Two-Center Black Holes Duality-Invariants for stu Model and its lower-rank Descendants

We classify 2-center extremal black hole charge configurations through duality-invariant homogeneous polynomials, which are the generalization of the unique invariant quartic polynomial for single-center black holes based on homogeneous symmetric cubic special Kaehler geometries. A crucial role is played by an horizontal SL(p,R) symmetry group, which classifies invariants for p-center black holes. For p = 2, a (spin 2) quintet of quartic invariants emerge. We provide the minimal set of independent invariants for the rank-3 N = 2, d = 4 stu model, and for its lower-rank descendants, namely the rank-2 st^2 and rank-1 t^3 models; these models respectively exhibit seven, six and five independent invariants. We also derive the polynomial relations among these and other duality invariants. In particular, the symplectic product of two charge vectors is not independent from the quartic quintet in the t^3 model, but rather it satisfies a degree-16 relation, corresponding to a quartic equation for the square of the symplectic product itself.Comment: 1+31 pages; v2: amendments in Sec. 9, App. C added, other minor refinements, Refs. added; v3: Ref. added, typos fixed. To appear on J.Math.Phy

Reduced Order and Surrogate Models for Gravitational Waves

We present an introduction to some of the state of the art in reduced order and surrogate modeling in gravitational wave (GW) science. Approaches that we cover include Principal Component Analysis, Proper Orthogonal Decomposition, the Reduced Basis approach, the Empirical Interpolation Method, Reduced Order Quadratures, and Compressed Likelihood evaluations. We divide the review into three parts: representation/compression of known data, predictive models, and data analysis. The targeted audience is that one of practitioners in GW science, a field in which building predictive models and data analysis tools that are both accurate and fast to evaluate, especially when dealing with large amounts of data and intensive computations, are necessary yet can be challenging. As such, practical presentations and, sometimes, heuristic approaches are here preferred over rigor when the latter is not available. This review aims to be self-contained, within reasonable page limits, with little previous knowledge (at the undergraduate level) requirements in mathematics, scientific computing, and other disciplines. Emphasis is placed on optimality, as well as the curse of dimensionality and approaches that might have the promise of beating it. We also review most of the state of the art of GW surrogates. Some numerical algorithms, conditioning details, scalability, parallelization and other practical points are discussed. The approaches presented are to large extent non-intrusive and data-driven and can therefore be applicable to other disciplines. We close with open challenges in high dimension surrogates, which are not unique to GW science.Comment: Invited article for Living Reviews in Relativity. 93 page

Analysis of the sum rate for massive MIMO using 10 GHz measurements

Orientador: Gustavo FraidenraichTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Este trabalho apresenta um conjunto de contribuições para caracterização e modelagem de canais reais de rádio abordando aspectos relacionados com as condições favoráveis de propagação para sistemas massive MIMO. Discutiremos como caracterizar canais de rádio em um ambiente real, processamento de dados e análise das condições favoráveis de propagação. Em uma segunda parte, focamos na determinação teórica de alguns aspectos da tecnologia de massive MIMO utilizando propriedades de distribuições matriciais Wishart. Inicialmente, apresentamos uma contribuição sobre a aplicação do algoritmo ESPRIT, para estimar parâmetros de um conjunto de dados multidimensional. Obtivemos dados por varredura em frequência de um Analisador Vetorial de Rede e os adaptamos para o algoritmo ESPRIT. Mostramos como remover a influência do ganho de padrão de antenas e como utilizar um gerador de modelo de canal baseado nas medidas reais de canal de rádio. As medidas foram feitas na frequência de 10.1 GHz com largura de faixa de 500 MHz. Utilizando um gerador de modelo de canal, fomos além do universo das simulações por distribuições Gaussianas. Introduzimos o conceito de propagação favorável e analisamos condições de linha-de-visada usando arranjos lineares uniformes e arranjos retangulares uniformes de antena. Como novidade da pesquisa, mostramos os benefícios de explorar um número extra de graus de liberdade devido à escolha dos formatos de arranjo de antenas e ao aumento do número de elementos. Esta propriedade é observada ao analisarmos a distribuição dos autovalores de matrizes Gramianas. Em seguida, estendemos o mesmo raciocínio para as matrizes de canal geradas a partir de informações reais e verificamos se as propriedades ainda permaneceriam válidas. Na segunda parte deste trabalho, incluímos mais de uma antena no terminal móvel e calculamos a probabilidade de indisponibilidade para várias configurações de antenas e número arbitrário de usuários. Esboçamos inicialmente a formulação para a informação mútua e, em seguida, calculamos os resultados exatos em uma situação com dois usuários e duas antenas, tanto na estação base (EB) como nos terminais de usuário(TU). Visto que as formulações para a derivação exata dos casos com mais antenas e mais usuários mostrou-se muito intrincada, propusemos uma aproximação Gaussiana para simplificar o problema. Esta aproximação foi validada por simulações Monte Carlo para diferentes relações sinal/ruídoAbstract: This thesis presents a set of contributions for channel modeling and characterization of real radio channels delineating aspects related with the favorable propagation for massive MIMO systems. We will discuss about how to proceed for characterizing radio channels in an real environment , data processing, and analysis of favorable conditions. In a second part, we focused on determination of some theoretical aspects of the Massive MIMO technology using properties of Wishart distribution matrices. We initially present a contribution on the application of ESPRIT algorithm for estimating a multidimensional set of measured data. We have obtained data by frequency sweep carried out by a vector network analyzer(VNA) and adapted it to fit in the ESPRIT algorithm. We show how to remove antenna pattern gain using virtual antenna arrays and how to use a channel model generator based on radio channel measurements of real environments. The measurements were conducted at the frequency of 10.1 GHz and 500 MHz bandwidth. By using a channel model generator, we have explored beyond the simulation of Gaussian Distributions. We will introduce the concept of favorable propagation and analyze the line-of-sight conditions using ULA and URA array shapes. As a research novelty, we will show the benefits of exploiting an extra degree of freedom due to the choice of the antenna shapes and amount of antenna elements. We observe these properties through the distribution of the Gramian Matrices. Next, we extend the same rationale to channel matrices generated from real channels and we verify that the properties are still valid. In a second part of the research work, we included more than one antenna in the mobile terminals and calculated the outage probability for several antenna configurations and arbitrary number users. We introduce a formulation for mutual information and then we calculate exact results in a case with two users with two antennas in both Base Station (BS) and User Terminals (UT). Since the formulations to the exact derivation for cases with more antennas and users seems to be intricate, we propose a Gaussian approximation solution to simplify the problem. We validated this approximation with Monte Carlo simulations for different signal-to-noise ratiosDoutoradoTelecomunicações e TelemáticaDoutor em Engenharia Elétrica248416/2013-8CNPQCAPE

