This Master's thesis relates to continuum mechanics and its connection with selected directions of modern differential geometry, which deal with geometric structures and objects. These are mainly tensors, bundles, varieties and jets. The first part is devoted to the mechanics of the continuum itself and its description in several areas, others deal with mathematical concepts and their possible application in mechanics.

In many applications that deal with high dimensional data, it is important to not store the high dimensional object itself, but its representation in a data sparse way. This aims to reduce the storage and computational complexity. There is a general scheme for representing tensors with the help of sums of elementary tensors, where the summation structure is defined by a graph/network. This scheme allows to generalize commonly used approaches in representing a large amount of numerical data (that can be interpreted as a high dimensional object) using sums of elementary tensors. The classification does not only distinguish between elementary tensors and non-elementary tensors, but also describes the number of terms that is needed to represent an object of the tensor space. This classification is referred to as tensor network (format). This work uses the tensor network based approach and describes non-linear block Gauss-Seidel methods (ALS and DMRG) in the context of the general tensor network framework. Another contribution of the thesis is the general conversion of different tensor formats. We are able to efficiently change the underlying graph topology of a given tensor representation while using the similarities (if present) of both the original and the desired structure. This is an important feature in case only minor structural changes are required. In all approximation cases involving iterative methods, it is crucial to find and use a proper initial guess. For linear iteration schemes, a good initial guess helps to decrease the number of iteration steps that are needed to reach a certain accuracy, but it does not change the approximation result. For non-linear iteration schemes, the approximation result may depend on the initial guess. This work introduces a method to successively create an initial guess that improves some approximation results. This algorithm is based on successive rank 1 increments for the r-term format. There are still open questions about how to find the optimal tensor format for a given general problem (e.g. storage, operations, etc.). For instance in the case where a physical background is given, it might be efficient to use this knowledge to create a good network structure. There is however, no guarantee that a better (with respect to the problem) representation structure does not exist.

Tensors occur in many areas of science and engineering. Especially, they are used to describe charge, mass and energy transport (i.e. electrical conductivity tensor, diffusion tensor, thermal conduction tensor resp.) If the locale transport pattern is complicated, usual second order tensor representation is not sufficient. So far, there are no appropriate visualization methods for this case. We point out similarities of symmetric higher order tensors and spherical harmonics. A spherical harmonic representation is used to improve tensor glyphs. This paper unites the definition of streamlines and tensor lines and generalizes tensor lines to those applications where second order tensors representations fail. The algorithm is tested on the tractography problem in diffusion tensor magnetic resonance imaging (DT-MRI) and improved for this special application.

Abstract Iris The power of quantum computing lies in its ability to perform certain calculations and solve complex problems exponentially faster than classical computers. This potential has profound implications for a wide range of fields, including particle physics. This thesis lays a fundamental foundation for understanding quantum computing. Particular emphasis is placed on the intricate process of quantum gate decomposition, an elementary lynchpin that underpins the development of quantum algorithms and plays a crucial role in this research. In particular, this concerns the implementation of quantum algorithms designed to simulate the dynamic evolution of multi-particle quantum systems - so-called Hamiltonian simulations. The concept of quantum gate decomposition is introduced and linked to quantum circuit optimisation. The decomposition of quantum gates plays a crucial role in fault-tolerant quantum computing in the sense that an optimal implementation of a quantum gate is essential to efficiently perform a quantum simulation, especially for near-term quantum computers. Part of this thesis aims to propose a new explicit tensorial notation of quantum computing. Two notations are commonly used in the literature. The first is the Dirac notation and the other standard formalism is based on the so-called computational basis. The main disadvantage of the latter is the exponential growth of vector and matrix dimensions and the fact that it hides some relevant quantum properties of the operations by increasing the apparent number of independent variables. A third possible notation is introduced here, which describes qubit states as tensors and quantum gates as multilinear or quasi-multilinear maps. Some advantages for the detection of separable and entangled systems and for measurement techniques are also shown. Finally, this thesis demonstrates the advantage of quantum computing in the description of multi-particle quantum systems by proposing a quantum algorithm to simulate collective neutrino oscillations. Collective flavour oscillations of neutrinos due to forward neutrino-neutrino scattering provide an intriguing many-body system for time evolution simulations on a quantum computer. These phenomena are of particular interest in extreme astrophysical settings such as core-collapse supernovae, neutron star mergers and the early universe. A detailed description of the physical phenomena and environments in which collective flavor oscillations occur is first reported, and the derivation of the Hamiltonian governing the evolution of flavor oscillations is detailed. The aim is to reproduce this evolution using a quantum algorithm. To manage the computational complexity, we use the Trotter approximation of the time evolution operator, which mitigates the exponential growth of circuit complexity. The quantum algorithm was designed to work on a trapped-ion based testbed (the theory of which is presented in detail). After machine-aware optimisation, the quantum circuit implementing the algorithm was run on the real quantum machine 'Quantinuum', and the results are presented and discussed.

