioned; since we use 1 -stable functions, the singular values are now accumulating at zero. his does not destroy the well-conditioning as

D 2 Λ � Λ T T G � Λ T D 2 Λ is spectrally equivalent to I , that is, x T D 2 Λ � Λ T T G � Λ T D 2 Λ x ≍ x T x , ∀x ∈ R Λ , (7.16)
and thus the spectrum is bounded from below. Since T Φ is a derivate operator, the Gram matrix

D Σ � Σ T G � Σ D Σ of the rescaled � must have integrative strength to precondition T Φ .
As it is true for integrative operators, the singular values of this Gram matrix cluster around zero, and since

D Σ � Σ T T � Σ D Σ is spectrally equivalent (up to a logarithmic term) to I , we can conclude that cond � D Σ � Σ T � C � Σ D Σ � ≲ log 2 � 1/ℎ 2 � .
(7.17) Summarizing, we have for the loop/hierarhical non-solenoidal basis preconditioner

cond � � Q 0 Λ, � Σ � T � CQ 0 Λ, � Σ � ≲ 1/ℎ 2 , (7.18)
for the hierarhical loop/hierarhical non-solenoidal basis preconditioner

cond � � Q 2 � Λ H , � Σ � T � CQ 2 � Λ H , � Σ � ≲ log 2 � 1/ℎ 2 � , (7.19)
and for the three-point hierarhical loop/hierarhical non-solenoidal basis preconditioner

cond � � Q 2 � Λ T , � Σ � T � CQ 2 � Λ T , � Σ � ≲ log 2 � 1/ℎ 2 � . (7.20)
Equivalently said, the combination loop/hierarhical non-solenoidal functions does not precondition the CFIE, while both combinations hierarhical loop/hierarhical non-solenoidal and three-point hierarhical loop/hierarhical non-solenoidal basis are a valid preconditioner for the CFIE. Both the hierarhical loops and the three-point hierarhical loops only operate on structured meshes. he hierarhical solenoidal basis � Λ presented in Chapter 5 might be an interesting alternative since it can be deined on unstructured meshes as well. It is based on an explicit inversion of the dual Haar basis transformation matrix and stability results for the 1/2 -space were obtained by using Oswald's result that the (dual) Haar basis is -1/2 -stable [START_REF] Oswald | Multilevel Norms for -1/2[END_REF] and by leveraging a discrete Calderón identity.

When it comes to the Laplace operator it is not immediately clear whether or not the dual Haar basis is applicable since there is no formal proof available that the Haar basis is -1 -stable. By anticipating a result from the next hapter, we assume that we have

x T �x ≍ x T W G -1 W x ≍ x T WW x /ℎ 2 , ∀x ∈ R . (7.21)
As shown in Chapter 5, the matrix

D -1 Λ H ‡ Λ G -1 W G -T � H ‡ Λ� T D -1 Λ (7.22)
is well-conditioned up to the log 2 (1/ℎ) perturbation and we note that the let preconditioned version

W G -T � H ‡ Λ� T D -2 Λ H ‡ Λ G -1 (7.23)
has the same eigenvalue spectrum due to the matrix similarity. hen clearly, we have

cond � G -T � H ‡ Λ� T D -2 Λ H ‡ Λ G -1 W W G -T � H ‡ Λ� T D -2 Λ H ‡ Λ G -1 � ≲ log 4 (1/ℎ) (7.24)
and by using (7.21), we ind

cond � G -T � H ‡ Λ� T D -2 Λ H ‡ Λ G -1 �G -T � H ‡ Λ� T D -2 Λ H ‡ Λ G -1 � ≲ log 4 (1/ℎ) (7.25)
Consequently, we can use

G -T � H ‡ Λ� T D -2 Λ H ‡ Λ G -1 (7.26)
as split preconditioner for � resulting in a condition number that grows with log 4 (1/ℎ). he message from this consideration is that it might be possible to modify � Λ suh that it becomes applicable to the CFIE, but at the price that the On the Hierarhical Preconditioning of the CFIE Chapter 7

upper bound of O(log 2 (1/ℎ)) on the condition number set by the Haar basis preconditioner for T Φ is deteriorated. Since we are interested in maintaining this bound, we do not further consider this basis.

In practical scenarios, the mesh is typically unstructured and thus hierarhical loop and three-point hierarhical loop functions are not available; yet, from the presented theory it is clear that the use of loop functions is not efective and a diferent strategy is necessary. To this end, we deine the transformation matrix

√ Λ ∶= Λ � Λ T Λ � -1/2 . (7.27)
he Gram matrix of these new loop functions is well-conditioned since

x T √ Λ T G √ Λx ≍ x T � Λ T Λ � -1/2 Λ T Λ � Λ T Λ � -1/2 x = x T x , ∀x ∈ R Λ . (7.28)
Using these orthogonalized loop functions, we obtain as hierarhically preconditioned system

� Q 0 √ Λ, � Σ � T � CQ 0 √ Λ, � Σ j = � Q 0 √ Λ, � Σ � T � -C e + (1 -C )G G -1 × , h� , (7.29) with cond � � Q 0 √ Λ, � Σ � T � CQ 0 √ Λ, � Σ � ≲ log 2 � 1/ℎ 2 � . (7.30)
While this approah is theoretically sound, it laks practical applicability due to the presence of square root matrices. Following the strategy outlined in Chapter 6, we modify the approah by considering a let instead of the split preconditioner in (7.29). hereby, we obtain

Q 0 √ Λ, � Σ � Q 0 √ Λ, � Σ � T = Λ � Λ T Λ � -1 Λ T + � Σ D 2 Σ � Σ T . (7.31)
As discussed in Section 4.d., the matrix Λ(Λ T Λ) -1 Λ T ≡ P Λ is a projector to the solenoidal Helmholz subspace [And+13] and compared with (Λ T Λ) 1/2 x = b, the system (Λ T Λ)x = b can be solved more rapidly [START_REF] Hale | Computing , log(), and Related Matrix Functions by Contour Integrals[END_REF]. he let preconditioner is motivated by the same argument as in Chapter 6: the let preconditioned system matrix is similar to the split precondition system matrix, so that they have the Section 7.b Spectral Analysis and Preconditioning of the CFIE same eigenvalue spectrum and the preconditioned conjugate gradient method would show the same iterative behavior if it were applicable. he conjugate gradient method is not applicable since � C is not HPD. But as in Chapter 6, we still expect a reasonable behavior of the iterative method given that the eigenvalue spectrum is the same. Albeit there is not strict proof for the well-conditioning, the numerical results corroborate our expectations.

Usually, best results are obtained when the (hierarhical) functions are rescaled by leveraging on a Jacobi preconditioner, that is, we use

� D Σ� = 1/ � � � Σ T � C � Σ � . (7.32)
For a fair comparison of the diferent bases discussed, solenoidal and non-solenoidal alike, eah basis is rescaled by using suh a Jacobi preconditioner.

When the conforming CFIE is used, the Gram matrices prohibit to obtain the Jacobi preconditioner eiciently in a direct manner. For the preconditioner presented in this work, this problem can be avoided by using

� D Σ� = 1/ � � � Σ T T � Σ � (7.33)
that is, the same procedure as for the EFIE can be used. If the basis from Chapter 5 is used, then (5.59) can be employed. his hoice cures the low-frequency breakdown (i.e., the condition is independent of the frequency); however, a further alignment of the singular value branhes associated with the solenoidal and the non-solenoidal Helmholz subspace improves the condition number. To this end, by selecting

= � � � P Λ � C � � �2 and = � � � � Σ D 2 Σ � Σ T � C � � �2
, we can deine

� P ∶= P Λ / + � Σ D 2 Σ � Σ T / (7.34)
resulting in the system

� P � Cj = � P � -C e + (1 -C )G G -1 × , h� . (7.35)
he norms can be estimated rapidly by using a power iteration method. While the CFIE is typically not used for extremely low frequencies, for suh a scenario it would be necessary to apply the new preconditioner to the frequency 119 On the Hierarhical Preconditioning of the CFIE Chapter 7 preconditioned P T � CP , where P was deined in (4.85), in order to avoid numerical cancellation in the current and in the right-hand side. he reader should note, however, two obstacles: suh a low-frequency sheme is only applicable when the MFIE is conformingly and carefully discretized. he reason is that for a standard MFIE the loop-loop part of the system matrix is not scaling in frequency. It does so for a conformingly discretized MFIE, but it requires a meticulous discretization [START_REF] Bogaert | Low Frequency Stability of the Mixed Discretization of the MFIE[END_REF].

c) Numerical Results

In the following, we compare our new formulation (7.35) with the unpreconditioned CFIE and when standard hierarhical basis preconditioners are applied. he standard hierarhical preconditioners we use always consist of the same hierarhical non-solenoidal basis complemented by loop, hierarhical loop, or three-point hierarhical loop functions, respectively. We do this for the conformingly discretized CFIE operator � C, and the standard CFIE C (denoted as S-CFIE). Furthermore, we used C = 0.9 for the CFIE parameter. By this hoice we proit from the fact that the EFIE usually yields a higher accuracy compared with the MFIE [START_REF] Oluoglu | Magnetic-Field Integral Equation [EM Programmer's Notebook[END_REF].

First, we analyzed the dense-discretization stability by using a cube with side length 1 m. We used a plane wave excitation and ixed the frequency at = 1 MHz with = /(2π), where is the speed of light. We varied the average edge length ℎ from 1.13 m to 0.07 m, and the number of iterations is displayed as a function of the spectral index 1/ℎ in Fig. 7.1a. he results conirm the presented theory: in particular, we can conclude from the igure that loop functions should not be used with the CFIE, and that hierarhical loops can safely be used with the CFIE but not with the EFIE. he new formulation (7.35) performs well both when applied to a conformingly discretized CFIE as well as a standard CFIE.

Next, we veriied the frequency stability. Figure 7.1b displays the condition number as a function of the frequency . We see that all preconditioned formulations are free from the low-frequency breakdown. Figure 7.1c shows that all preconditioned CFIEs are resonance-free. We also see that in terms of the condition number, the unpreconditioned CFIE works beter than the preconditioned counterparts. his is not unexpected since hierarhical basis preconditioners need to be adapted to high-frequency problems.

Part III.

Calderón Multiplicative Preconditioners 125

Chapter 8

On a Reinement-Free Calderón Multiplicative Preconditioner for the EFIE his chapter presents a Calderón preconditioner for the EFIE, which does not require a barycentric reinement of the mesh, where the condition number can be bounded independently from the frequency and average edge length of the mesh, and which yields an Hermitian, positive deinite (HPD) system matrix allowing for the usage of the CG solver. Diferent from existing Calderón preconditioners, no second discretization of the EFIE operator with BC functions is necessary. We obtain this preconditioner by leveraging spectral equivalences between (scalar) integral operators, namely the single layer and hypersingular operator, and the Laplace-Beltrami operator. Since our approach incorporates Helmholz projectors, there is no search for global loops necessary and our method remains stable on multiply connected geometries. he numerical results demonstrate the efectiveness of our approach for both canonical and realistic (multi-scale) problems. Preliminary results have been presented at conferences [AAE14a; AAE15].

