In this thesis, three problems have been considered and new coding schemes have been devised for each of them. The first is related to distributed function computation, the second to coding for distributed storage and the final problem is based on locally correctable codes. A common theme of the first two problems considered is distributed computation. The first problem is motivated by the problem of distributed function computation considered by Korner and Marton, where the goal is to compute XOR of two binary sources at the receiver. It has been shown that linear encoders give better sum rates for some source distributions as compared to the usual Slepian-Wolf scheme. We generalize this distributed function computation setting to the case of more than two sources and the receiver is interested in computing multiple linear combinations of the sources. Consider `m' random variables each of which takes values from a finite field and are associated with a certain joint probability distribution. The receiver is interested in the lossless computation of `s' linear combinations of the m random variables. By considering the set of all linear combinations of m random variables as a vector space V , this problem can be interpreted as a subspace-computation problem. For this problem, we develop three increasingly refined approaches, all based on linear encoders. The first two approaches which are termed as common code approach and selected subspace approach, use a common matrix to encode all the sources. In the common code approach, the desired subspace W is computed at the receiver, whereas in the selected subspace approach, possibly a larger subspace U which contains the desired subspace is computed. The larger subspace U which gives the minimum sum rate itself is based on a decomposition of vector space V into a chain of subspaces. The chain of subspaces is determined by the joint probability distribution of m random variables and a notion of normalized measure of entropy. The third approach is a nested code approach, where all the encoding matrices are nested and the same subspace U which is identified in the selected subspace approach is computed. We characterize the sum rates under all the three approaches. The sum rate under nested code approach is no larger than both selected subspace approach and Slepian-Wolf approach. For a large class of joint distributions and subspaces W , the nested code scheme is shown to improve upon Slepian-Wolf scheme. Additionally, a class of source distributions and subspaces are identified, for which the nested code approach is sum-rate optimal. In the second problem, we consider a distributed storage network, where data is stored across nodes in a network which are failure-prone. The goal is to store data reliably and efficiently. For a required level of reliability, it is of interest to minimise storage overhead and also of interest to perform node repair efficiently. Conventionally replication and maximum distance separable (MDS) codes are employed in such systems. Though replication is very efficient in terms of node repair, the storage overhead is high. MDS codes have low storage overhead but even the repair of a single failed node requires contacting a large number of nodes and downloading all their data. We consider two coding solutions that have recently been proposed, which enable efficient node repair in case of single node failure. The first solution called regenerating codes seeks to minimize the amount of data downloaded for node repair, while codes with locality attempt to minimize the number of helper nodes accessed. We extend these results in two directions. In the first one, we introduce the notion of codes with locality where the local codes have minimum distance more than 2 and hence can recover a code symbol locally even in the presence of multiple erasures. These codes are termed as codes with local erasure correction. We say that a code has information locality if there exists a set of message symbols, each of which is covered by local codes. A code is said to have all-symbol locality if all the code symbols are covered by local codes. An upper bound on the minimum distance of codes with information locality is presented and codes that are optimal with respect to this bound are constructed. We make a connection between codes with local erasure correction and concatenated codes. The second direction seeks to build codes that combine the advantages of both codes with locality as well as regenerating codes. These codes, termed here as codes with local regeneration, are codes with locality over a vector alphabet, in which the local codes themselves are regenerating codes. There are two well known classes of regenerating codes known as minimum storage regenerating (MSR) codes and minimum bandwidth regenerating (MBR) codes. We derive two upper bounds on the minimum distance of vector-alphabet codes with locality, one for the case when the local codes are MSR codes and the second for the case when the local codes are MBR codes. We also provide several optimal constructions of both classes of codes which achieve their respective minimum distance bounds with equality. The third problem deals with locally correctable codes. A block code of length `n' is said to be locally correctable, if there exists a randomized algorithm such that any one of the coordinates of the codeword can be recovered by querying at most `r' coordinates, even in presence of some fraction of errors. We study the local correctability of linear codes whose duals contain 4-designs. We also derive a bound relating `r' and fraction of errors that can be tolerated, when each instance of the randomized algorithm is `t'-error correcting instead of simple parity computation.