The topic of this dissertation lies at the interface between the areas of Harmonic Analysis, Partial Differential Equations, and Geometric Measure Theory, with an emphasis on the study of singular integral operators associated with second and higher order elliptic boundary value problems in non-smooth domains. The overall aim of this work is to further the development of a systematic treatment of second and higher order elliptic boundary value problems using singular integral operators. This is relevant to the theoretical and numerical treatment of boundary value problems arising in the modeling of physical phenomena such as elasticity, incompressible viscous fluid flow, electromagnetism, anisotropic plate bending, etc., in domains which may exhibit singularities at all boundary locations and all scales. Since physical domains may exhibit asperities and irregularities of a very intricate nature, we wish to develop tools and carry out such an analysis in a very general class of non-smooth domains, which is in the nature of best possible from the geometric measure theoretic point of view. The dissertation will be focused on three main, interconnected, themes: A. A systematic study of the poly-Cauchy operator in uniformly rectifiable domains in $\mathbb{C}$; B. Solvability results for the Neumann problem for the bi-Laplacian in infinite sectors in ${\mathbb{R}}^2$; C. Connections between spectral properties of layer potentials associated with second-order elliptic systems and the underlying tensor of coefficients. Theme A is based on papers [16, 17, 18] and this work is concerned with the investigation of polyanalytic functions and boundary value problems associated with (integer) powers of the Cauchy-Riemann operator in uniformly rectifiable domains in the complex plane. The goal here is to devise a higher-order analogue of the existing theory for the classical Cauchy operator in which the salient role of the Cauchy-Riemann operator $\overline{\partial}$ is now played by $\overline{\partial}^m$ for some arbitrary fixed integer $m\in{\mathbb{N}}$. This analysis includes integral representation formulas, higher-order Fatou theorems, Calderón-Zygmund theory for the poly-Cauchy operators, radiation conditions, and higher-order Hardy spaces. Theme B is based on papers [3, 19] and this regards the Neumann problem for the bi-Laplacian with $L^p$ data in infinite sectors in the plane using Mellin transform techniques, for $p\in(1,\infty)$. We reduce the problem of finding the solvability range of the integrability exponent $p$ for the $L^{p}$ biharmonic Neumann problem to solving an equation involving quadratic polynomials and trigonometric functions employing the Mellin transform technique. Additionally, we provide the range of the integrability exponent for the existence of a solution to the $L^{p}$ biharmonic Neumann problem in two-dimensional infinite sectors. The difficulty we are overcoming has to do with the fact that the Mellin symbol involves hypergeometric functions. Finally regarding theme C, based on the ongoing work in [2], the emphasis is the investigation of coefficient tensors associated with second-order elliptic operators in two dimensional infinite sectors and properties of the corresponding singular integral operators, employing Mellin transform. Concretely, we explore the relationship between distinguished coefficient tensors and $L^{p}$ spectral and Hardy kernel properties of the associated singular integral operators. / Mathematics

Dans cette thèse, nous nous focalisons sur le problème de la décomposition polyadique minimale de tenseurs de dimension trois, problème auquel on se réfère généralement sous différentes terminologies : « Polyadique Canonique » (CP en anglais), « CanDecomp », ou encore « Parafac ». Cette décomposition s'avère très utile dans un très large panel d'applications. Cependant, nous nous concentrons ici sur la spectroscopie de fluorescence appliquée à des données environnementales particulières de type échantillons d'eau qui pourront avoir été collectés en divers endroits ou différents moments. Ils contiennent un mélange