31 research outputs found

    Algorithms for curve design and accurate computations with totally positive matrices

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    Esta tesis doctoral se enmarca dentro de la teoría de la Positividad Total. Las matrices totalmente positivas han aparecido en aplicaciones de campos tan diversos como la Teoría de la Aproximación, la Biología, la Economía, la Combinatoria, la Estadística, las Ecuaciones Diferenciales, la Mecánica, el Diseño Geométrico Asistido por Ordenador o el Álgebra Numérica Lineal. En esta tesis nos centraremos en dos de los campos que están relacionados con matrices totalmente positivas.This doctoral thesis is framed within the theory of Total Positivity. Totally positive matrices have appeared in applications from fields as diverse as Approximation Theory, Biology, Economics, Combinatorics, Statistics, Differential Equations, Mechanics, Computer Aided Geometric Design or Linear Numerical Algebra. In this thesis, we will focus on two of the fields that are related to totally positive matrices.<br /

    Lattice paths and branched continued fractions II. Multivariate Lah polynomials and Lah symmetric functions

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    We introduce the generic Lah polynomials Ln,k(ϕ), which enumerate unordered forests of increasing ordered trees with a weight ϕi for each vertex with i children. We show that, if the weight sequence ϕ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials Ln(ϕ,y) is coefficientwise Hankel-totally positive. Upon specialization we obtain results for the Lah symmetric functions and multivariate Lah polynomials of positive and negative type. The multivariate Lah polynomials of positive type are also given by a branched continued fraction. Our proofs use mainly the method of production matrices; the production matrix is obtained by a bijection from ordered forests of increasing ordered trees to labeled partial Łukasiewicz paths. We also give a second proof of the continued fraction using the Euler–Gauss recurrence method

    Local preconditioning for parallel iterative solvers

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    This thesis aims at improving the convergence of iterative solvers, used for algebraic systems coming from the discretization of partial differential equations (PDE), in the context of large scale simulations and high performance computing (HPC). The methodology followed consists in adapting some existing preconditiong techniques to the physics and numerics of convection-dominated transport and boundary layer problems in flows. For convection-dominated flows, a physics-based permutation algorithm is presented, which consists in renumbering the mesh in the direction of convection. This renumbering is then used together with a Gauss-Seidel preconditioner to propagate the result of the matrix-vector products along the convection. The robutsness and effectiveness of this preconditioner is proved in several test cases solving the heat equation as well as the Navier-Stokes equations in both sequential and in parallel using the Message Passing Interface library MPI. Additionally, the composition of preconditioners is proposed to solve cases where different local physical behaviors co-exist in the same flow. In particular, we focus on such problems where of a highly convective flow encounters an obstacle. Such problems involve a zone with high convection far from the obstacle and the development of a boundary layer in the vicinity of the obstacle. In numerical terms, these local behaviors translate into specific matrix structures that we will take advantage of to adapt the preconditioner locally. On the one hand, the linelet preconditioner is a well-known efficient preconditioner for boundary layers where the mesh is highly anisotropic, in particular to solve the Poisson equation. On the other hand, the streamline linelet that we propose in this thesis (Gauss-Seidel together with a mesh renumbering in the convection direction) is well adapted for locally hyperbolic flows. Both preconditioners will be composed (combined) in different ways to investigate their robustness in terms of convergence as well as their costs to solve the proposed transport problems. We will study as well their performances in terms of parallelization.Esta tesis tiene como objetivo mejorar la convergencia de los métodos iterativos utilizados para resolver sistemas de ecuaciones algebraicas provenientes de la discretización de ecuaciones diferenciales en derivadas parciales (EDP), en el contexto de las simulaciones a gran escala y computación de altas prestaciones (HPC). La metodología seguida consiste en adaptar algunas técnicas de precondicionamiento existentes, a la física y la numérica en flujos que presentan una alta convección y flujos que presentan una capa límite. Para los flujos dominados por convección, se presenta un algoritmo de permutación basado en la física, que consiste en la renumeración de la malla en la dirección de la convección. Esta renumeración se usa luego junto con el precondicionador Gauss-Seidel para propagar el resultado de los productos matriz-vector a lo largo de la convección. La robustez y eficiencia de este precondicionador se demuestra en varios ejemplos en los que se resuelve la ecuación de calor y las ecuaciones de Navier-Stokes tanto en secuencial como en paralelo utilizando la librería interfaz de paso de mensajes (MPI). Además, se propone la composición de precondicionadores para resolver casos donde diferentes comportamientos físicos locales coexisten en el mismo flujo. En particular, nos enfocamos en los casos donde un flujo altamente convectivo se encuentra un obstáculo. En este tipo de problemas nos encontramos dos zonas: una con alta convección lejos del obstáculo y otra donde se desarrolla una capa límite en los alrededores del obstáculo. En términos numéricos, estos comportamientos locales se traducen en estructuras matriciales específicas que aprovecharemos para adaptar localmente el precondicionador. Por un lado, sabemos que el linelet es un precondicionador eficiente para resolver problemas de capa límite donde la malla es altamente anisótropa. En particular resulta eficiente para resolver la ecuación de Poisson. Por otro lado, sabemos que el linelet aerodinámico, el precondicionador que proponemos en esta tesis (precondicionador Gauss-Seidel junto con una renumeración de malla en la dirección de la convección) está bien adaptado para flujos localmente hiperbólicos. Con todo esto, proponemos también una composición de los dos precondicionadores (combinación de ambos) de distintas formas para investigar su robustez en términos de convergencia, así como sus costes para resolver los problemas de transporte propuestos. Estudiaremos también el rendimiento en cuanto a la paralelización se refiere.Matemàtica aplicad