Diese Arbeit beschäftigt sich mit der Berechnung singulärer Punkte nichtlinearer Gleichungssysteme F(x)=0. Dazu werden minimal erweiterte Systeme der Form F(x)+D*s=0, f(x)=0 betrachtet. Die allgemeine Vorgehensweise zur Berechnung singulärer Punkte mit solchen erweiterten Systemen wird geschlossen dargestellt. Dazu werden zuerst die (teilweise verallgemeinerten Ljapunov-Schmidt-)reduzierten Funktionen von Golubitsky und Schaeffer, Beyn, Jepson und Spence, Griewank und Reddien, Kunkel bzw. Govaerts verallgemeinert und zusammengefasst. Es wird die verallgemeinerte Kontaktäquivalenz all dieser verallgemeinerten reduzierten Funktionen und die Gleichheit der benötigten Regularitätsannahmen bewiesen. Für eine weitere, neu eingeführte reduzierte Funktion wird die in dieser Arbeit definierte Ableitungsäquivalenz zu den anderen reduzierten Funktionen gezeigt. Mit dieser neuen reduzierten Funktion wird eine Reihe singulärer Punkte klassifiziert. Aus dieser Klassifikation ergeben sich Funktionen f aus Ableitungen der neuen reduzierten Funktion. Mit den so eingeführten Funktionen f kann das zweistufiges Newtonverfahren nach Pönisch und Schwetlick effektiv angewendet werden. Alle benötigten Ableitungen werden mittels Automatischer Differentiation bestimmt. Die numerischen Ergebnisse für eine Reihe von Beispielen zeigen die Effizienz dieses Verfahrens. Beim Newtonverfahren werden lineare Gleichungssysteme mit geränderten Matrizen B gelöst. Es wird gezeigt, für welche Ränderungen die Konditionszahl von B minimal ist. Dies ist z.B. für gewisse Vielfache der Singulärvektoren zu den kleinsten Singulärwerten der Fall. Zur Bestimmung dieser Ränderungen wird die inverse Teilraumiteration für Singulärwerte in verschiedenen Algorithmen angewendet. Die Konvergenzeigenschaften werden untersucht. Für einen Algorithmus wird bewiesen, dass die Konditionszahlen der iterierten geränderten Matrizen monoton fallen. Die numerischen Experimente bestätigen diese Eigenschaften.

To meet the exponentially increasing traffic data driven by the rapidly growing mobile subscriptions, both industry and academia are exploring the potential of a new genera- tion (5G) of wireless technologies. An important 5G goal is to achieve high data rate. Small cells with spectrum sharing and multiple-input multiple-output (MIMO) techniques are one of the most promising 5G technologies, since it enables to increase the aggregate data rate by improving the spectral efficiency, nodes density and transmission bandwidth, respectively. However, the increased interference in the densified networks will in return limit the achievable rate performance if not properly managed. The considered setup can be modeled as MIMO interference networks, which can be classified into the K-user MIMO interference channel (IC) and the K-cell MIMO interfering broadcast channel/multiple access channel (MIMO-IBC/IMAC) according to the number of mobile stations (MSs) simultaneously served by each base station (BS). The thesis considers two physical layer (PHY) resource allocation problems that deal with the interference for both models: 1) Pareto boundary computation for the achiev- able rate region in a K-user single-stream MIMO IC and 2) grouping-based interference alignment (GIA) with optimized IA-Cell assignment in a MIMO-IMAC under limited feedback. In each problem, the thesis seeks to provide a deeper understanding of the system and novel mathematical results, along with supporting numerical examples. Some of the main contributions can be summarized as follows. It is an open problem to compute the Pareto boundary of the achievable rate region for a K-user single-stream MIMO IC. The K-user single-stream MIMO IC models multiple transmitter-receiver pairs which operate over the same spectrum simultaneously. Each transmitter and each receiver is equipped with multiple antennas, and a single desired data stream is communicated in each transmitter-receiver link. The individual achievable rates of the K users form a K-dimensional achievable rate region. To find efficient operating points in the achievable rate region, the Pareto boundary computation problem, which can be formulated as a multi-objective optimization problem, needs to be solved. The thesis transforms the multi-objective optimization problem to two single-objective optimization problems–single constraint rate maximization problem and alternating rate profile optimization problem, based on the formulations of the ε-constraint optimization and the weighted Chebyshev optimization, respectively. The thesis proposes two alternating optimization algorithms to solve both single-objective