C alderón identity based preconditioners, unlike the hierarhical ba- sis preconditioners, allow to obtain an EFIE, where the condition number has an upper bound independent from ℎ [CN02; BC07; And+08; Con+02; Ada04]. In the static limit, however, the Calderón strategies stop working due to numerical cancellation in both the right-hand side excitation vector and the unknown current since solenoidal and non-solenoidal components scale differently in [START_REF] Zhang | Magnetic Field Integral Equation at Very Low Frequencies[END_REF]. Explicit quasi-Helmholz decompositions do not sufer from this cancellation since the solenoidal and non-solenoidal components are stored separately. To make Calderón preconditioners stable in the static limit, one could combine the Calderón multiplicative preconditioner (CMP) with an explicit quasi-Helmholz decomposition.

hese approahes sufer, however, from the same defect that explicit quasi-Helmholz decompositions share: if the geometry is multiply connected, then the quasi-harmonic global loop functions have to be added to the basis of the decomposition [START_REF] Vechi | Loop-Star Decomposition of Basis Functions in the Discretization of the EFIE[END_REF]. Diferent from loop, star, or tree functions, (or any of the hierarhical bases mentioned in this thesis) the construction of the global loops is, in general, a costly operation making the overall problem O( 2 log( )), where is the number of unknowns [AAE14b] (see also Chapter 6). In order to avoid the construction of the global loops, a modiied CMP has been presented whih leverages an implicit quasi-Helmholz decomposition based on projectors [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF]And+13]. hese projectors require the application of the inverse primal (i.e., cell-based) and the inverse dual (i.e., vertex-based) graph Laplacian, a task for whih blakbox-like preconditioners suh as algebraic multigrid methods can be used for obtaining the inverse rapidly. We note that a sheme based on an explicit loop-star quasi-Helmholz decomposition does not ofer an alternative since the inverse Gram matrices appearing therein are all spectrally equivalent to discretized Laplace-Beltrami operators, however, with the additional hallenge that the Gram matrices are not symmetric since the loop-star basis is applied to a mixed Gram matrix, that is, BC functions are used as expansion and rotated RWG functions are used as testing functions [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF]And+13].

he work in [And+08; And+13] demonstrated a Calderón sheme that can be integrated relatively easily in existing codes. Instead of discretizing the operator on the standard mesh with RWG and BC functions, only a single discretization with RWG functions on the barycentrically reined mesh is necessary. Yet, there are caveats: (i) the memory consumption as well as the costs for a single-matrix vector product are increased by a factor of six and (ii) for open structures the codes must be modiied suh that half-RWG functions on the boundary edges are included for the discretization of the EFIE.

In this work, we propose a reinement-free Calderón multiplicative preconditioner (RF-CMP) for the EFIE. Diferent from existing Calderón preconditioners, no BC functions are employed, so that a standard discretization of the EFIE with RWG functions can be used. No modiications for open structures are necessary, though similar to the CMP, we do not have a constant upper bound of the Section 8.a Bakground condition number. What is more, we get a system matrix, whih is HPD. We obtain this result by leveraging spectral equivalences between the single layer and hypersingular operator known from electrostatics and the Laplace-Beltrami operator. Similar to [And+13], graph Laplacians need to be inverted. Since the new system matrix is HPD, we are allowed to employ the CG solver. In contrast to other Krylov subspace methods, it guarantees convergence and has the least computational overhead. he numerical results corroborate the new formulation.

he hapter is structured as follows. Section 8.a discusses the bakground; Section 8.b introduces the new formulation and provides implementational details, while Section 8.c provides the theoretical apparatus behind the new formulation. For the implementation of the new preconditioner, it is not necessary to study Section 8.c. Numerical results demonstrating the efectiveness of the new approah are shown in Section 8.d. a) Bakground he matrix T is ill-conditioned both in and ℎ [START_REF] Christiansen | A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas[END_REF]. An optimal preconditioner is given by T itself: the Calderón identity

T 2 = -I/4 + K 2 , (8.1)
where K is a compact operator, dictates that its discretization

G -T × , � T G -1 × , T (8.2)
is well-conditioned with � � T � = ( × , T ) 2 , where ∈ are functions dual to suh as the BC functions [START_REF] Bufa | A Dual Finite Element Complex on the Barycentric Reinement[END_REF].

he matrix in (8.2) is, however, not numerically stable down to the static limit due to numerical cancellation in the excitation e and the unknown current vector j , and on multiply connected geometries it comprises a null space associated with the harmonic Helmholz subspace [START_REF] Cools | Nullspaces of MFIE and Calderón Preconditioned EFIE Operators Applied to Toroidal Surfaces[END_REF]. A irst approah to overcome the numerical cancellation could be to use an explicit quasi-Helmholz decomposition. While this could succeed in preventing the numerical cancellation and in preserving the quasi-harmonic Helmholz subspace, it also comes with several drawbaks as will be discussed in the following.

In more detail, let

Q = � Λ/ √ i H / √ i Σ √ i � (as deined in Section 4.d.
). If we were to eliminate loop and star functions so that Q ∈ C × has full rank (i.e., this follows the classical loop-star preconditioner approah), then Q T T Q is well-conditioned in frequency and hence

� Q T G × , Q � -T Q T � T Q � Q T G × , Q � -1 Q T T Q (8.3)
is stable in frequency down to the static limit. here are two drawbaks: First, the global loops have to be constructed, where currently available algorithms have, in general, a complexity of O( 2 ) for ≍ , where is the genus of geometry, and H is dense (a sparse matrix H can be obtained, but then the costs for inding the global loop is O( 3 ), see Chapter 6 and references therein). Second, the Gram matrix Q T G × , Q is ill-conditioned with a condition number that grows as O(1/ℎ 2 ). he reason for this is that loop and star functions are not 2 -stable, their Gram matrices are spectrally equivalent to discretized Laplace-Beltrami operators, for whih the condition number grows with 1/ℎ 2 , that is, we have for the Gram matrices [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF] Λ T G Λ = � (8.4) and

Σ T G Σ = � � , (8.5)
where

� � � = (∇ , ∇ ) 2 (8.6)
is the Laplace-Beltrami operator discretized with piecewise linear nodal functions

() = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 for ∈ , 0 for ∈ ≠ , linear elsewhere, (8.7)
where ∈ is the th vertex of mesh, and

� � � � = � ∇ , ∇ � 2 (8.8)
On a Reinement-Free Calderón Multiplicative Preconditioner for the EFIE Chapter 8

where

P o = P ΛH / + iP Σ / , (8.11)
P m = P mΛ + P mΣ , (8.12)

P mΛ = ΛG -1 Λ T / 2 + P ΛH / , (8.13)

P mΣ = Σ � Σ T Σ � + G ‡ � Σ T Σ � + Σ T / 2 , (8.14) G ‡ = G -1 -G -1 1 Σ 1 T Σ G -1 / � 1 T Σ G -1 1 Σ � (8.15)
with j = P o i , and the coeicients

= 4 � � � � P ΛH T † A ΛG -1 Λ T T A P ΛH � � �2 , (8.16) = 4 � � � � P Σ T † Φ P mΣ T Φ P Σ � � �2 , (8.17)
= � � � (P ΛH /)T † A P ΛH T A (P ΛH /) � � �2 . (8.18)
hese coeicients are necessary to cure the low-frequency breakdown. A power iteration method can be used to compute the norms. he coeicients could be replaced by functions of as shown in Section 8.c, that is, we could have used

= √ , (8.19) = 1/ √ , (8.20)
= , (8.21) but typically the condition number obtained by using norms is lower; thereby, the number of iterations used by a Krylov subspace method is reduced (and this saving usually outweighs the costs for estimating the norms). he use of the imaginary unit +i in the deinition P o is motivated for the same reason as it was for P : to prevent the numerical cancellation due to diferent scaling of the solenoidal and non-solenoidal components in e and j (for a detailed analysis, see [And+13]).

In Section 8.c, we derive the new formulation in a rigorous way. Given that this section is rather tehnical, we shall provide some intuitive ideas about the Section 8.b New Formulation approah. In a irst step, we decompose the EFIE into two scalar operators as shown in Chapter 4.

Equation (8.1) is not the only Calderòn identity that exists. For the single layer operator V and the hypersingular operator W, we have the identities [START_REF] Sauter | Boundary Element Methods[END_REF] (see also Section 3.c)

W•V = -I/4 + K 1 , (8.22) V•W = -I/4 + K 2 . (8.23)
Here, and in the following, K , ∈ N, denotes operators, whih are compact on a smooth geometry. hese two equations imply that the hypersingular operator can be used as preconditioner for the single layer potential and vice versa [START_REF] Steinbah | he Construction of Some Eficient Preconditioners in the Boundary Element Method[END_REF]. We are, however, not interested in doing this, since we would still need to use dual basis functions so that there is no real advantage compared with the standard Calderón tehniques. By combining (8.22) and (8.23), we yield

V•W•W•V = I/16 + K 3 , (8.24)
W•V•V•W = I/16 + K 4 . (8.25)
hese expressions can be simpliied, if we take into account the identities [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF] -W = (-Δ ) 1/2 + K 5 , (8.26)

V = (-Δ ) -1/2 + K 6 , (8.27)
where Δ is the Laplace-Beltrami operator. Combining (8.24) with (8.26) and (8.25) with (8.27), we obtain

V•(-Δ )•V = I/16 + K 7 , (8.28) W•(-Δ ) -1 •W = I/16 + K 8 . (8.29)
hen we could consider

G -1 V G -1 � �G -1 V (8.30)
and

G -1 W � + W (8.31)
as discretizations of (8.28) and (8.29). For deriving the inal formulation, it is beter to use symmetric, positive deinite versions of these matrices, that is, we use

G -1/2 V G -1 � �G -1 V G -1/2 (8.32)
and

G -1/2 W � + W G -1/2 , ( 8.33)
whih are well-conditioned as well since they are similar matrices. he RWG and BC functions are 2 -stable so the discretization of the identity operator is well-conditioned, that is, we have [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF] x

T G x ≍ x T G x ≍ x T x , ∀x ∈ R , (8.34)
and thus

x T � �x ≍ x T Σ T Σ x , ∀x ∈ R C , (8.35)
and

x T �x ≍ x T Λ T Λx , ∀x ∈ R V . (8.36)
While the mixed Gram matrix is necessary to obtain a rigorous proof for the well-conditioning, numerical evidence suggests that it has no practical impact. Since its omission allows numerical savings, we leave it out. hen (8.32) simpliies to

G -1/2 V Σ T ΣV G -1/2
(8.37) and (8.33) simpliies to

G -1/2 W � Λ T Λ � + W G -1/2 . (8.38)
Considering the deinition of P Σ and substituting bak from V to T Φ , then (8.37) reads

G -1/2 � Σ T Σ � Σ T T 0 Φ P Σ T 0 Φ Σ � Σ T Σ � G -1/2 . (8.39)
If a matrix A T A is well-conditioned, then so is AA T , whih can be seen by considering the SVD of A. Hence, the matrix

P Σ T 0 Φ Σ � Σ T Σ � G -1 � Σ T Σ � Σ T T 0 Φ P Σ (8.40)
is well-conditioned on the non-solenoidal Helmholz subspace, where we used that P 2 Σ = P Σ . Likewise, we ind for (8.38)

G -1/2 Λ T T 0 A Λ � Λ T Λ � + Λ T T 0 A ΛG -1/2
(8.41) and using the deinition of P Λ , we obtain

G -1/2 Λ T T 0 A P Λ T 0 A ΛG -1/2 . (8.42)
If this matrix is well-conditioned on the solenoidal Helmholz subspace, then so is

P Λ T 0 A ΛG -1 Λ T T 0 A P Λ . (8.43)
On the other hand, for → 0, (8.10) reduces to

P Λ T 0 A ΛG -1 Λ T T 0 A P Λ + P † Σ T 0 Φ Σ � Σ T Σ � + G ‡ � Σ T Σ � + T 0 Φ P Σ . (8.44) he matrix G ‡ is a variant of G -1
, where we introduced a null space to math with the null space of Σ T Σ (this results in a beter conditioning for non-uniform meshes, whih is motivated in Section 8.c.). We see that (8.10) is the sum of (8.42) and (8.41), it is therefore well-conditioned, and it can be interpreted as discretization of the scalar Calderón identities. he dynamic kernel introduces a compact perturbation, and so we can conclude that for ≠ 0 the formulation of (8.10) is still well-conditioned. In addition, since any matrix of the form A † A is HPD, then so is (8.10).

his derivation has, however, some caveats and due to these we provide a diferent derivation in Section 8.c, a derivation whih is alas less intuitive and more complicated. For example, (8.27) and (8.26) were derived in [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF] under the assumption that the surface is smooth (this is a considerable limitation, a simple cube is not a smooth geometry). his means that for the case that is a Lipshiz polyhedral domain, we cannot prove that (8.28) and (8.29) hold.

