No description available.

No / Spectrum sensing is used in cognitive radio systems to detect the availability of spectrum holes for secondary usage. The simplest and most famous spectrum sensing techniques are based either on energy detection or eigenspace analysis from Random Matrix Theory (RMT) such as using the Marchenko-Pastur law. These schemes suffer from uncertainty in estimating the noise variance which reduces their performance. In this paper we propose a new method to evaluate the noise variance that can eliminate the limitations of the aforementioned schemes. This method estimates the noise variance from a measurement set of noisy signals or noise-only signals. Extensive simulations show that the proposed method performs well in estimating the noise variance. Its performance greatly improves with increasing numbers of measurements and also with increasing numbers of samples taken per measurement.

No / Dynamic Spectrum Access (DSA) for secondary usage of underutilized radio spectrum is currently of great interest for radio regulatory authorities and for cellular network operators. However, the co-existence of multiple devices operating in the same bands, such as wireless microphones which also operate in TV bands, poses a challenge to DSA. Efficient and proactive spectrum sensing could prevent harmful interference between collocated devices, but existing blind spectrum sensing schemes such as energy detection and schemes based on Random Matrix Theory (RMT) have performance limitations. We propose a new blind spectrum sensing scheme for cognitive radio. The proposed scheme uses a new formula for the estimation of noise variance. The scheme has been evaluated through extensive simulations on wireless microphone signals and shows higher performance as compared to energy detection and RMT-based sensing schemes such as MME and EME. It also shows higher performance in terms of probability of detection (Pd).

<p dir="ltr">Continuous optimization is a fundamental topic both in theoretical computer science and applications of machine learning. Meanwhile, an important idea in the development modern algorithms it the use of randomness to achieve empirical speedup and improved theoretical runtimes. Stochastic gradient descent (SGD) and matrix-multiplication time linear program solvers [1] are two important examples of such achievements. Matrix sketching and related ideas provide a theoretical framework for the behavior of random matrices and vectors that arise in these algorithms, thereby provide a natural way to better understand the behavior of such randomized algorithms. In this dissertation, we consider three general problems in this area.</p>

Computational fluid dynamics (CFD) has been widely used to simulate turbulent flows. Although an increased availability of computational resources has enabled high-fidelity simulations (e.g. large eddy simulation and direct numerical simulation) of turbulent flows, the Reynolds-Averaged Navier-Stokes (RANS) equations based models are still the dominant tools for industrial applications. However, the predictive capability of RANS models is limited by potential inaccuracies driven by hypotheses in the Reynolds stress closure. With the ever-increasing use of RANS simulations in mission-critical applications, the estimation and reduction of model-form uncertainties in RANS models have attracted attention in the turbulence modeling community. In this work, I focus on estimating uncertainties stemming from the RANS turbulence closure and calibrating discrepancies in the modeled Reynolds stresses to improve the predictive capability of RANS models. Both on-line and off-line data are utilized to achieve this goal. The main contributions of this dissertation can be summarized as follows: First, a physics-based, data-driven Bayesian framework is developed for estimating and reducing model-form uncertainties in RANS simulations. An iterative ensemble Kalman method is employed to assimilate sparse on-line measurement data and empirical prior knowledge for a full-field inversion. The merits of incorporating prior knowledge and physical constraints in calibrating RANS model discrepancies are demonstrated and discussed. Second, a random matrix theoretic framework is proposed for estimating model-form uncertainties in RANS simulations. Maximum entropy principle is employed to identify the probability distribution that satisfies given constraints but without introducing artificial information. Objective prior perturbations of RANS-predicted Reynolds stresses in physical projections are provided based on comparisons between physics-based and random matrix theoretic approaches. Finally, a physics-informed, machine learning framework towards predictive RANS turbulence modeling is proposed. The functional forms of model discrepancies with respect to mean flow features are extracted from the off-line database of closely related flows based on machine learning algorithms. The RANS-modeled Reynolds stresses of prediction flows can be significantly improved by the trained discrepancy function, which is an important step towards the predictive turbulence modeling. / Ph. D.