optimization problems. The convergence of both algorithms is guaranteed. Also, a heuristic initialization scheme is provided for each algorithm to achieve a high-quality solution. By varying the weights in each single-objective optimization problem, numerical results show that both algorithms provide an inner bound very close to the Pareto boundary. Furthermore, the thesis also computes some key points exactly on the Pareto boundary in closed-form. A framework for interference alignment (IA) under limited feedback is proposed for a MIMO-IMAC. The MIMO-IMAC well matches the uplink scenario in cellular system, where multiple cells share their spectrum and operate simultaneously. In each cell, a BS receives the desired signals from multiple MSs within its own cell and each BS and each MS is equipped with multi-antenna. By allowing the inter-cell coordination, the thesis develops a distributed IA framework under limited feedback from three aspects: the GIA, the IA-Cell assignment and dynamic feedback bit allocation (DBA), respec- tively. Firstly, the thesis provides a complete study along with some new improvements of the GIA, which enables to compute the exact IA precoders in closed-form, based on local channel state information at the receiver (CSIR). Secondly, the concept of IA-Cell assignment is introduced and its effect on the achievable rate and degrees of freedom (DoF) performance is analyzed. Two distributed matching approaches and one centralized assignment approach are proposed to find a good IA-Cell assignment in three scenrios with different backhaul overhead. Thirdly, under limited feedback, the thesis derives an upper bound of the residual interference to noise ratio (RINR), formulates and solves a corresponding DBA problem. Finally, numerical results show that the proposed GIA with optimized IA-Cell assignment and the DBA greatly outperforms the traditional GIA algorithm.

This thesis consists of two main parts. First of all, the condition number of rectangular Vandermonde matrices with nodes on the complex unit circle is studied. The first time quantitative bounds for the extreme singular values are proven in the multivariate setting and when nodes of the Vandermonde matrix form clusters. In the second part, an optimized presentation of the deterministic stability analysis of the subspace method ESPRIT is given and results from the first part are applied.

Mit dem Begriff Wellenleiterquantenelektrodynamik (WQED) wird gemeinhin die Physik des quantisierten und in eindimensionalen Wellenleitern geführten Lichtes in Wechselwirkung mit einzelnen Emittern bezeichnet. In dieser Arbeit untersuche ich Effekte der WQED für einzelne Dreiniveausysteme (3NS) bzw. Paare von Zweiniveausystemen (2NS), die in den Wellenleiter eingebettet sind. Hierzu bediene ich mich hauptsächlich numerischer Methoden und betrachte die Modellsysteme im Rahmen der Drehwellennäherung. Ich untersuche die Dynamik der Streuung einzelner Photonen an einzelnen, in den Wellenleiter eingebetteten 3NS. Dabei analysiere ich den Einfluss dunkler bzw. nahezu dunkler Zustände der 3NS auf die Streuung und zeige, wie sich mit Hilfe stationärer elektrischer Treibfelder gezielt auf die Streuung einwirken lässt. Ich quantifiziere Verschränkung zwischen dem Lichtfeld im Wellenleiter und den Emittern mit Hilfe der Schmidt-Zerlegung und untersuche den Einfluss der Form der Einhüllenden eines Einzelphotonpulses auf die Ausbeute der Verschränkungserzeugung bei der Streuung des Photons an einem einzelnen Lambda-System im Wellenleiter. Hier zeigt sich, dass die Breite der Einhüllenden im k-Raum und die Emissionszeiten der beiden Übergänge des 3NS die maßgeblichen Parameter darstellen. Abschließend ergründe ich die Emissionsdynamik zweier im Abstand L in den Wellenleiter eingebetteter 2NS. Diese Dynamik wird insbesondere durch kavitätsartige und polaritonische Zustände des Systems aus Wellenleiter und Emitter ausschlaggebend beeinflusst. Bei der kollektiven Emission der 2NS treten - abhängig vom Abstand L - Sub- bzw. Superradianz auf. Dabei nimmt die Intensität dieser Effekte mit längerem Abstand L zu. Diese Eigenart lässt sich auf die Eindimensionalität des Wellenleiters zurückführen. / The field of waveguide quantum electrodynamics (WQED) deals with the physics of quantised light in one-dimensional (1D) waveguides coupled to single emitters. In this thesis, I investigate WQED effects for single three-level systems (3LS) and pairs of two-level systems (2LS), respectively, which are embedded in the waveguide. To this end, I utilise numerical techniques and consider all model systems within the rotating wave approximation. I