So far, we have only discussed simply connected geometries. For multiply connected ones, we note that H T T 0 A H is well-conditioned since the global loops are associated with the geometry of and hence and subsequently � , T A � 2 remains the same when ℎ → 0. hus we have

x T H T T 0 A H x ≍ x T H T H x ≍ x T x , ∀x ∈ R H (8.45)
and since H = P ΛH H , we have

x T P ΛH T 0 A P ΛH x ≍ x T x , (8.46) for all x ∈ X H ∶= � x ∈ R | ∃y ∈ R H ∶ x = H y �
, and thus

x T P ΛH T 0 A P ΛH T 0 A P ΛH x ≍ x T x , ∀x ∈ X H . (8.47)
c) heoretical Apparatus he formulation for whih we are going to show the well-conditioning in the static limit reads

Po † T † Pm T Po ȋ = -Po † T † Pm e , (8.48)
where

Po = P ΛH / √ + iP gΣ √ , (8.49)
P gΣ = Σ � Σ T Σ � + G -1 Σ T , (8.50)
Pm = PmΛ / + PmΣ , (8.51)

PmΛ = ΛG -1 Λ T + P ΛH , (8.52) PmΣ = Σ � Σ T Σ � + G -1 � Σ T Σ � + Σ T . (8.53)
Two diferences compared with the formulation in (8.10) are noticable: (i) instead of using the norms, we make the frequency treatment explicit by using the wavenumber , (ii) and we use P gΣ and not P Σ . For obtaining a rigorous statement on the conditioning, we cannot omit the Gram matrix in P gΣ . In terms of the number of iterations used by a Krylov subspace method, there is only a minor diference observable between using P gΣ or P Σ . Since the usage of P gΣ is computationally more expensive, we recommend to use P Σ . If, however, the implementation of (8.48) is desired, then this can still be trivially ahieved since and

� = � + G T 1 Λ 1 T Λ G . (8.57)
If are used for the discretization, the resulting matrices read

� W = � W + G T 1 Σ 1 T Σ G (8.58) and � � = � � + G T 1 Σ 1 T Σ G . (8.59)
In addition, we deine

W ∶= W + 1 Λ 1 T Λ ℎ 4 (8.60)
and

� ∶= � + 1 Λ 1 T Λ ℎ 4 . (8.61)
In the following, we assume the spectral equivalences

x T Ŵ G -1 Ŵ x ≍ x T �x , ∀x ∈ R V (8.62) and x T � W G -1 � W x ≍ x T � �x , ∀x ∈ R C . (8.63)
As shown in the Appendix for (8.62), suh a spectral equivalence can be established when we have a nested sequence of function spaces. Similarly to preconditioning strategies suh as algebraic multigrid, the new formulation remains efective as can be seen from the numerical results in Section 8.d.

) Vector Potential Operator

Here, we prove for the case of a simply connected geometry that

P Λ � T A � † ΛG -1 Λ T T A P Λ (8.64)
is well-conditioned up to its null space. To this end, we irst need to establish several spectral equivalences.

Section 8.c heoretical Apparatus Lemma 8.1. We have the spectral equivalences

x T �x ≍ x T �x , ∀x ∈ R V , (8.65) and x T Ŵ x ≍ x T W x , ∀x ∈ R V .
(8.66)

Proof. Here we prove (8.65); the proof for (8.66) follows analogously. We note that the null space of � is spanned by 1 Λ . Furthermore, we note

1 T Λ G T 1 Λ 1 T Λ G 1 Λ = 2 ≍ 1 , (8.67)
where is the area of , and‖1 Λ ‖ 2 = √ V ≍ 1/ℎ and thus

1 T Λ 1 Λ 1 T Λ 1 Λ ℎ 4 ≍ 1 . (8.68)
First, consider that we have

x T 1 Λ 1 T Λ ℎ 4 x ≲ x T x ℎ 2 , ∀x ∈ R V , (8.69)
and

x T G T 1 Λ 1 T Λ G x ≲ x T x ℎ 2 , ∀x ∈ R V , (8.70)
where the last inequality follows from the well-known equivalence

x T G x ≍ x T x ℎ 2 , ∀x ∈ R V , (8.71)
and the submultiplicativity of the matrix norm, that is,

� � � G T 1 Λ 1 T Λ G � � �2 ≤ � � � G T � � �2 � � � 1 Λ 1 T Λ � � �2 � � G � �2 ≲ ℎ 2 . (8.72) Let x = x ∥ + x ⟂ be an orthogonal spliting with x ∥ ∈ span 1 Λ . If x ⟂ = 0 Λ , then x T �x = x T G T 1 Λ 1 T Λ G x ≍ x T 1 Λ 1 T Λ ℎ 4 x = x T �x (8.73)
due to (8.67), (8.68) (noting that x ∥ is a multiple of 1 Λ ). Since we have [START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF] x we note that for x ⟂ ≠ 0 Λ the leading contribution x T �x scales at least O(ℎ 2 ) and at most O(1). he contribution from x T G T 1 Λ 1 T Λ G x and x T 1 Λ 1 T Λ x adds a positive quantity that scales at most O(ℎ 2 ) due to (8.70), (8.71), and the fact that G T 1 Λ 1 T Λ G and 1 Λ 1 T Λ are positive semi-deinite rank-1 matrices. Hence for ℎ → 0, we can conclude that the eigenvalues of � and � whih scale with O(ℎ ), 0 ≤ < 2 (and their associated eigenvectors) are spectrally identical and the eigenvalues scaling by O(ℎ 2 ) are shited at most by a constant factor. hus (8.65) follows.

T x ℎ 2 ≲ x T �x ≲ x T x , ∀x ∈ � span 1 Λ� ⟂ , ( 8
For (8.66), the same argumentation can be used noting that

x T x ℎ 2 ≲ x T W x ≲ x T x ℎ , ∀x ∈ � span 1 Λ� ⟂ , (8.75) holds.
Remark. his lemma will be frequently used in order to replace

G T 1 Λ 1 T Λ G by 1 Λ 1 T Λ .
In essence, we are allowed to do so if the matrix accompanying G T

1 Λ 1 T Λ G has a null space spanned by 1 Λ and where the smallest non-zero singular value scales as O(ℎ ) with ≤ 2. Lemma 8.2. We have the spectral equivalences

x T Ŵ G -1 Ŵ x ≍ x T � W G -1 W + G T 1 Λ 1 T Λ G � x ≍ x T � W G -1 W + 1 Λ 1 T Λ ℎ 4 � x , ∀x ∈ R V . (8.76) Proof. We have Ŵ G -1 Ŵ = � W G -1 + G 1 Λ 1 T Λ �� W + G 1 Λ 1 T Λ G � = W G -1 W + G 1 Λ 1 T Λ G 1 Λ 1 T Λ G (8.77) using W 1 Λ = 0 and 1 T Λ W = 0 T . Since 1 T Λ G 1 Λ = � d() = is a constant, we yield x T Ŵ G -1 Ŵ x ≍ x T � W G -1 W + G 1 Λ 1 T Λ G � x , ∀x ∈ R V , (8.78)
Section 8.c heoretical Apparatus whih proves the irst equivalence. For the second equivalence, we have due to (8.71)

x T W G -1 W x ≍ x T WW x /ℎ 2 , ∀x ∈ R V (8.79)
and using (8.75), we have

x T x ℎ 2 ≲ x T WW x /ℎ 2 ≲ x T x , ∀x ∈ � span 1 Λ� ⟂ . (8.80)
hen using the argumentation of Lemma 8.1, we obtain

x T � W G -1 W + G T 1 Λ 1 T Λ G � x ≍ x T � W G -1 W + 1 Λ 1 T Λ ℎ 4 � x , ∀x ∈ R V . (8.81)
Lemma 8.3. We have the spectral equivalence

x T �x ≍ x T � Λ T Λ + 1 Λ 1 T Λ ℎ 4 � x , ∈ R V . (8.82) Proof. Since � = Λ T G Λ and [And12a] x T G x ≍ x T x ∀x ∈ R , (8.83)
we have

T �x ≍ x T Λ T Λx , ∀x ∈ R V . (8.84)
Equation (8.84) remains true when we add 1 Λ 1 T Λ ℎ 4 to the matrices since all matrices appearing are positive, semi-deinite.

Proposition 8.1. We have the spectral equivalence

x T P Λ � T A � † ΛG -1 Λ T T A P Λ x ≍ x T P Λ x , ∀x ∈ R . (8.85)
Proof. By combining the previous lemmas we can establish that is in more detail, we have

x T � W G -1 W + 1 Λ 1 T Λ ℎ 4 � x ≍ x T � Λ T Λ + 1 Λ 1 T Λ ℎ 4 � x , ∀x ∈ R V , ( 8
x T � W G -1 W + 1 Λ 1 T Λ ℎ 4 � x Lemma 8.2 ≍ x T Ŵ G -1 Ŵ x Proposition A.1 ≍ x T �x Lemma 8.1 ≍ x T �x Lemma 8.3 ≍ x T � Λ T Λ + 1 Λ 1 T Λ ℎ 4 � x , ∀x ∈ R V . (8.87)
We apply the substitution x = (Λ T Λ) + Λ T y and obtain

y T Λ � Λ T Λ � + W G -1 W � Λ T Λ � + Λ T y ≍ y T P Λ y , ∀y ∈ R , (8.88)
where P Λ ≡ Λ(Λ T Λ) + Λ T . We note that Λ T T 0 A Λ = W = W † ; for ≠ 0, the dynamic kernel introduces a compact perturbation, whih does not afect the ℎ-conditioning, and we can freely hoose T † A instead of T A resulting in (8.85). Remark. he matrix P Λ (T A ) † ΛG -1 Λ T T A P Λ is Hermitian and positive semi-deinite, whih can be seen by considering

P Λ � T A � † ΛG -1 Λ T T A P Λ = � G -1/2 Λ T T A P Λ � † G -1/2 Λ T T A P Λ . (8.89)
Now, we are ready to consider the case that is multiply connected: we have to establish that

P ΛH � T A � † � ΛG -1 Λ T + P ΛH � T A P ΛH (8.90)
is well-conditioned up to its null space. For deriving this result some preliminary considerations are necessary, and again, we start with considering the static limit → 0. Proposition 8.2. We have the spectral equivalence

x T � ΛT Λ� -1/4 ΛT T 0 A Λ� ΛT Λ� -1/4 x ≍ x T x , ∀x ∈ � span 1 Λ� ⟂ , (8.91)
where we used the substitution Λ ∶= Λ/ℎ.