Cette thèse s'inscrit dans la théorie des matrices aléatoires, à l'intersection avec la théorie des probabilités libres et des algèbres d'opérateurs. Elle s'insère dans une démarche générale qui a fait ses preuves ces dernières décennies : importer les techniques et les concepts de la théorie des probabilités non commutatives pour l'étude du spectre de grandes matrices aléatoires. On s'intéresse ici à des généralisations du théorème de liberté asymptotique de Voiculescu. Dans les Chapitres 1 et 2, nous montrons des résultats de liberté asymptotique forte pour des matrices gaussiennes, unitaires aléatoires et déterministes. Dans les Chapitres 3 et 4, nous introduisons la notion de fausse liberté asymptotique pour des matrices déterministes et certaines matrices hermitiennes à entrées sous diagonales indépendantes, interpolant les modèles de matrices de Wigner et de Lévy. / The thesis fits into the random matrix theory, in intersection with free probability and operator algebra. It is part of a general approach which is common since the last decades: using tools and concepts of non commutative probability in order to get general results about the spectrum of large random matrices. Where are interested here in generalization of Voiculescu's asymptotic freeness theorem. In Chapter 1 and 2, we show some results of strong asymptotic freeness for gaussian, random unitary and deterministic matrices. In Chapter 3 and 4, we introduce the notion of asymptotic false freeness for deterministic matrices and certain random matrices, Hermitian with independent sub-diagonal entries, interpolating Wigner and Lévy models.

Dans cette thèse, nous considérons le problème de la localisation de source dans les grands réseaux de capteurs, quand le nombre d'antennes du réseau et le nombre d'échantillons du signal observé sont grands et du même ordre de grandeur. Nous considérons le cas où les signaux source émis sont déterministes, et nous développons un algorithme de localisation amélioré, basé sur la méthode MUSIC. Pour ce faire, nous montrons de nouveaux résultats concernant la localisation des valeurs propres des grandes matrices aléatoires gaussiennes complexes de type information plus bruit / In this thesis, we consider the problem of source localization in large sensor networks, when the number of antennas of the network and the number of samples of the observed signal are large and of the same order of magnitude. We also consider the case where the source signals are deterministic, and we develop an improved algorithm for source localization, based on the MUSIC method. For this, we fist show new results concerning the position of the eigen values of large information plus noise complex gaussian random matrices

Três ensembles, respectivamente relacionados com as distribuições Gaussiana, Lognormal e de Levy, são abordados neste trabalho primordialmente do ponto de vista da termodinâmica de seus espectros. Novas expressões para as grandezas termodinâmicas sao encontradas para os ensembles de Stieltjes e de Bertuola-Pato, e a conexão destes com os ensembles Gaussianos e estabelecida. Esta tese também se compromete com a continuação do desenvolvimento e aprimorarão do ensemble generalizado de Bertuola-Pato, estendendo alguns resultados para os ensembles simplifico e unitário generalizados, alem do ortogonal generalizado já introduzido anteriormente por A. C. Bertuola e M. P. Pato. / Three ensembles, related to the Gaussian, the Lognormal and the L´evy distributions respectively, have been studied in this work and were investigated most of all in what concerns their spectral thermodynamics. New expressions for the thermodynamics quantities were found for the Stieltjes and the Bertuola-Pato ensembles, and the connection with the gaussian ensembles is established. This work concerned with the development continuity and with the improvement of Bertuola-Pato generalized ensemble, extending some of the results to the simplectic and unitary generalized ensembles, besides the orthogonal generalized ensemble introduced before by A. C. Bertuola and M. P. Pato.