investigate the dynamics of single-photon scattering by single, embedded 3LS. In doing so, I analyse the influence of dark and almost-dark states of the 3LS on the scattering dynamics. I also show, how stationary electrical driving fields can control the outcome of the scattering. I quantify entanglement between the waveguide''s light field and single emitters by utilising the Schmidt decomposition. I apply this formalism to a lambda-system embedded in a 1D waveguide and study the generation of entanglement by scattering single-photon pulses with different envelopes on the emitter. I show that this entanglement generation is mainly determined by the photon''s width in k-space and the 3LS''s emission times. Finally, I explore the emission dynamics of a pair of 2LS embedded by a distance L into the waveguide. These dynamics are primarily governed by bound states in the continuum and by polaritonic atom-photon bound-states. For collective emission processes of the two 2LS, sub- and superradiance appear and depend strongly on the 2LS''s distance: the effects increase for larger L. This is an exclusive property of the 1D nature of the waveguide.

Nous nous intéressons dans cette thèse, à l'étude de certaines méthodes numériques de type krylov dans le cas symplectique, en utilisant la technique de blocs. Ces méthodes, contrairement aux méthodes classiques, permettent à la matrice réduite de conserver la structure Hamiltonienne ou anti-Hamiltonienne ou encore symplectique d'une matrice donnée. Parmi ces méthodes, nous nous sommes intéressés à la méthodes d'Arnoldi symplectique par bloc que nous appelons aussi bloc J-Arnoldi. Notre but essentiel est d’étudier cette méthode de façon théorique et numérique, sur la nouvelle structure du K-module libre ℝ²nx²s avec K = ℝ²sx²s où s ≪ n désigne la taille des blocs utilisés. Un deuxième objectif est de chercher une approximation de l'epérateur exp(A)V, nous étudions en particulier le cas où A est une matrice réelle Hamiltonnienne et anti-symétrique de taille 2n x 2n et V est une matrice rectangulaire ortho-symplectique de taille 2n x 2s sur le sous-espace de Krylov par blocs Km(A,V) = blockspan {V,AV,...,Am-1V}, en conservant la structure de la matrice V. Cette approximation permet de résoudre plusieurs problèmes issus des équations différentielles dépendants d'un paramètre (EDP) et des systèmes d'équations différentielles ordinaires (EDO). Nous présentons également une méthode de Lanczos symplectique par bloc, que nous nommons bloc J-Lanczos. Cette méthode permet de réduire une matrice structurée sous la forme J-tridiagonale par bloc. Nous proposons des algorithmes basés sur deux types de normalisation : la factorisation S R et la factorisation Rj R. Dans une dernière partie, nous proposons un algorithme qui généralise la méthode de Greville afin de déterminer la pseudo inverse de Moore-Penros bloc de lignes par bloc de lignes d'une matrice rectangulaire de manière itérative. Nous proposons un algorithme qui utilise la technique de bloc. Pour toutes ces méthodes, nous proposons des exemples numériques qui montrent l'efficacité de nos approches. / We study, in this thesis, some numerical block Krylov subspace methods. These methods preserve geometric properties of the reduced matrix (Hamiltonian or skew-Hamiltonian or symplectic). Among these methods, we interest on block symplectic Arnoldi, namely block J-Arnoldi algorithm. Our main goal is to study this method, theoretically and numerically, on using ℝ²nx²s as free module on (ℝ²sx²s, +, x) with s ≪ n the size of block. A second aim is to study the approximation of exp (A)V, where A is a real Hamiltonian and skew-symmetric matrix of size 2n x 2n and V a rectangular matrix of size 2n x 2s on block Krylov subspace Km (A, V) = blockspan {V, AV,...Am-1V}, that preserve the structure of the initial matrix. this approximation is required in many applications. For example, this approximation is important for solving systems of ordinary differential equations (ODEs) or time-dependant partial differential equations (PDEs). We also present a block symplectic structure preserving Lanczos method, namely block J-Lanczos algorithm. Our approach is based on a block J-tridiagonalization procedure of a structured matrix. We propose algorithms based on two normalization methods : the SR factorization and the Rj R factorization. In the last part, we proposea generalized algorithm of Greville method for iteratively computing the Moore-Penrose inverse of a rectangular real matrix. our purpose is to give a block version of Greville's method. All methods are completed by many numerical examples.