de plusieurs molécules organiques et l'objectif des traitements numériques mis en œuvre est de parvenir à séparer et à ré-estimer ces composés présents dans les échantillons étudiés. Par ailleurs, dans plusieurs applications comme l'imagerie hyperspectrale ou justement, la chimiométrie, il est intéressant de contraindre les matrices de facteurs recherchées à être réelles et non négatives car elles sont représentatives de quantités physiques réelles non négatives (spectres, fractions d'abondance, concentrations, ...etc.). C'est pourquoi tous les algorithmes développés durant cette thèse l'ont été dans ce cadre (l'avantage majeur de cette contrainte étant de rendre le problème d'approximation considéré bien posé). Certains de ces algorithmes reposent sur l'utilisation de méthodes proches des fonctions barrières, d'autres approches consistent à paramétrer directement les matrices de facteurs recherchées par des carrés. / In this manuscript, we focus on the minimal polyadic decomposition of third order tensors, which is often referred to: “Canonical Polyadic” (CP), “CanDecomp”, or “Parafac”. This decomposition is useful in a very wide panel of applications. However, here, we only address the problem of fluorescence spectroscopy applied to environment data collected in different locations or times. They contain a mixing of several organic components and the goal of the used processing is to separate and estimate these components present in the considered samples. Moreover, in some applications like hyperspectral unmixing or chemometrics, it is useful to constrain the wanted loading matrices to be real and nonnegative, because they represent nonnegative physical data (spectra, abundance fractions, concentrations, etc...). That is the reason why all the algorithms developed here take into account this constraint (the main advantage is to turn the approximation problem into a well-posed one). Some of them rely on methods close to barrier functions, others consist in a parameterization of the loading matrices with the help of squares. Many optimization algorithms were considered: gradient approaches, nonlinear conjugate gradient, that fits well with big dimension problems, Quasi-Newton (BGFS and DFP) and finally Levenberg-Marquardt. Two versions of these algorithms have been considered: “Enhanced Line Search” version (ELS, enabling to escape from local minima) and the “backtracking” version (alternating with ELS).

Das Hauptresultat dieser Arbeit ist ein regularisierter arithmetischer Satz von Riemann-Roch für ein hermitesches Geradenbündel, die isometrisch zum Geradenbündel den Spitzenformen vom geraden Gewicht ist, auf eine arithmetische Fläche, deren komplexe Faser isometrisch zu einer hyperbolischen Riemannschen Fläche ohne elliptische Punkte ist. Der Beweis des Resultats erfolgt durch metrische Degeneration: Wir regularisieren die betreffenden Metriken in einer Umgebung der Singularitäten, wenden dann den arithmetischen Riemann-Roch-Satz von Gillet und Soulé an und lassen schließlich den Parameter gegen Null gehen. Durch die metrische Degeneration entsteht auf beiden Seiten der Formel ein divergenter Term. Die asymptotische Entwicklung der Divergenz berechnet sich auf der einen Seite direkt aus der Definition der glatten arithmetischen Selbstschnittzahlen. Der divergente Term auf der anderen Seite ist die zeta-regularisierte Determinante des zu den regularisierten Metriken assoziierten Laplace-Operators, der auf den 1-Formen mit Werten in dem betrachteten hermitischen Geradenbündel operiert. Wir definieren und berechnen zuerst eine Regularisiereung des entsprechenden zu den singulären Metriken assoziierten Laplace-Operators; diese wird später im regularisierten Riemann-Roch-Satz auftauchen. Zu diesem Zweck passen wir Ideen von Jorgenson-Lundelius, D'Hoker-Phong und Sarnak auf die vorliegende Situation an und verallgemeinern diese. Schließlich beweisen wir eine Formel für den zum betrachteten hermitischen Geradenbündel assoziierten Wärmeleitungskern auf der Diagonalen bei einer Modellspitze. Diese Darstellung steht im Zusammenhang mit einer Entwicklung nach