Simetrias compõe parte integral da análise na Teoria das Matrizes Aleatórias (RMT). As simetrias de inversão temporal e rotacional são aspectos-chave do Ensemble Gaussiano Ortogonal (GOE), enquanto esta última é quebrada no Ensemble Gaussiano Simplético (GSE) e ambas são quebradas no Conjunto Unitário Gaussiano (GUE). Desde o final da década de 1990, o crescente interesse no campo dos sistemas quânticos PT-simétricos levou os pesquisadores a considerar o efeito, em matrizes aleatórias, dessa classe de simetrias, bem como simetrias pseudo-hermitianas. A principal questão a ser respondida pela pesquisa apresentada nesta tese é se a simetria PT ou, de forma mais geral, a pseudo-Hermiticidade implica alguma distribuição de probabilidade específica para os autovalores. Ou, em outras palavras, se há um aspecto comum transmitido por tal simetria que pode ser usada para modelar alguma classe particular de sistemas físicos. A abordagem inicial considerada consistiu na introdução de um conjunto pseudo-hermitiano, isospectral ao conjunto -Hermite, que apresentaria o tipo de quebra de realidade típico dos sistemas PT-simétricos. Nesse modelo, a primeira abordagem adotada foi a introdução de perturbações que quebraram a realidade dos espectros. Os resultados obtidos permitem concluir que a transformação em seu similar pseudo-hermitiano conduz a um sistema assintoticamente instável. Esse modelo foi extendido ao considerar um pseudo-hermitiano não positivo, que leva a uma quebra similar na realidade dos espectros. Este caso apresenta um comportamento mais próximo do típico dos sistemas PT-simétricos presentes na literatura. Um modelo denso geral baseado em projetores foi proposto, e duas realizações particulares deste modelo receberam atenção mais detalhada. O comportamento espectral também foi similar àquele típico da simetria PT para as duas realizações consideradas, e seus limites assintóticos foram conectados a conjuntos clássicos de teoria de matriz aleatória. Além disso, as propriedades de seus polinômios característicos médios foram obtidas e os limites assintóticos desses polinômios também foram considerados e relacionados a polinômios clássicos. O comportamento estatístico deste conjunto foi estudado e comparado com o destes polinômios. Impor a pseudo-Hermiticidade não parece implicar qualquer distribuição particular de autovalores, sendo a característica comum a quebra da realidade dos autovalores comumente encontrados na literatura de simetria PT. O resultado mais notável dos estudos apresentados nesta tese é o fato de que uma interação pseudo-hermitiana pode ser construída de tal forma que o comportamento espectral médio possa ser controlado calibrando-se o mecanismo de interação, bem como sua intensidade. / The role of symmetries is an integral part of the analysis in Random Matrix Theory (RMT). Time reversal and rotational symmetries are key aspects of the Gaussian Orthogonal Ensemble (GOE), whereas the latter is broken in the Gaussian Sympletic Ensemble (GSE) and both are broken in the Gaussian Unitary Ensemble (GUE). Since the late 1990s, growing interest in the field of PT symmetric quantum systems has led researchers to consider the effect, in random matrices, of this class of symmetries, as well as that of pseudo-Hermitian symmetries. The primary question to be answered by the research presented in this thesis is whether PT-symmetry or, more generally, pseudo-Hermiticity implies some specific probability distribution for the eigenvalues. Or, in other words, whether there is a common aspect imparted by such a symmetry which may be used to model some particular class of physical systems. The initial approach considered consisted of introducing an pseudo-Hermitian ensemble, isospectral to the -Hermite ensemble, which would present the type of reality-breaking typical of PT-symmetrical systems. In this model, the first approach taken was to introduce perturbation which broke the reality of the spectra. The results obtained allow the conclusion that the transformation into its pseudo-Hermitian similar leads into a system which is asymptotically unstable. An extension of this model was to consider a non-positive pseudo-Hermitian , which lead to similar breaking in the reality of the spectra. This case displays behavior closer to that typical of the PT-symmetric systems present in the literature. A general dense projector model was proposed, and two particular realizations of this model were given more detailed attention. The spectral behavior was also similar to that typical of PT-symmetry for the two realizations considered, and their asymptotic limits were shown to connect to classical ensembles of random matrix theory. Furthermore, the properties of their average characteristic polynomials were obtained and the asymptotic limits of these polynomials were also considered and were related to classical polynomials. The statistical behavior of this ensemble was studied and compared to that of these polynomials. Imposing the pseudo-Hermitian does seem not imply any particular eigenvalue distribution, the common feature being the breaking of the reality of the eigenvalues commonly found in PT-symmetry literature. The most notable result of the studies presented herein is the fact that a pseudo-Hermitian interaction may be constructed such that the average spectral behavior may be controlled by calibrating the mechanism of interaction as well as its intensity.

Le sujet principal de cette thèse est la physique des verres de spin. Les verres de spin ont été introduits au début des années 70 pour décrire alliages magnétiques diluées. Ils ont désormais été considerés pour comprendre le comportement de liquides sousrefroidis. Parmis les systèmes qui peuvent être décrits par le langage des systèmes desordonnés, on trouve les problèmes d'optimisation combinatoire. Dans la première partie de cette thèse, nous considérons les modèles de verre de spin avec intéraction de Kac pour investiguer la phase de basse température des liquides sous-refroidis. Dans les chapitres qui suivent, nous montrons comment certaines caractéristiques des modèles de verre de spin peuvent être obtenues à partir de résultats de la théorie des matrices aléatoires en connection avec la statistique des valeurs extrêmes. Dans la dernière partie de la thèse, nous considérons la connexion entre la théorie desverres de spin et la science computationnelle, et présentons un nouvel algorithme qui peut être appliqué à certains problèmes dans le domaine des finances.