Compressed sensing has been proposed as a computationally efficient method to estimate the finite-dimensional signals. The idea is to develop an undersampling operator that can sample the large but finite-dimensional sparse signals with a rate much below the required Nyquist rate. In other words, considering the sparsity level of the signal, the compressed sensing samples the signal with a rate proportional to the amount of information hidden in the signal. In this dissertation, first, we employ compressed sensing for physical layer signal processing of directional millimeter-wave communication. Second, we go through the theoretical aspect of compressed sensing by running a comprehensive theoretical analysis of compressed sensing to address two main unsolved problems, (1) continuous-extension compressed sensing in locally convex space and (2) computing the optimum subspace and its dimension using the idea of equivalent topologies using Köthe sequence. In the first part of this thesis, we employ compressed sensing to address various problems in directional millimeter-wave communication. In particular, we are focusing on stochastic characteristics of the underlying channel to characterize, detect, estimate, and track angular parameters of doubly directional millimeter-wave communication. For this purpose, we employ compressed sensing in combination with other stochastic methods such as Correlation Matrix Distance (CMD), spectral overlap, autoregressive process, and Fuzzy entropy to (1) study the (non) stationary behavior of the channel and (2) estimate and track channel parameters. This class of applications is finite-dimensional signals. Compressed sensing demonstrates great capability in sampling finite-dimensional signals. Nevertheless, it does not show the same performance sampling the semi-infinite and infinite-dimensional signals. The second part of the thesis is more theoretical works on compressed sensing toward application. In chapter 4, we leverage the group Fourier theory and the stochastical nature of the directional communication to introduce families of the linear and quadratic family of displacement operators that track the join-distribution signals by mapping the old coordinates to the predicted new coordinates. We have shown that the continuous linear time-variant millimeter-wave channel can be represented as the product of channel Wigner distribution and doubly directional channel. We notice that the localization operators in the given model are non-associative structures. The structure of the linear and quadratic localization operator considering group and quasi-group are studied thoroughly. In the last two chapters, we propose continuous compressed sensing to address infinite-dimensional signals and apply the developed methods to a variety of applications. In chapter 5, we extend Hilbert-Schmidt integral operator to the Compressed Sensing Hilbert-Schmidt integral operator through the Kolmogorov conditional extension theorem. Two solutions for the Compressed Sensing Hilbert Schmidt integral operator have been proposed, (1) through Mercer's theorem and (2) through Green's theorem. We call the solution space the Compressed Sensing Karhunen-Loéve Expansion (CS-KLE) because of its deep relation to the conventional Karhunen-Loéve Expansion (KLE). The closed relation between CS-KLE and KLE is studied in the Hilbert space, with some additional structures inherited from the Banach space. We examine CS-KLE through a variety of finite-dimensional and infinite-dimensional compressible vector spaces. Chapter 6 proposes a theoretical framework to study the uniform convergence of a compressible vector space by formulating the compressed sensing in locally convex Hausdorff space, also known as Fréchet space. We examine the existence of an optimum subspace comprehensively and propose a method to compute the optimum subspace of both finite-dimensional and infinite-dimensional compressible topological vector spaces. To the author's best knowledge, we are the first group that proposes continuous compressed sensing that does not require any information about the local infinite-dimensional fluctuations of the signal.