zur Whittaker-Gleichung assoziierten Eigenfunktionen, die im Anhang bewiesen wird. Weitere Abschätzungen des zum betrachteten hermitischen Geradenbündel gehörigen Wärmeleitungskern auf der komplexe Faser der arithmetischen Fläche schließen den Beweis des Hauptresultats ab. / The main result of the dissertation is an arithmetic Riemann-Roch theorem for the hermitian line bundle of cusp form of given even integer weights on an arithmetic surface whose complex fiber is isometric to an hyperbolic Riemann surface without elliptic points. The proof proceeds by metric degeneration: We regularize the metric under consideration in a neighborhood of the singularities, then we apply the arithmetic Riemann-Roch theorem of Gillet and Soulé, and finally we let the parameter go to zero. Both sides of the formula blow up through metric degeneration. On one side the exact asymptotic expansion is computed from the definition of the smooth arithmetic intersection numbers. The divergent term on the other side is the zeta-regularized determinant of the Laplacian acting on 1-forms with values in the chosen hermitian line bundle associated to the regularized metrics. We first define and compute a regularization of the determinant of the corresponding Laplacian associated to the singular metrics, which will later occur int he regularized arithmetic Riemann-Roch theorem. To do so we adapt and generalize ideas od Jorgenson-Lundelius, D'Hoker-Phong, and Sarnak. Then, we prove a formula for the on-diagonal heat kernel associated to the chosen hermitian line bundle on a model cusp, from which its behavior close to a cusp is transparent. This expression is related to an expansion in terms of eigenfunctions associated to the Whittaker equation, which we prove in an appendix. Further estimates on the heat kernel associated to the chosen hermitian line bundle on the complex fiber of the arithmetic surface prove the main theorem.

Laboratoire I3S/CNRS / Os sistemas de telecomunicaÃÃes atuais oferecem servios que demandam taxas de transmissÃo muito elevadas. O problema da identificaÃÃo de canal aparece nesse contexto com um problema da maior importÃncia. O uso de tÃcnicas cegas tem sido de grande interesse na busca por um melhor compromisso entre uma taxas binÃria adequada e a qualidade da informaÃÃo recuperada. Apoiando-se em propriedades especiais dos cumulantes de 4a ordem dos sinais à saÃda do canal, esta tese introduz novas ferramentas de processamento de sinais com aplicaÃÃes em sistemas de comunicaÃÃo rÃdio-mÃveis. Explorando a estrutura simÃtrica dos cumulantes de saÃda, o problema da identificaÃÃo cega de canais à abordado a partir de um modelo multilinear do tensor de cumulantes 4a ordem, baseado em uma decomposiÃÃo em fatores paralelos (Parafac). No caso SISO, os componentes do novo modelo tensorial apresentam uma estrutura Hankel. No caso de canais MIMO sem memÃria, a redundÃncia dos fatores tensoriais à explorada na estimaÃÃo dos coeficientes dos canal. Neste contexto, novos algoritmos de identificaÃÃo cega de canais sÃo desenvolvidos nesta tese com base em um problema de otimizaÃÃo de mÃnimos quadrados de passo Ãnico (SS-LS). Os mÃtodos propostos exploram plenamente a estrutura multilinear do tensor de cumulantes bem como suas simetrias e redundÃncias, evitando assim qualquer forma de prÃ-processamento. Com efeito, a abordagem SS-LS induz uma soluÃÃo baseada em um Ãnico procedimento de minimizaÃÃo, sem etapas intermediÃrias, contrariamente ao que ocorre na maior parte dos mÃtodos existentes na literatura. Utilizando apenas os cumulantes de ordem 4 e explorando o conceito de Arranjo Virtual, trata-se tambÃm o problema da localizaÃÃo de fontes, num contexto multiusuÃrio. Uma contribuÃÃo original consiste em aumentar o nÃmero de sensores virtuais com base em uma decomposiÃÃo particular do tensor de cumulantes, melhorando assim a resoluÃÃo do arranjo, cuja estrutura à tipicamente obtida quando se usa estatÃsticas de ordem 6. Considera-se ainda a estimaÃÃo dos parÃmetros fÃsicos de um canal de comunicaÃÃo MIMO com muti-percursos. AtravÃs de uma abordagem completamente cega, o canal