Section 8.c heoretical Apparatus

Proof. Proposition 8.1 states that

x T P Λ � T 0 A � † ΛG -1 Λ T T 0 A P Λ x ≍ x T P Λ x , ∀x ∈ X Λ (8.92) with X Λ ∶= � x ∈ R | x = P Λ x � (8.93)
holds. We apply the substitution y

= (Λ T Λ) -1/2 Λ T x noting that (Λ T Λ) -1/2 Λ T ∶ X Λ → � span 1 Λ� ⟂
is one-to-one and onto and that

P Λ = Λ(Λ T Λ) -1/2 (Λ T Λ) -1/2 Λ T so we obtain y T � Λ T Λ � -1/2 Λ T � T A � † ΛG -1 Λ T T A Λ � Λ T Λ � -1/2 y ≍ y T y , ∀y ∈ � span 1 Λ� ⟂ . (8.94)
By using (8.71), we obtain

x T � Λ T Λ � -1/2 Λ T T 0 A Λ � ℎ -2 � Λ T T 0 A Λ � Λ T Λ � -1/2 x ≍ x T x , ∀x ∈ � span 1 Λ� ⟂ . (8.95)
We deine Λ = Λ/ℎ noting that

� Λ T Λ � -1/2 Λ T T 0 A Λ � ℎ -2 � Λ T T 0 A Λ � Λ T Λ � -1/2 = � ΛT Λ� -1/2 ΛT � T 0 A � † Λ ΛT T 0 A Λ� ΛT Λ� -1/2
. (8.96) Furthermore, we notice that the singular values of

ΛT T 0 A Λ� ΛT Λ� -1/2 (8.97)
are by deinition of the SVD the square roots of the singular values of he absolute value of the largest eigenvalue can always be bounded from above by the largest singular value and the smallest eigenvalue can always be bounded from below by the smallest singular value. he second half of this statement is not entirely helpful since the smallest eigenvalue and singular value are both zero. However, since the let null space and the right null space of (8.97) are identical, we can show that the smallest non-zero absolute eigenvalue min can be bounded from below by the smallest non-zero singular value min .

� ΛT Λ� -1/2 ΛT � T 0 A � † Λ ΛT T 0 A Λ� ΛT Λ� -1/2 , ( 8
To see this, let v be the unit eigenvector associated with min and we use the abbreviation A = ΛT T 0 A Λ( ΛT Λ) -1/2 . We have

| min | 2 = v T A T Av ≥ min ‖x‖ 2 =1∧x⟂1 Λ x T A T Ax = 2 min (8.99)
following the properties of the SVD noting that‖v ‖ 2 = 1 and v ⟂ 1 Λ . Similar matrices have the same eigenvalues and thus A and (Λ T Λ) -1/4 A(Λ T Λ) 1/4 have the same eigenvalues. Since

� ΛT Λ� -1/4 A � ΛT Λ� 1/4 = � ΛT Λ� -1/4 ΛT T 0 A Λ� ΛT Λ� -1/4 (8.100)
is a symmetric, positive semideinite matrix, the eigenvalues and singular values coincide and thus

� ΛT Λ� -1/4 ΛT T 0 A Λ� ΛT Λ� -1/4 (8.101)
is well-conditioned up to its null space.

Given Proposition 8.2, we can conclude that if the matrix Λ( ΛT Λ) -1/4 is used as solenoidal basis, we obtain a well-conditioned matrix with bounded norm. hus if we were to pursue a classical explicit quasi-Helmholz decomposition sheme, we could use the basis � Λ( ΛT Λ) -1/4 H � as preconditioner for T A on multiply connected geometries. Using (8.45), it follows that the matrix

� � ΛT Λ� -1/4 ΛT H T � T 0 A � Λ� ΛT Λ� -1/4 H � (8.102)
is well-conditioned in ℎ up to the null space of the loop functions since the basis transformation matrix has full rank (up to the null space of the loop functions) and since the bloks on the main diagonal are well-conditioned and all bloks are Section 8.c heoretical Apparatus bounded: the boundedness of H T T 0 A Λ( ΛT Λ) -1/4 and of ( ΛT Λ) -1/4 ΛT T 0 A H follows from the boundedness of the bloks on the main diagonal. Now, it remains to return from the explicit quasi-Helmholz decomposition of (8.102) to the new formulation in (8.90).

Proposition 8.3. We have the spectral equivalence

x T P ΛH � T A � † � ΛG -1 Λ T + P ΛH � T A P ΛH x ≍ x T P ΛH x , ∀x ∈ R . (8.103)
Proof. We deine

T ΛH ∶= � ΛT H T � T 0 A� Λ H � , (8.104) Q ΛH ∶= � � ΛT Λ� -1/4 I � , (8.105)
and observe

� � ΛT Λ� -1/4 ΛT H T � T 0 A � Λ� ΛT Λ� -1/4 H � = Q ΛH T ΛH Q ΛH . (8.106)
In other words, we can interpret Q ΛH as a preconditioner for the standard loop/global loop discretized T ΛH . We note that T ΛH and Q ΛH are symmetric matrices and that they have the same null space (i.e., the null space due to the linear dependency of the loop functions). Summarizing, we have the Rayleigh quotient

x T Q ΛH T ΛH Q ΛH x ≍ x T x , ∀x ∈ � null Q ΛH � ⟂ . (8.107)
By using the substitution y = Q ΛH x , we obtain and thus

y T T ΛH y ≍ y T Q -2 ΛH y , ∀y ∈ � null Q ΛH � ⟂ (8.108) from whih immediately y T T 2 ΛH y ≍ y T Q -4 ΛH y , ∀y ∈ � null Q ΛH � ⟂ ( 8
x T Q 2 ΛH T 2 ΛH Q 2 ΛH x ≍ x T x , ∀x ∈ � null Q ΛH � ⟂ (8.110)
follows.

Moreover, we notice that

T 2 ΛH = � ΛT H T � T 0 A� Λ H � � ΛT H T � T 0 A� Λ H � = � ΛT H T � T 0 A � Λ ΛT + H H T � T 0 A� Λ H � (8.111)
using that H T Λ = 0. he global loop transformation matrix is not uniquely deined, but a possible transformation matrix can always be constructed from P H ∶= I -P Λ -P Σ by using its SVD so that H is the column space of it. Hence, we can always obtain

P H = H H T .
(8.112)

By using (8.111) and (8.112), we have

Q 2 ΛH T 2 ΛH Q 2 ΛH = � � ΛT Λ� -1/2 ΛT H T � T 0 A � Λ ΛT + P H � T 0 A � Λ� ΛT Λ� -1/2 H � . (8.113)
We also note that the transformation � Λ( ΛT Λ) -1/2 H � is well-conditioned, in fact,

� Λ� ΛT Λ� -1/2 H � � � ΛT Λ� -1/2 ΛT H T � = P ΛH . (8.114) hus we have x T � Λ� ΛT Λ� -1/2 H � Q 2 ΛH T 2 ΛH Q 2 ΛH � � ΛT Λ� -1/2 ΛT H T � x T ≍ x T P ΛH x , ∀x ∈ R , (8.115) Section 8.c heoretical Apparatus
where we note that the preconditioned system matrix can be expressed as

� Λ� ΛT Λ� -1/2 H � Q 2 ΛH T 2 ΛH Q 2 ΛH � � ΛT Λ� -1/2 ΛT H T � x T = P ΛH T 0 A � Λ ΛT + P H � T 0 A P ΛH . (8.116)
We can replace P H by P ΛH since the matrix P ΛH T 0 A P Λ T 0 A P ΛH is symmetric, positive deinite and

� � � P ΛH T 0 A P Λ T 0 A P ΛH � � �2 ≲ ‖P ΛH ‖ 2 � � � T 0 A � � �2 ‖P Λ ‖ 2 � � � T 0 A � � �2 ‖P ΛH ‖ 2 ≲ 1 (8.117) is bounded, where � � � T 0 A � � �2
≲ 1 follows from the compactness of T A . Likewise, the dynamic kernel is a compact perturbation and by substituting bak from Λ to Λ and G -1

, we obtain that the matrix

P ΛH � T A � † � ΛG -1 Λ T + P ΛH � T A P ΛH x ≍ x T P ΛH x , x ∈ R (8.118)
is well-conditioned (up to its null space).

) Scalar Potential Operator

In this section, we are going to establish that the matrix

P T gΣ � T Φ � † PmΣ T Φ P gΣ (8.119)
is well-conditioned (up to its null space).

As for the vector potential operator, we need some lemmas and auxiliary matrices. We deine the matrix

V ∶= � Σ T Σ � + Σ T ΣV Σ T Σ � Σ T Σ � + .
(8.120) his matrix is important since it is connected to the scalar potential by (see (5.20))

V ≡ � Σ T Σ � + Σ T T 0 Φ Σ � Σ T Σ � + . (8.121)
Lemma 8.4. We have the spectral equivalence

x T V G -1 V x ≍ x T V G -1 V x , ∀x ∈ � span 1 Σ� ⟂ . (8.122)
Proof. If x is suh that 1 T Σ x = 0, then we have

Σ T Σ � Σ T Σ � + x = x (8.123)
and thus

x T V x = x T V x . (8.124)
Clearly, we have for suh x also

x T VV x = x T V V x , (8.125)
and since y T G -1 y ≍ y T y holds for all y ∈ R C , we have

x T V G -1 V x ≍ x T V G -1 V x . (8.126)
Corollary 8.1. We have the spectral equivalence

x T � �x ≍ x T � Σ T Σ + 1 Σ 1 T Σ ℎ 4 � x , ∀x ∈ R C . (8.127)
Proof. Follows from Lemma 8.3.

Lemma 8.5. he vector 1 Σ is a right eigenvector of G -T

, that is,

1 Σ = G -T 1 Σ . Proof. If y = x = 1 , ∀ ∈ , ( 8
G 1 Σ = G x = 1 Σ . (8.129) Since G = G T , we have 1 Σ = G -T 1 Σ . Section 8.c heoretical Apparatus
Corollary 8.2. For any mean value free vector x , that is, 1 T Σ x = 0, we have that the vector G -1

x is mean value free as well.