Optimisation of condition number for eigenstructure problems

This thesis deals with a new solution for the problem of eigenstructure assignment in control systems design. A wide range of challenging issues is examined involving the problem of eigenstructure assignment and associated system properties through different forms of complexity which are strongly related to control system design. In this thesis, specific attention has been given to the issue of skewness of the closedloop eigenframe of the state matrix. In fact, the aim is to develop a new methodology for determining the best angle between closed-loop eigenvectors by optimising the minimal condition number of the closed-loop eigenvector matrix. This problem is strongly linked to sensitivity of eigenvalues to parameter uncertainty, perturbations to model parameter uncertainty. The importance of this methodology can be expressed in terms of results related to the Sensitivity of eigenvalues, Relative measures of controllability and observability and also deviations from strong stability to overshooting behaviour. Among this variety of eigenstructure assignment methods, special consideration has been paid to Geometric Theory, which introduces an alternative solution to the assignability of spectrum of controllability subspaces (cs) based on an eigenvector approach and then develops a new pole assignment algorithm based on openloop/ closed-loop spectra as a practical application of this approach. In order to tackle the problem of measuring the skewness of angles between closed-loop eigenvalues, some measures for Eigenframe skewness have been defined in general and so the necessary and efficient conditions have been derived for the angle between some subspaces in a direct sum decomposition to be maximized. This has been done via three metrics; Condition Number, Determinant of Gram Matrix and Singular Values. The thesis presents the parametrisation of closed-loop eigenframes result by the method generated in [4]. Within this thesis, a non-smooth algorithm has been developed in order to select the most orthogonal closed-loop eigenframe and so the influence of selected closed-loop spectra. Also, the parametrisation of controllability subspaces in a standard direct sum decomposition using matrix fraction description (MFD) has been derived. Within this thesis the construction and the existence of controllability subspaces connected to (A,B)-invariant subspaces, has also been studied. In addition, an algebraic description of the total system behaviour which leads to an algebraic characterisation of the total input, state and output behaviour in an implicit formulation is given based on properties of MFD descriptions, a topic which remains open for future studies

Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective

The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of problems, e.g., nonnegativity or sparsity-constrained factorization, we take a {\it top-down} approach: we start with general optimization theory (e.g., inexact and accelerated block coordinate descent, stochastic optimization, and Gauss-Newton methods) that covers a wide range of factorization problems with diverse constraints and regularization terms of engineering interest. Then, we go `under the hood' to showcase specific algorithm design under these introduced principles. We pay a particular attention to recent algorithmic developments in structured tensor and matrix factorization (e.g., random sketching and adaptive step size based stochastic optimization and structure-exploiting second-order algorithms), which are the state of the art---yet much less touched upon in the literature compared to {\it block coordinate descent} (BCD)-based methods. We expect that the article to have an educational values in the field of structured factorization and hope to stimulate more research in this important and exciting direction.Comment: Final Version; to appear in IEEE Signal Processing Magazine; title revised to comply with the journal's rul