multi-percurso à primeiramente tratado como um modelo convolutivo e uma nova tÃcnica à proposta para estimar seus coeficientes. Esta tÃcnica nÃo-paramÃtrica generaliza os mÃtodos previamente propostos para os casos SISO e MIMO (sem memÃria). Fazendo uso de um formalismo tensorial para representar o canal de multi-percursos MIMO, seus parÃmetros fÃsicos podem ser obtidos atravÃs de uma tÃcnica combinada de tipo ALS-MUSIC, baseada em um algoritmo de subespaÃo. Por fim, serà considerado o problema da determinaÃÃo de ordem de canais FIR, particularmente no caso de sistemas MISO. Um procedimento completo à introduzido para a detecÃÃo e estimaÃÃo de canais de comunicaÃÃo MISO seletivos em freqÃÃncia. O novo algoritmo, baseado em uma abordagem de deflaÃÃo, detecta sucessivamente cada fonte de sinal, determina a ordem de seu canal de transmissÃo individual e estima os coeficientes associados. / Les systÃmes de tÃlÃcommunications modernes exigent des dÃbits de transmission trÃs ÃlevÃs. Dans ce cadre, le problÃme dâidentification de canaux est un enjeu majeur. Lâutilisation de techniques aveugles est dâun grand intÃrÃt pour avoir le meilleur compromis entre un taux binaire adÃquat et la qualità de lâinformation rÃcupÃrÃe. En utilisant les propriÃtÃs des cumulants dâordre 4 des signaux de sortie du canal, cette thÃse introduit de nouvelles mÃthodes de traitement du signal tensoriel avec des applications pour les systÃmes de communication radio-mobiles. En utilisant la structure symÃtrique des cumulants de sortie, nous traitons le problÃme de lâidentification aveugle de canaux en introduisant un mod`ele multilinÃaire pour le tenseur des cumulants dâordre 4, basà sur une dÃcomposition de type Parafac. Dans le cas SISO, les composantes du modÃle tensoriel ont une structure de Hankel. Dans le cas de canaux MIMO instantanÃs, la redondance des facteurs tensoriels est exploitÃe pour lâestimation des coefficients du canal. Dans ce contexte, nous dÃveloppons des algorithmes dâidentification aveugle basÃs sur une minimisation de type moindres carrÃs à pas unique (SS-LS). Les mÃthodes proposÃes exploitent la structure multilinÃaire du tenseur de cumulants aussi bien que les relations de symÃtrie et de redondance, ce qui permet dâÃviter toute sorte de traitement au prÃalable. En effet, lâapproche SS-LS induit une solution basÃe sur une seule et unique procÃdure dâoptimisation, sans les Ãtapes intermÃdiaires requises par la majorità des mÃthodes existant dans la littÃrature. En exploitant seulement les cumulants dâordre 4 et le concept de rÃseau virtuel, nous abordons aussi le problÃme de la localisation de sources dans le cadre dâun rÃseau dâantennes multiutilisateur. Une contribution originale consiste à augmenter le nombre de capteurs virtuels en exploitant un arrangement particulier du tenseur de cumulants, de maniÃre à amÃliorer la rÃsolution du rÃseau, dont la structure Ãquivaut à celle qui est typiquement issue de lâutilisation des statistiques dâordre 6. Nous traitons par ailleurs le problÃme de lâestimation des paramÃtres physiques dâun canal de communication de type MIMO à trajets multiples. Dans un premier temps, nous considÂerons le canal à trajets multiples comme un modÃle MIMO convolutif et proposons une nouvelle technique dâestimation des coefficients. Cette technique non-paramÃtrique gÃnÃralise les mÃthodes proposÃes dans les chapitres prÃcÃdents pour les cas SISO et MIMO instantanÃ. En reprÃsentant le canal multi-trajet à lâaide dâun formalisme tensoriel, les paramÃtres physiques sont obtenus en utilisant une technique combinÃe de type ALS-MUSIC, basÃe sur un algorithme de sous-espaces. Enfin, nous considÃrons le problÃme de la dÂetermination dâordre de canaux de type RIF, dans le contexte des systÃmes MISO. Nous introduisons une procÃdure complÃte qui combine la dÃtection des signaux avec lâestimation des canaux de communication MISO sÃlectifs en frÃquence. Ce nouvel algorithme, basà sur une technique de dÃflation, est capable de dÃtecter successivement les sources, de dÃterminer lâordre de chaque canal de transmission et dâestimer les coefficients associÂes.