Proof. his follows from Lemma 8.5 since if we have 1 T Σ x = 0, then

1 T Σ G -1 x = 1 T Σ x = 0 . (8.130)
Proposition 8.4. We have the spectral equivalence

x T P T gΣ � T Φ � † PmΣ T Φ P gΣ x ≍ x T P Σ x , ∀x ∈ R . (8.131)
Proof. We start with (8.63)

x T � W G -1 � W x ≍ x T � �x , ∀x ∈ R C (8.132)
and applying the substitution x = G -1/2 y yields

y T G -1/2 � W G -1 � W G -1/2 y = y T � G -1/2 � W G -1/2 � 2 y ≍ y T G -1/2 � �G -1/2 y , ∀y ∈ R C , (8.133)
and hence

y T G 1/2 � W -1 G 1/2 y ≍ y T � G 1/2 � �G 1/2 � -1/2 y , ∀y ∈ R C . (8.134)
From the Calderón identities and the theory outlined in [SW98; BC07; Hip06], we have

x T � W x ≍ x T G -T V -1 G -1 x , ∀x ∈ R C . (8.135)
Inserting this in (8.134) and applying the bak-substitution y = G 1/2 x , we obtain he right-hand side can be simpliied: it was shown in [And+13] that

x T G -T V G -1 G G -T V G -1 x ≍ x T � � -1 x , ∀x ∈ R C . ( 8
� Σ T Σ + 1 Σ 1 T Σ / C � -1 = � Σ T Σ � + + 1 Σ 1 T Σ / C (8.137)
holds. In addition with C ≍ 1/ℎ 2 and Corollary 8.1, we can simplify the right-hand side of (8.136) yielding

x T G -T V G -1 G G -T V G -1 x ≍ x T � � Σ T Σ � + + 1 Σ 1 T Σ � x , ∀x ∈ R C . (8.138)
From [SW98; BC07], we can obtain

x T G -1 x ≍ x T G -1 G G -T x , ∀x ∈ R C . (8.139)
Inserting this into (8.138) yields

x T G -T V G -1 V G -1 x ≍ x T � � Σ T Σ � + + 1 Σ 1 T Σ � x , ∀x ∈ R C . (8.140)
hen we use the substitution x = Σ T y and obtain

y T ΣG -T V G -1 V G -1 Σ T y ≍ y T P Σ y , ∀y ∈ R , (8.141)
since Σ 1 Σ = 0. Due to this relationship, it is clear that all vectors Σ T y with y ∈ R are mean value free, that is, we have 1 T Σ Σ T y = 0. hus we can invoke Lemma 8.4 and obtain

y T ΣG -T V G -1 V G -1 Σ T y ≍ y T P Σ y , ∀y ∈ R . (8.142)
Inserting the right-hand side from (8.121), we obtain

y T P T gΣ T 0 Φ PmΣ T 0 Φ P gΣ y ≍ y T P Σ y ≍ y T P Σ y , ∀y ∈ R . (8.143)
As in Proposition 8.1, we note that the dynamic kernel only introduces a compact perturbation, and that by using (T Φ ) † for the let scalar potential operator matrix in (8.131), we yield a symmetric, positive semi-deinite system. Section 8.c heoretical Apparatus ) Preconditioned Electric Field Integral Equation Proposition 8.5. he new formulation is well-conditioned in the static limit, that is, the matrix in (8.48) satisies

lim →0 x T Po † T † Pm T Po x ≍ x T x , ∀x ∈ R . (8.144)
Proof. We have

Po † T † Pm T Po = P † ΛH� T A � † PmΛ T A P ΛH + P † gΣ� T Φ � † PmΣ T Φ P gΣ + 2 P † gΣ� T A � † PmΛ T A P gΣ + 4 P † gΣ� T A � † PmΣ T A P gΣ iP † ΛH� T A � † PmΛ T A P gΣ -iP † gΣ� T A � † PmΛ T A P ΛH i 3 P † ΛH� T A � † PmΣ T A P gΣ -i 3 P † gΣ� T A � † PmΣ T A P ΛH iP † ΛH� T A � † PmΣ T Φ P gΣ -iP † gΣ� T Φ � † PmΣ T A P ΛH . (8.145) hus we ind lim →0 Po † T † Pm T Po = P † ΛH� T 0 A � † PmΛ T A P ΛH + P † gΣ� T 0 Φ � † PmΣ T Φ P gΣ . (8.146)
Clearly, the new formulation is low-frequency stable and the well-conditioning in ℎ follows from Proposition 8.3 and Proposition 8.4 so that we have

x T � P † ΛH� T 0 A � † PmΛ T 0 A P ΛH + P † gΣ� T 0 Φ � † PmΣ T 0 Φ P gΣ � x ≍ x T x , ∀x ∈ R . (8.147)
For the dynamic case, we note that the additional terms appearing in (8.145) have at least up to a certain frequency a smaller norm than then principal terms in 151 On a Reinement-Free Calderón Multiplicative Preconditioner for the EFIE Chapter 8 (8.147). Numerical evidence suggests that even for geometries spanning several wavelengths, the new preconditioner shows a beneicial behavior compared with an unpreconditioned system. he derivation also holds for the case that the mesh is non-uniform, that is, the condition number is still bounded ater subsequent structured mesh reinements. If, however, only a local reinement is performed then the condition number can still grow. If in suh a process only the conditioning of G and G is increased but not of G and G , then this can be prevented.

In fact, only the preconditioner for the scalar potential part is afected. he reason for this is that we have used that

� � W G -1 � W � -1 = � W + G � W -1 . (8.148)
For W , the same statement (with pseudo-inverses) is not true:

� � W G -1 � W � + ≠ � W + G � W + , (8.149)
where we note that in our analysis � W + does not appear since we have already moved to V and later on to V in the proof of Proposition 8.4.

What we have implicitly obtained in the proof is a spectral equivalence between

( � W G -1 � W ) + and � W + G � W +
, where however the bounding constants tend to deteriorate with an increasing ill-conditioning of G . If G is modiied suh that it has a null space spanned by 1 Σ resulting in the matrix G , then we would have

� � W G+ � W � + = � W + G � W + , (8.150)
and thus no deterioration would occur. Since in the end we are not interested in � W but in preconditioning V , we replace G -1 by G ‡ , whih has a null space spanned by 1 Σ . Section 6.d conirms the efectiveness of this approah.

We have now assumed that G and G are reasonably conditioned. A study for the case that the condition number assumes extreme values is not within the scope of this work. For the non-canonical numerical examples, the condition number of G or G is up to around 1 ⋅ 10 3 . Section 8.d Numerical Results

d) Numerical Results

First, we considered a sphere, radius 1 m, to conirm the low-frequency stability by computing the condition number obtained by the new formulation and compared it with a loop-tree preconditioned system. Figure 8.1 shows that the new formulation is frequency stable and Figure 8.2 that the bistatic radar cross section can be accurately computed down to 1 ⋅ 10 -30 Hz. he saturation of the condition number in the case "no preconditioner" stems from numerical cancellation: the null space of T Φ exists only up to numerical precision and when becomes too small, the (numerical) norm of the null space of T Φ is larger than the norm of T A so that T A completely vanishes in numerical noise. To verify the dense-discretization stability, we computed the condition number for the new formulation and the loop-tree preconditioned system for an increasing spectral index 1/ℎ. We can see from Figure 8.3 that the new formulation is dense-discretization stable, whereas the loop-tree preconditioner is not.

In addition, we considered a plate as an example for an open structure. Similar to Calderón preconditioners, the condition number shows a slight growth in 1/ℎ as displayed in Figure 8.4; however, it remains small compared with a loop-tree preconditioner. In order to conirm the low-frequency stability in the case for multiply connected geometries, we considered a closed structure with two global loops shown in Figure 8.5. Evidently, the new formulation remains stable.

Next, we considered more realistic structures. To compress the system matrix, we used an ACA with tolerance 1 ⋅ 10 -4 . As iterative solver, we used the CG method for the new formulation and the CGS method for the other formulations since the CG method is only applicable if the matrix is HPD. We note that a single iteration step of CGS requires two matrix-vector products. We employed the AGMG library [Not;[START_REF] Napov | An Algebraic Multigrid Method with Guaranteed Convergence Rate[END_REF] for the fast inversion of the graph Laplacians with solver tolerance 1 ⋅ 10 -14 to demonstrate that even for extreme small tolerances our preconditioner remains eicient.

As irst realistic example, we considered as open and multiply connected structure the model of a MiG-15 displayed in Figure 8.6, whih has one global loop. As excitation we used a plane wave and its electric size is 2 ⋅ 10 -2 , where is the wavelength. Several unstructured reinements where conducted ranging from 1518 to 306 036 unknowns. Figure 8.7 and Figure 8.8 show the number of iterations and total time, respectively, as a function of 1/ℎ. Clearly, the reduction in the number of iterations relects in a saving of computational time. Tab. 8.1.: MiG-15: the number of iterations and the time used by the solver to obtain a residual error below 1 ⋅ 10 -4 .

tion. Table 8.2 shows the number of iterations and total time. We can see that in this case the number of iterations grows also for the new formulation. his is not unexpected since the condition number of G has grown for the irst reinements. Further reinements would lead to a converging condition number of G and hence, we can expect that the new formulation would converge likewise. Lastly, as an example of a multiply connected and closed structure, we have hosen an model of the Fokker Dr.I depicted in Figure 8.10, whih has 390 global loops. As excitation we considered both a plane wave and a voltage gap excitation. he model is discretized with 294 420 unknowns resulting in a non-uniform mesh with cond G ≈ 3 ⋅ 10 3 and cond G ≈ 7 ⋅ 10 4 . Its electric length is 1 ⋅ 10 -3 . From Table 8.3, we can see that there is not only a signiicant saving in the number of iterations, but also in time. Chapter 9

A Hermitian, Positive Deinite, and Well-Conditioned CFIE his last technical chapter sketches some preliminary results on a new preconditioner for the CFIE that gives rise to a Hermitian, positive deinite system of linear equations. Diferent from other Calderón strategies, this scheme necessitates a standard discretization of the EFIE with RWG basis functions (i.e., no dual EFIE matrix required), is free from spurious resonances, and is stable down to the static limit for both simply and multiply connected geometries. he fact that the new system matrix is Hermitian, positive deinite, and well-conditioned makes it amenable for fast iterative solvers. Numerical results demonstrate the efectiveness of the proposed approach. his work is based on the conference article [AAE16a].

F or closed, electrically large problems, the CFIE can be used to avoid interior resonances. Classical Calderón preconditioners have been extended to the CFIE [Bag+09; And+12]. However, these shemes demand the use of the Yukawa potential (i.e., an EFIE where the wavenumber in the Green's function is replaced by +i), thereby requiring the discretization of another operator and, furthermore, they do not give rise to Hermitian formulations.

In the previous hapter, a sheme has been presented that allows to precondition the EFIE without the need of dual basis functions and a barycentric reinement of the mesh. Based on this formulation, we present in this hapter a well-conditioned CFIE that has three key properties: (i) At the best of our knowledge it is the irst CFIE formulation giving rise to a Hermitian, positive deinite system matrix at every frequency. To obtain this, we use the RF-CMP for preconditioning the EFIE part of the CFIE leveraging on the fact that the matrix P o is, up to the frequency scaling, a well-conditioned matrix. herefore, Chapter 9 we can apply it to the MFIE part without jeopardizing its well-conditioning. (ii) Diferent from other Calderón-like preconditioning shemes, we can prove the resonance-freeness of the discretized equation under the assumption that the unpreconditioned CFIE is resonance-free. (iii) When compared with standard Calderón tehniques, the computational costs of the new formulation presented here are substantially reduced since, while dual basis functions are still necessary for the MFIE, the EFIE can be discretized solely with standard RWG functions (no dual EFIE matrix is required). his hapter will deine the new equation and analyze its theoretical properties in detail. Finally, numerical results will corroborate the theory and show the practical impact of the new sheme on real case scenarios.

a) New Formulation

We deine the abbreviation

M = G G -1 × , � M .
(9.1)

For resonance frequencies, T and M are rank deicient, while the linear combination T + M has full rank, but is not well-conditioned due to T (i.e., in the case of multiply connected geometries, it is also not well-conditioned due to M ). We propose the following preconditioned CFIE:

P † o � T † P m T / + � M † � M / � P o i = P † o � -T † P m e + � M † h/ � , (9.2)
where P o and P m are the same as deined in Chapter 8, that is, as deined in (8.11) and (8.12), and we used the normalization factors

= � � � P † o T † P m T P o � � � , (9.3) = � � � P † o � M † � M P o � � � . (9.4)
Since � M is well-conditioned except when is multiply connected: the part quasi-harmonic Helmholz subspace associated with the toroidal loops becomes a null space in the static limit [START_REF] Bogaert | Low Frequency Stability of the Mixed Discretization of the MFIE[END_REF]. As shown in the previous hapter, the since T † P m T and � M † � M are positive deinite on R ⧵ � nullT ∪ null � M � , that is,

x † T † P m T x / > 0 (9.6) and

x † � M † � M x / > 0 (9.7) using that > 0. Clearly, when either x ∈ nullT or x ∈ null � M , we still have

x † � T † P m T / + � M † � M / � x > 0 , (9.8) and hence, the matrix is positive deinite and thus has full rank at resonance frequencies.

b) Numerical Results

To verify the presented theory, we irst tested the frequency stability. We used a sphere and compared the new formulation with an EFIE, CFIE, and a looptree preconditioned CFIE. Figures 9.1a and 9.1b show that from the standard tehniques only the loop-tree preconditioned is frequency stable (the CFIE deteriorates for low-frequencies, the EFIE for low and high-frequencies), however, with a relatively high condition number compared with the new formulation. We can see that the new formulation does not sufer from interior resonances, though we observe a growth of the condition number with increasing frequency, whih stems from the compact perturbation of the dynamic kernel. Tab. 9.1.: Space Shutle: the number of iterations; solver tolerance 1 ⋅ 10 -4 .