Interpolation with the polynomial kernels

The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their lack of strict positive definiteness. In particular they do not enjoy the usual property of unisolvency for arbitrary point sets, which is one of the key properties used to build kernel-based interpolation methods. This paper is devoted to establish some initial results for the study of these kernels, and their related interpolation algorithms, in the context of approximation theory. We will first prove necessary and sufficient conditions on point sets which guarantee the existence and uniqueness of an interpolant. We will then study the Reproducing Kernel Hilbert Spaces (or native spaces) of these kernels and their norms, and provide inclusion relations between spaces corresponding to different kernel parameters. With these spaces at hand, it will be further possible to derive generic error estimates which apply to sufficiently smooth functions, thus escaping the native space. Finally, we will show how to employ an efficient stable algorithm to these kernels to obtain accurate interpolants, and we will test them in some numerical experiment. After this analysis several computational and theoretical aspects remain open, and we will outline possible further research directions in a concluding section. This work builds some bridges between kernel and polynomial interpolation, two topics to which the authors, to different extents, have been introduced under the supervision or through the work of Stefano De Marchi. For this reason, they wish to dedicate this work to him in the occasion of his 60th birthday

Effizientes Lösen von großskaligen Riccati-Gleichungen und ein ODE-Framework für lineare Matrixgleichungen

This work considers the iterative solution of large-scale matrix equations. Due to the size of the system matrices in large-scale Riccati equations the solution can not be calculated directly but is approximated by a low rank matrix ZYZ^*. Herein Z is a basis of a low-dimensional rational Krylov subspace. The inner matrix Y is a small square matrix. Two ways to choose this inner matrix are examined: By imposing a rank condition on the Riccati residual and by projecting the Riccati residual onto the Krylov subspace generated by Z. The rank condition is motivated by the well-known ADI iteration. The ADI solutions span a rational Krylov subspace and yield a rank-p residual. It is proven that the rank-p condition guarantees existence and uniqueness of such an approximate solution. Known projection methods are generalized to oblique projections and a new formulation of the Riccati residual is derived, which allows for an efficient evaluation of the residual norm. Further a truncated approximate solution is characterized as the solution of a Riccati equation, which is projected to a subspace of the Krylov subspace generated by Z. For the approximate solution of Lyapunov equations a system of ordinary differential equations (ODEs) is solved via Runge-Kutta methods. It is shown that the space spanned by the approximate solution is a rational Krylov subspace with poles determined by the time step sizes and the eigenvalues of the matrices of the Butcher tableau of the used Runge-Kutta method. The method is applied to a model order reduction problem. The analytical solution of the system of ODEs satisfies an algebraic invariant. Those Runge-Kutta methods which preserve this algebraic invariant are characterized by a simple condition on the corresponding Butcher tableau. It is proven that these methods are equivalent to the ADI iteration. The invariance approach is transferred to Sylvester equations.Diese Arbeit befasst sich mit der numerischen Lösung hochdimensionaler Matrixgleichungen mittels iterativer Verfahren. Aufgrund der Größe der Systemmatrizen in großskaligen algebraischen Riccati-Gleichung kann die Lösung nicht direkt bestimmt werden, sondern wird durch eine approximative Lösung ZYZ^* von geringem Rang angenähert. Hierbei wird Z als Basis eines rationalen Krylovraums gewählt und enthält nur wenige Spalten. Die innere Matrix Y ist klein und quadratisch. Es werden zwei Wege untersucht, die Matrix Y zu wählen: Durch eine Rang-Bedingung an das Riccati-Residuum und durch Projektion des Riccati-Residuums auf den von Z erzeugten Krylovraum. Die Rang-Bedingung wird durch die wohlbekannten ADI-Verfahren motiviert. Die approximativen ADI-Lösungen spannen einen Krylovraum auf und führen zu einem Riccati-Residuum vom Rang p. Es wird bewiesen, dass die Rang-p-Bedingung Existenz und Eindeutigkeit einer solchen approximativen Lösung impliziert. Aus diesem Ergebnis werden effiziente iterative Verfahren abgeleitet, die eine solche approximative Lösung erzeugen. Bisher bekannte Projektionsverfahren werden auf schiefe Projektionen erweitert und es wird eine neue Formulierung des Riccati-Residuums hergeleitet, die eine effiziente Berechnung der Norm erlaubt. Weiter wird eine abgeschnittene approximative Lösung als Lösung einer Riccati-Gleichung charakterisiert, die auf einen Unterraum des von Z erzeugten Krylovraums projiziert wird. Um die Lösung der Lyapunov-Gleichung zu approximieren wird ein System gewöhnlicher Differentialgleichungen mittels Runge-Kutta-Verfahren numerisch gelöst. Es wird gezeigt, dass der von der approximativen Lösung aufgespannte Raum ein rationaler Krylovraum ist, dessen Pole von den Zeitschrittweiten der Integration und den Eigenwerten der Koeffizientenmatrix aus dem Butcher-Tableau des verwendeten Runge-Kutta-Verfahrens abhängen. Das Verfahren wird auf ein Problem der Modellreduktion angewendet. Die analytische Lösung des Differentialgleichungssystems erfüllt eine algebraische Invariante. Diejenigen Runge-Kutta-Verfahren, die diese Invariante erhalten, werden durch eine Bedingung an die zugehörigen Butcher-Tableaus charakterisiert. Es wird gezeigt, dass diese speziellen Verfahren äquivalent zur ADI-Iteration sind. Der Invarianten-Ansatz wird auf Sylvester-Gleichungen übertragen