Cette thèse développe des méthodes de diagonalisation conjointe de matrices et de tenseurs d’ordre trois, et son application à la séparation MIMO de sources de télécommunications numériques. Après un état, les motivations et objectifs de la thèse sont présentés. Les problèmes de la diagonalisation conjointe et de la séparation de sources sont définis et un lien entre ces deux domaines est établi. Par la suite, plusieurs algorithmes itératifs de type Jacobi reposant sur une paramétrisation LU sont développés. Pour chacun des algorithmes, on propose de déterminer les matrices permettant de diagonaliser l’ensemble considéré par l’optimisation d’un critère inverse. On envisage la minimisation du critère selon deux approches : la première, de manière directe, et la seconde, en supposant que les éléments de l’ensemble considéré sont quasiment diagonaux. En ce qui concerne l’estimation des différents paramètres du problème, deux stratégies sont mises en œuvre : l’une consistant à estimer tous les paramètres indépendamment et l’autre reposant sur l’estimation indépendante de couples de paramètres spécifiquement choisis. Ainsi, nous proposons trois algorithmes pour la diagonalisation conjointe de matrices complexes symétriques ou hermitiennes et deux algorithmes pour la diagonalisation conjointe d’ensembles de tenseurs symétriques ou non-symétriques ou admettant une décomposition INDSCAL. Nous montrons aussi le lien existant entre la diagonalisation conjointe de tenseurs d’ordre trois et la décomposition canonique polyadique d’un tenseur d’ordre quatre, puis nous comparons les algorithmes développés à différentes méthodes de la littérature. Le bon comportement des algorithmes proposés est illustré au moyen de simulations numériques. Puis, ils sont validés dans le cadre de la séparation de sources de télécommunications numériques. / This thesis develops joint diagonalization of matrices and third-order tensors methods for MIMO source separation in the field of digital telecommunications. After a state of the art, the motivations and the objectives are presented. Then the joint diagonalisation and the blind source separation issues are defined and a link between both fields is established. Thereafter, five Jacobi-like iterative algorithms based on an LU parameterization are developed. For each of them, we propose to derive the diagonalization matrix by optimizing an inverse criterion. Two ways are investigated : minimizing the criterion in a direct way or assuming that the elements from the considered set are almost diagonal. Regarding the parameters derivation, two strategies are implemented : one consists in estimating each parameter independently, the other consists in the independent derivation of couple of well-chosen parameters. Hence, we propose three algorithms for the joint diagonalization of symmetric complex matrices or hermitian ones. The first one relies on searching for the roots of the criterion derivative, the second one relies on a minor eigenvector research and the last one relies on a gradient descent method enhanced by computation of the optimal adaptation step. In the framework of joint diagonalization of symmetric, INDSCAL or non symmetric third-order tensors, we have developed two algorithms. For each of them, the parameters derivation is done by computing the roots of the considered criterion derivative. We also show the link between the joint diagonalization of a third-order tensor set and the canonical polyadic decomposition of a fourth-order tensor. We confront both methods through numerical simulations. The good behavior of the proposed algorithms is illustrated by means of computing simulations. Finally, they are applied to the source separation of digital telecommunication signals.

Abstract in English below.