To beter assess the impact of the dynamical kernel on the iterative solver, we considered a plane wave excitation and solved the system by using the generalized minimal residual (GMRES) and CG method with solver tolerance 1 ⋅ 10 -4 . Figure 9.1c displays the results. We can see a correspondence between the condition number and the number of iterations, though we note that for the highest frequency the ratio of the condition numbers of the conformingly discretized CFIE and the new formulation is around 18, while for the number of iterations the ratio is around 1.45.

Next, we veriied the dense-discretization stability. To this end, we reined the toroidal structure (i.e., a multiply connected geometry, whih has a harmonic subspace) depicted in Figure 9.2a. Figure 9.2b shows that for suh a geometry even the MFIE is ill-conditioned, whereas the new formulation is stable. Eventually, we used a more realistic structure, the model of a Space Shutle discretized with 3780 unknowns, where we employed a plane wave excitation with frequency 1 MHz and 10 MHz, respectively. For the later, the electric length is 4. he results are summarized in Figure 7.2; they conirm the observation from Figure 9.1c: for high frequencies the dynamic kernel deteriorates the conditioning of (9.2). If, however, a iner mesh would be used for the discretization, then the conforming CFIE would have a growing condition number, while the condition number of the matrix in (9.2) can be bounded.

c) Conclusion

It was shown that it is possible to extend the RF-CMP to a conformingly discretized CFIE: the low-frequency and the dense-discretization breakdown are Plane wave excitation and GMRES method as solver for the unpreconditioned conforming CFIE and the loop-tree preconditioned conforming CFIE. Conjugate gradient method as solver for the new formulation. Part IV.

Finale

Chapter 10

Concluding Scientiic Postscript

For small erections may be inished by their irst arhitects; grand ones, true ones, ever leave the copestone to posterity.

Moby-Dick; or, he Whale Herman Melville P reconditioners for the EFIE and the CFIE based on hierarhical bases and on Calderón identities have been presented in this dissertation that sensibly advanced the state of the art: we have obtained preconditioned EFIEs and CFIEs where in the case of the hierarhical bases preconditioners the condition number is logarithmically bounded in the number of unknowns and where in the case of the RF-CMP preconditioner the condition number is bounded independently of . Diferent from existing shemes, the results were obtained without the explicit use of dual basis functions. We did consider multiply connected geometries for both the hierarhical basis and the RF-CMP. he later even yields an HPD system matrix, whih allows to apply the CG method as solver, whih, at least in theory, guarantees convergence.

A great German philosopher once said, "Ater the game, is before the game." While the presented preconditioners have already a certain degree of maturity, further work needs to be carried out to make them admissible for the usage in commercial sotware:

• In general, geometries are not necessarily Lipshiz polyhedral, for example, when they include T-junctions. he hierarhical basis needs extension to this scenario. he RF-CMP, is when it comes to the formulation, already applicable to structures with junctions since loop and star functions have

been deined on these structures before. It is, however, not clear how strong the preconditioning efect on a junction is. his needs careful evaluation and maybe further amendments.

• Oten enough, we are also interested in solving scatering problems involving penetrable objects. he Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) equation is due to the EFIE part of the operator ill-conditioned as well. Hierarhical basis preconditioners have been adapted to the PMCHWT [Guz+17]. In a next step, the RF-CMP should be extended to this equation.

• Many algebraic multigrid preconditioners have been presented in the past [Not; LB12; BOS15]. A study and comparison of the diferent methods would be desirable. Particular atention should be paid to the case of multi-body problems, where the null spaces are more complicated (as it was discussed in Section 4.d.). In addition, it would be desirable to have a (parallelized) algebraic multigrid solver at hand, developed under a liberal license both for commercial and academic use.

Further targets to takle and questions to answer include:

• hroughout this thesis, it was assumed that the condition number of the RWG Gram matrix is reasonably large. What happens, however, if the RWG Gram matrix is highly ill-conditioned? To what degree can this problem be ixed?

• When the geometry is open, one observes for the Calderón preconditioner a growth of the condition number, whih is of logarithmic nature. Even though this is not a critical issue, it would be interesting to obtain an optimal preconditioner for open problems.

• For non-linear problems, frequency domain methods cannot be used. hus it would be desirable to extend the preconditioners to time-domain ield integral equations.

• If the frequency is increased to the limit that ℎ ≈ /10, the number of iterations used by an iterative solver grows. Modiied Calderón preconditioners have been presented to reduce this increase [START_REF] Boubendir | Well-Conditioned Boundary Integral Equation Formulations for the Solution of High-Frequency Electromagnetic Scatering Problems[END_REF]. It should be possible to amend the RF-CMP in a similar way.

• It would be desirable to obtain a well-conditioned CFIE or CSIE that does not require dual basis functions.

x T x ≍ x T D+2 � λ T T G � λ T D+2 x ≍ x T D+2 � λ T T � λ T D+2 x ℎ 2 , ∀x ∈ R V , (A.6)
where we assumed without loss of generality that

� � � � λ T T � λ T � � �
≍ 1 (due to the deinition we have used here, the functions � T are, if not rescaled at all, 1 -stable). hus we ind by substitution

x T � λ T T � λ T x ℎ 2 ≍ x T D-4 x , ∀x ∈ R V . (A.7)
Since the matrices are invertible, we also have

x T � λ -1 T � λ -T T x ≍ x T D4 x ℎ 2 , ∀x ∈ R V . (A.8)
Using (A.6) in (A.3) and (A.4), we ind

x T � λ T T �� λ T x ≍ x T x , ∀x ∈ R V , (A.9) and x T � λ T T Ŵ � λ T x ≍ x T D-2 x , ∀x ∈ R V . (A.10) By using the substitution y = � λ T x in (A.9) and (A.10), we obtain

x T �x ≍ x T � λ -T T � λ -1 T x , ∀x ∈ R V , (A.11) and

x T Ŵ x ≍ x T � λ -T T D-2 � λ -1 T x , ∀x ∈ R V .
(A.12) Summarizing, we obtain the spectral equivalence

x T Ŵ G -1 Ŵ x (8.71) ≍ x T Ŵ 2 x /ℎ 2 (A.12) ≍ x T � � λ -T T D-2 � λ -1 T ℎ 2 � 2 x /ℎ 2 ≍ x T � λ -T T D-2 � λ -1 T � λ -T T D4 ℎ 2 D-2 � λ -1 T x /ℎ 2 (A.8) ≍ x T � λ -T T � λ -1 T x ≍ x T �x , ∀x ∈ R V . (A.

  Deinition 2.1 (Domain). A domain is a connected, open subset of a inite dimensional vector space.

14

  In fact, if we postulate harge conservation, then (3.3) and (3.4) are a consequence of this postulate. 15 Conducting materials, whih are not considered in this thesis, are introduced by a third constitutive relation Equations and heir Solution useful mathematical tool. Together with the constitutive relations and a normalization of and by the wave impedance

  Equations and heir Solution his means that we can replace the original problem by solving Maxwell's equations only in c subject to the boundary conditions (3.29) and (3.30). Likewise, the internal problem can be solved (with zero ields outside) by enforcing the boundary condition = × , ∀ ∈ , (3.33) = -× , ∀ ∈ . (3.34)

  38) into (3.13) yields curl curl e = -i e + (3.41) and applying the vector identity curl curl e = grad div ee [Bro+08], we obtain grad div e + e = i e . (3.42) Combining (3.40) and (3.42), we yield + grad div e + e + 2 e =grad e . (3.43) Equation (3.38) only deines the curl of e , so we are free to hoose div e = i e ; this hoice is called the Lorenz gauge. hus we get e + 2 e = -(3.44)and we ind for the electric ield e = i e -1/(i) grad div e .

  ( s , s ) must satisfy the Silver-Müller radiation condition [Sil84; Mül48] 19 lim →∞ � s × -s � = 0 , (3.70) where = ‖‖. Instead of solving the partial diferential equations (3.66) and (3.67) directly, we use the mixed potential formulas (3.58) and (3.59) and solve for the surface current density . he scatered wave ( s , s ) is obtained by evaluating (3.56) and (3.57).

  3.79) subject to the boundary condition e = , ∀ ∈ (3.80) for the Dirihlet problem, ∂ ∂ e = , ∀ ∈ (3.81) for the Neumann problem , and the decay condition [SS11] e () ≤ /‖‖ for ‖‖ → ∞ . (3.82) A solution for the Dirihlet problem can be obtained by solving V = (3.83)

  Fig. 4.1.: Deinition of the RWG functions. he vector denotes the directed edge, + and - denote the domains of the cells, + and - denote vertices on the edge , and +and - are the vertices opposite to the edge .

Fig

  Fig. 5.1.: he standard (thik lines) and the barycentrically reined (thin lines) mesh. he grey shaded area is a dual cell.

  all = 1, … , 12 with bar being the area of the support of bar and � = ∑ 12 =1 bar is the area of the support of . Now we have all the functions spaces necessary to construct the primal and dual Haar bases. he Haar bases are obtained in the form of transformation matrices ĤΣ and ĤΛ . he columns of the matrices ĤΣ and ĤΛ carry the expansion coeicients of the Haar functions in terms of piecewise constant and dual piecewise constant Primal and Dual Haar Bases on Unstructured Meshes for the EFIE Chapter 5 functions , respectively. We refer to the generalized Haar bases ĤΣ and ĤΛ as primal and dual generalized Haar basis.

  he initial macro cell. (b) he macro cells ater the irst partitioning using METIS. (c) he next macro cells ater the application of METIS to a macro cell from Figure 5.2b.

Fig. 5

 5 Fig. 5.2.: Space Shutle: macro cells generated by using METIS.

  .2b and Figure 5.2c). his gives rise to new macro cells, on whih we deine new macro piecewise constant functions; we obtain a new matrix B and a new matrix R, and every time the new matrix B R is added to ĤΣ . In this way, we obtain C generalized Haar functions. Section 5.b Construction of the Generalized Haar Basis

  Haar Bases on Unstructured Meshes for the EFIE Chapter 5

  Fig. 5.3.: Cube: spectral analysis of the diferent preconditioners for varying ℎ and .

Chapter 5

 5 Tab. 5.1.: Space Shutle: the number of iterations and the time used by the solver to obtain a residual error below 1 ⋅ 10 -4 . :52 04:01:41 * he relative error is with respect to the solution obtained by using the loop-tree preconditioner. † his includes also the time for constructing the matrix T using the ACA. Loop-tree = 0°: Standard hier. pre. = 0°: his work = 90°: Loop-tree = 0°: Standard hier. pre. = 90°: his work (a) Plane wave excitation. = 0°: Loop-tree = 0°: Standard hier. pre. = 0°: his work = 90°: Loop-tree = 0°: Standard hier. pre. = 90°: his work (b) Voltage gap excitation.

Fig. 5

 5 Fig.5.4.: Space Shutle: bistatic scatering cross section. he system matrix is compressed with the ACA (precision 1 ⋅ 10 -4 ).

  Fig. 6.1.: Cube with six holes: open surface with ive global loops.

HBFig. 6

 6 Fig.6.2.: Cube: number of iterations for a CGS solver with precision 1 ⋅ 10 -6 and ACA accuracy 1 ⋅ 10 -7 .

Fig. 6

 6 Fig. 6.3.: Cube with six holes: the condition number as a function of the frequency.

Fig. 6

 6 Fig. 6.5.: Plates: relative error of the iteratively obtained solutions compared with the solution obtained by a direct inversion of the matrix.

Fig. 6

 6 Fig. 6.6.: Plate: model with 64 holes. he surface current density excited by a plane wave with frequency 1 MHz is shown. © IEEE 2014.

Fig. 6

 6 Fig. 6.7.: MiG-15: open surface with one hole.

Fig. 6 HB

 6 Fig. 6.8.: MiG-15: current density for a plane wave impinging when the frequency is 1 MHz. © IEEE 2014.

  .128) then y = 1 Σ and � x � = , where y = ∑ V =0 � y � and x = ∑ C =1 � x � . Testing this equation with yields

( a )

 a Real part of excited by an incident plane wave.(b) Non-uniform mesh due to hose.

Fig. 8

 8 Fig. 8.9.: Rafale.

Fig. 8 .

 8 Fig. 8.10.: Fokker Dr.I: real part of excited by an incident plane wave.

  he condition number as a function of the frequency. Average edge length ℎ = 0.he condition number as a function of the frequency. Average edge length ℎ = 0.2 m.A Hermitian, Positive Deinite, and Wellhe number of iterations as a function of the frequency. Average edge length ℎ = 0.2 m.

Fig. 9

 9 Fig. 9.1.: Sphere: study of the impact of the frequency on the condition number and the number of iterations.

  Paradigmen zur Vorkonditionierung von Integralgleihungen zur Lösung elektromagnetisher Streu-und Ausstrahlungsprobleme vor. Integralgleihungsverfahren werden häuig für Probleme dieser Art eingesezt, da-neben einer Reihe weiterer Vorteile-ofene Randbedingungen niht aufwendig modelliert werden müssen und keine numerishe Dispersion autrit. Für die Problembeshreibung werden typisherweise die elektrishe Feldintegralgleihung und die kombinierte Feldintegralgleihung verwendet. Leider sind diese shleht konditioniert: Wird die Anzahl der Unbekannten erhöht oder die Frequenz verringert, dann wähst die Konditionszahl der Systemmatrix, die durh die Diskretisierung der jeweiligen Integralgleihung entsteht.

	Präkonditionen [pʀɛːkɔndiːt͡ sɪoːˈnɛn]: restlose Abklärung von Vorbedingun-gen; eine absurdum geführte Transpa-renz; das Röntgen von Mietern durh Immobilienmakler oder von Menshen durh die Krankenkassen * Zusammenfassung Der große Polt: Ein Konversationslexikon Gerhard Polt Diese Dissertation stellt neue

  We are faced with the variational problem to ind ∈ suh that(, ) = ⟨ , ⟩ ′ × ∀ ∈ with ∈ ′ .

	Discretization of Boundary Integral Operators and Equations	Chapter 4
	with > 0. (4.3)
	Equation (4.1) (or equivalently (4.3)) has a unique solution if 23
	sup ∈	� � (, ) � � > 0 , ∀ ≠ 0 ,	(4.5)
	inf ∈	sup ∈	(, ) ‖‖ ‖‖	≥ ALBB > 0	(4.6)
	are satisied, where ALBB is a positive constant. hese conditions are referred to as inf-sup conditions for the analytic problem [Bab71; Bre74; XZ03]. 24 For a numerical approximation, we must use inite-dimensional subspaces of and , that is, we employ = span � � =1 ⊂ and = span

  :53:03 * his is the total time including the setup costs for the preconditioners.

	Section 8.e			Numerical Results
	/ℎ		Preconditioner
		None		his work
		Iterations	Time * (h:m:s)	Iterations	Time * (h:m:s)
	Voltage gap excitation 0.48 25 13 0.6 20 16 0.8 15 24	13 189 17 546 15 793	89:03:11 180:17:37 1055 705 159:54:04 1943	12:10:09 29:18:19 59:18:11
	Plane wave excitation 0.48 25 13 0.6 20 16 0.8 15 24	10 842 16 654 14 151	73:25:57 171:08:41 1544 1413 136:22:36 3223	24:08:18 42:14:42

79

  A discretized with RWG functions, deined (4.22) T Transformation matrix of a solenoidal hierarhical basis. Deinition varies from hapter to hapter. � Σ Transformation matrix of a non-solenoidal hierarhical basis. Definition varies from hapter to hapter.

	13) Discretized Δ operator with dual basis functions, deined in (8.5) Sobolev space of vector functions on a surface, deined in Equa-tion (2.82) See (2.84) See (2.90) (curl , ) See (2.91) ( ) 2 ( ) (div , ) Operators D α Derivative operator, see (2.1) grad Surface gradient, see Deinition 2.5 curl Surface curl, deined in Deinition 2.6 Δ Laplace-Beltrami operator, see Deinition 2.7 div Surface divergence, see Deinition 2.9 Surfacic curl, see Deinition 2.10 T EFIE operator, deined in (3.61) T A Vector potential operator, deined in (3.62) T Φ Scalar potential operator, deined in (3.64) M MFIE operator, deined in (3.74) C CFIE operator, deined in (3.78) K Deined in (3.60) I, I Identity operators (vector and scalar functions) V Single layer operator, deined in (3.84) W Hypersingular operator (3.87) Matrices G Gram matrix of and functions T T operator discretized with RWG functions, see (4.20) T Φ T Φ discretized with RWG functions, deined in (4.24) e × i discretized, deined in (4.25) j Unknown vector, see (4.19) M Discretized M operator, deined in (4.28) h × i discretized, deined in (4.32) � M Discretized M operator, tested with rotated dual functions, de-ined in (4.34) h × i discretized, deined in (4.38) C Standard CFIE system matrix, deined in (4.40) � C Conforming CFIE system matrix, deined in (4.42) V Discretized V operator, deined in (4.43) W Discretized W operator, deined in (4.44) � Discretized Δ operator, deined in (8.4) � � 1 Λ Index All-one vector, deined in (4.74) 1 Σ All-one vector, deined in (4.75) Λ Loop transformation matrix, deined in (4.66) Σ Star transformation matrix, deined in (4.67) H Globel loop transformation matrix P Λ Projector to solenoidal Helmholz subspace, deined in (4.76) P Σ Projector to non-solenoidal Helmholz subspace, deined in (4.77) P H Projector to quasi-harmonic Helmholz subspace, deined in (4.78) Q Loop-star transformation matrix, deined in (4.68) P Helmholz projector based preconditioner, see (4.85) � Λ Basis functions Piecewise constant functions, deined in (4.14) Piecewise linear functions, deined in (4.15) Rao-Wilton-Glisson (RWG) functions, deined in (4.16) Functions dual to the RWG functions, suh as the BC functions � Dual piecewise constant functions, deined in (5.1) � Bufa-Christiansen piecewise linear functions Variables Number of unknowns C Number of cells of the mesh V Number of (inner) vertices of the mesh Λ Number of linearly independent solenoidal functions, whih can be formed with RWG functions Σ Number of linearly independent non-solenoidal functions, whih can be formed with RWG functions A atlas • 18 distribution • 25 tempered • 26 domain • 16 B ball B () • 17 basis functions dual piecewise constant • 68 E equivalence principle • 34 equivalent norm • 29 Haar • 68 piecewise constant • 47 piecewise linear • 48 RWG • 48 bilinear form • 45 boundary condition Dirihlet • 42 Neumann • 42 C Calderón identity scalar • 43, 78, 133 vector • 129 F function space ∞ • 17 • 17 • 27 1 loc () • 24 (div , ) • 30 () • 29 (curl , ) • 30 () • 24 test functions • 24 topological dual • 25 harts • 18 constitutive relations • 32 D G Gram matrix • 48 derivative distribution • 25 weak • 24 difeomorphism • 17 H Huygens' principle • see equivalence principle

A T

Important contributions to the formalism we are using today stem from Oliver Heaviside[START_REF] Heaviside | Electrical Papers[END_REF][START_REF] Heaviside | Electromagnetic heory[END_REF] and Heinrih Herz[START_REF] Herz | Ueber die Beziehungen zwishen den Maxwell'shen electrodynamishen Grundgleihungen und den Grundgleihungen der gegnerishen Electrodynamik[END_REF][START_REF] Herz | Ueber die Grundgleihungen der Electrodynamik für ruhende Körper[END_REF]. he irst textbook on Maxwell's equations, whih according to[START_REF] Arthur | he Evolution of Maxwell's Equations from 1862 to the Present Day[END_REF] presented the theory in a clear and instructive manner, was published by August Föppl[START_REF] Föppl | Einführung in die Maxwellsche heorie der Elektrizität: mit einem einleitenden Abschnite über das Rechnen mit Vektorgrössen in der Physik[END_REF]. For a detailed discussion on the evolution of Maxwell's equations, see[START_REF] Arthur | he Evolution of Maxwell's Equations from 1862 to the Present Day[END_REF].

A cornucopia of numerical methods has been presented in the past decades for the solution of Maxwell's equations. Hence, the methods discussed in this section are mere samples to illuminate the context in whih integral equation methods are situated. For a more complete discussion of the diferent tehniques, the reader may consider standard textbooks suh as[START_REF] Jin | heory and Computation of Electromagnetic Fields[END_REF].

In fact, a discretization of the Calderón identity could have been obtained before the introduction of the BC functions since the Chen-Wilton (CW) functions would haven been applicable as well[START_REF] Chen | Electromagnetic Scatering by hree-Dimensional Arbitrary Complex Material/Conducting Bodies[END_REF].

Also[START_REF] Andriulli | Solving the EFIE at Low Frequencies with a Conditioning hat Grows Only Logarithmically with the Number of Unknowns[END_REF] yields a logarithmic growth of the condition number, but it is limited to structured meshes.

hey can be applied eiciently if more computational complexity is spend on their construction.

For the sake of simplicity, we only consider single-body problems. An extension to multibody problems is possible.

Clearly, these deinitions can be generalized to the -dimensional case. For the sake of simplicity, we constrain ourselves here to the cases whih are relevant for the modeling of realistic scatering scenarios.

A difeomorphism is a bijective, continuously diferentiable function whose inverse function is continuously diferentiable as well.

he cotangent plane is the dual space of the tangent plane. For the deinition of dual space, see Deinition 2.13.

An example for the limitation imposed by the strong derivative is the one-dimensional scalar wave equation 2 2 / 2 -2 / 2 = 0, where in the distributional sense every function of the form ( ± ) is a solution even though the solution might not be twice continuously diferentiable.

Formally, we should write that the space () contains all equivalence classes of measurable functions deined on suh that|| is integrable for 1 ≤ < ∞. he introduction of equivalence classes is necessary to ensure that () has only one element whose norm is zero. See[START_REF] Rudin | Functional Analysis. 2nd ed. International series in pure and applied mathematics[END_REF].

Virtually any vector space we encounter in this thesis will at least be a normed or even be a Hilbert space (and so they are also topological vector spaces). he reason why we have to mention topological vector spaces here, is that D is not a normed space (in fact, it is not even a metric space).

he radiation condition is necessary to enforce that only solutions are obtained whih are waves traveling from a source to ininity, where the ratio of the electric and magnetic ield is the wave impedance 2 , and where the electric ield, the magnetic ield, and propagation vector form a trihedron.

We note that the terminology of what is to be called CFIE is not consistent in literature. What is called CFIE in[START_REF] Bufa | A Coercive Combined Field Integral Equation for Electromagnetic Scatering[END_REF] is typically called combined source integral equation in the engineering community. Diferent from the CFIE, one does not solve directly for the (real) electric current, but for some equivalent current. he combined source integral equation was irst presented for the Helmholz equation in[START_REF] Brakhage | Über das dirihletshe Außenraumproblem für die Helmholzshe Shwingungsgleihung[END_REF] and is therefore oten referred to as the Brakhage-Werner trik.

his is by no means the only ansaz for inding a solution of the Dirihlet problem. It is an example of the indirect approahes, where the quantity , we solve for, has no physical meaning. For an overview of the diferent approahes, we refer the reader to[START_REF] Steinbah | Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements[END_REF].

See[START_REF] Rudin | Functional Analysis. 2nd ed. International series in pure and applied mathematics[END_REF] for a deinition of quotient space.

In literature, they are also referred to as path functions.

Oten referred to as pyramid or nodal functions.

In literature, the Gram matrix is also referred to as mass matrix.

he CW functions, whih have been developed in a diferent context many years before the BC functions[START_REF] Chen | Electromagnetic Scatering by hree-Dimensional Arbitrary Complex Material/Conducting Bodies[END_REF], could be used as well.

-1/2 (curl , ). A discretization with × is doomed to fail since discrete inf-sup conditions are not satisied. Instead, a diferent set of basis functions must be used suh that the discrete inf-sup conditions are satisied. his issue was discussed in[START_REF] Cools | Accurate and Conforming Mixed Discretization of the MFIE[END_REF], where the use of BC functions was proposed. In this

( )-stable, then the upper bound of the condition number of � Λ T G � Λ or � Σ T G � Σ can still be prohibitively large for practical applications.

If the preconditioner from Chapter 5 is applied, where the power iteration method is used to cure the low-frequency breakdown, it is more customary to employ the power iteration method for the quasi-harmonic part of the spectrum as well.

Due to the elimination of a loop function, the conditioning is actually worse: there is an isolated singular value going astray[START_REF] Andriulli | Loop-Star and Loop-Tree Decompositions: Analysis and Eicient Algorithms[END_REF]. In order not to complicate the situation, we ignore this efect.

-stable satisfying (8.71), and the 2 -stability of � T (properly rescaled by D+2 ) can be expressed as

To apply our new method to a more realistic structure, we employed the model of a Space Shutle shown in Fig. 7.2. he electric size of the Space Shutle is 1/2, where is the wavelength. Table 7.1 summarizes our results, where the solver tolerance was 1 ⋅ 10 -6 .

d) Conclusion

First, we can conclude that the hierarhical loop functions, whih fail to precondition the vector potential operator T A of the EFIE, can be successfully applied to the CFIE (though the results obtained by using three-point hierarhical loop functions are usually beter, and they work for the EFIE as well). he best results were obtained with our new formulation in (7.35), whih diferent from all other hierarhical preconditioners, works on both structured and unstructured meshes. It is part of ongoing researh to obtain a hierarhical preconditioner whih remains eicient for electrically large problems.

Section 8.b New Formulation is the Laplace-Beltrami operator discretized with dual piecewise linear nodal functions as deined in [START_REF] Bufa | A Dual Finite Element Complex on the Barycentric Reinement[END_REF]. he matrix Q T G × , Q is even more diicult to invert since unlike � it is not a symmetric matrix anymore due to G × , .

Recently, a sheme has been presented that leverages the quasi-Helmholz projectors [And+13], whih we have discussed in Section 4.d.. In addition to the projectors that were deined therein, we also need P ΣH ∶= I -P Λ . his allows to deine the primal P ∶= P ΛH / √ + iP Σ √ and the dual decomposition operator

is well-conditioned [And+13]. Diferent from (8.3), the costly global loop inding and construction of the (dense) matrix H is avoided. Instead of dealing with the non-symmetric matrix Q T G × , Q , in (8.9) only symmetric, positive semi-deinite graph Laplacians Λ T Λ (vertex-based) and Σ T Σ (cell-based) appear. As pointed out in [And+13], a plethora of (blak box) algorithms exists for inverting these matrices eiciently.

b) New Formulation his section introduces and motivates the new formulation and provides the implementational details. For the interested reader, a theoretical derivation of this formulation is provided in the next section.

Similar to the formulation in (8.9), the new formulation uses quasi-Helmholz projectors. Diferent from (8.9), no second discretization of T with dual functions is required and thus no (barycentric) reinement of the mesh. Instead, we exploit the fact that Λ and Σ are ill-conditioned and use them to precondition T .

We propose the new formulation we have the analytic formula

, if cells and share edge , ,

where the function NoC() returns the number of cells atahed to th vertex of the mesh, the function VoC(, ) returns the global index of the th vertex of the th cell, and + and -are the indices of the vertices of the th edge.

In order to show the well-conditioning of (8.48) in the static limit, we need to establish spectral equivalences between W and � as well as V and � �. hese equivalences will be established by using Rayleigh quotients. What makes it diicult is that for example W possesses a null space and if we need to form the inverse of a product of matrices where some matrices have a null space and some matrices have not, then the inverse of suh a product cannot be simpliied easily. To avoid the null space issue, we follow a standard approah (as we have done already in Chapter 5) by deining operators that are identical to W and Δ for mean-value free functions, but have no null space [START_REF] Steinbah | Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements[END_REF]: we introduce the operator Ŵ ∶ 1/2 → -1/2 deined by the bilinear form

for all , ∈ 1/2 ( ). We note that the unique solution of Ŵ = is also a solution of W = when satisies the solvability condition � d( ′ ) = 0. his can be seen when = 1 in (8.54), whih reduces to (1, ) 2 ( ) (1, 1) 2 ( ) = 0. Likewise, let Δ ∶ 1 /R → -1 be the Laplace-Beltrami operator [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF]; we consider the 1 -elliptic modiied Laplace-Beltrami operator Δ deined by the bilinear form

When are used for the discretization of Ŵ and Δ , the resulting matrices are Since the MiG-15 is an open structure, we were also interested to see how the new formulation behaves for high frequencies, that is, when the electric size of the model spans several wave lengths . To this end, we used as discretization of the MiG-15 with 306 036 unknowns, where the maximum edge length is 0.07 m and the average edge length ℎ is 0.025 m (standard deviation is 0.0023 m). We varied the frequency from 0.48 GHz, where ℎ = /25, to 0.8 GHz, where ℎ = /15. his frequency range corresponds to an electric size varying from 13 to 24. We considered a voltage gap and a plane wave excitation and compared the new formulation with an unpreconditioned EFIE. Table 8.1 summarizes or indings. We see that the new formulation behaves favorably even for higher frequencies, though it seems that its advantage is waning the higher the frequency becomes.

As an example of a simply connected and closed structure, we considered the model of a Rafale ighter shown in Figure 8.9a, whih is discretized with a non-uniform mesh (see Figure 8.9b). We employed a coarse initial discretization with 18 171 unknowns and performed two structured reinement steps ending up with 290 736 unknowns (by using a structured reinement the geometry is not hanged). We note that cond G ≈ 1.8 ⋅ 10 5 remains constant for all reinements, but G varies from 276 of the initial discretization to 769 of the inest discretiza- Spectral index 1/ℎ in 1 /m Tab. 8.2.: Rafale: the number of iterations and the time used by the solver to obtain a residual error below 1 ⋅ 10 -4 .

e) Conclusion

We presented a preconditioner for the EFIE that yields a Hermitian, positive deinite, and well-conditioned system matrix without requiring the use of a barycentrically reined mesh. While it outperforms a standard tehnique suh as the loop-tree preconditioner in the low-and mid-frequency range, further investigations are necessary for electrically large problems. both cured. We observed, however, that the condition number starts to grow when the frequency is increased once a geometry dependent threshold is passed.

Appendix A he Discretized Laplace-Beltrami and the Hypersingular Operator

In the following, we give a proof for the spectral equivalence of Ŵ and � in the case that we have a nested sequence of piecewise linear function spaces.

Proposition A.1. Let , ⊂ ,+1 , = 0, … , -1, denote a nested sequence of piecewise linear function spaces, ∈ , , and V = dim � , � is the number of vertices of the mesh. Let � be the discretization of the modiied Laplace-Beltrami operator Δ with piecewise linear functions ∈ , , that is,

(A.1)

Proof. To prove this proposition, we leverage the stability results that Stevenson obtained for his three-point hierarhical wavelets [START_REF] Stevenson | Stable hree-Point Wavelet Bases on General Meshes[END_REF] (see also Chapter 7, where we have discussed the preconditioning efect of this basis). Let � λ T ∈ R V × V be the transformation matrix that maps from the three point hierarhical wavelets �

T to the piecewise linear basis . hen the following equivalences follow from [START_REF] Stevenson | Stable hree-Point Wavelet Bases on General Meshes[END_REF]:

where

he Discretized Laplace-Beltrami and the Hypersingular Operator Chapter A and the function Λ () returns the level on whih the function � T was deined. Equations (A.3) and (A.4) can be further simpliied by considering that are List of Variables and Other Mathematical Symbols he following list summarizes frequently used variables and mathematical symbols.

Symbol Description

Accents and Operations � he wide hat symbols denotes the involvement of a hierarhical basis ̂he standard hat denotes a unit vector, or that a null space of an operator has been delected � he wide tilde denotes the involvement of dual basis functions

Inverse of the matrix A (if it exists)

A + Moore-Penrose pseudo-inverse of the matrix A A T Transpose of the matrix A A, () Conjugate of the matrix A or the function , respectively Abstract : The electric field integral equation (EFIE) and the combined field integral equation (CFIE) suffer from the dense-discretization and the low-frequency breakdown: if the average edge length of the mesh is reduced, or if the frequency is decreased, then the condition number of the system matrix grows. This leads to slowly or non-converging iterative solvers. This dissertation presents new paradigms for rapidly converging integral equation solvers: to overcome the illconditioning, we advance and extend the state of the art both in hierarchical basis and in Calderón preconditioning techniques. For the EFIE, we introduce a hierarchical basis for structured and unstructured meshes based on generalized primal and dual Haar prewavelets. Furthermore, a framework is introduced which renders the hierarchical basis able to efficiently precondition the EFIE in the case that the scatterer is multiply connected. The applicability of hierarchical basis preconditioners to the CFIE is analyzed and an efficient preconditioning scheme is derived. In addition, we present a refinement-free Calderón multiplicative preconditioner (RF-CMP) that yields a system matrix which is Hermitian, positive definite (HPD), and well-conditioned. Different from existing Calderón preconditioners, no dual basis functions and thus no refinement of the mesh is required. Since the matrix is HPD-in contrast to standard discretizations of the EFIE-we can apply the conjugate gradient (CG) method as iterative solver, which guarantees convergence. Eventually, the RF-CMP is extended to the CFIE.