    Local preconditioning for parallel iterative solvers

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    This thesis aims at improving the convergence of iterative solvers, used for algebraic systems coming from the discretization of partial differential equations (PDE), in the context of large scale simulations and high performance computing (HPC). The methodology followed consists in adapting some existing preconditiong techniques to the physics and numerics of convection-dominated transport and boundary layer problems in flows. For convection-dominated flows, a physics-based permutation algorithm is presented, which consists in renumbering the mesh in the direction of convection. This renumbering is then used together with a Gauss-Seidel preconditioner to propagate the result of the matrix-vector products along the convection. The robutsness and effectiveness of this preconditioner is proved in several test cases solving the heat equation as well as the Navier-Stokes equations in both sequential and in parallel using the Message Passing Interface library MPI. Additionally, the composition of preconditioners is proposed to solve cases where different local physical behaviors co-exist in the same flow. In particular, we focus on such problems where of a highly convective flow encounters an obstacle. Such problems involve a zone with high convection far from the obstacle and the development of a boundary layer in the vicinity of the obstacle. In numerical terms, these local behaviors translate into specific matrix structures that we will take advantage of to adapt the preconditioner locally. On the one hand, the linelet preconditioner is a well-known efficient preconditioner for boundary layers where the mesh is highly anisotropic, in particular to solve the Poisson equation. On the other hand, the streamline linelet that we propose in this thesis (Gauss-Seidel together with a mesh renumbering in the convection direction) is well adapted for locally hyperbolic flows. Both preconditioners will be composed (combined) in different ways to investigate their robustness in terms of convergence as well as their costs to solve the proposed transport problems. We will study as well their performances in terms of parallelization.Esta tesis tiene como objetivo mejorar la convergencia de los métodos iterativos utilizados para resolver sistemas de ecuaciones algebraicas provenientes de la discretización de ecuaciones diferenciales en derivadas parciales (EDP), en el contexto de las simulaciones a gran escala y computación de altas prestaciones (HPC). La metodología seguida consiste en adaptar algunas técnicas de precondicionamiento existentes, a la física y la numérica en flujos que presentan una alta convección y flujos que presentan una capa límite. Para los flujos dominados por convección, se presenta un algoritmo de permutación basado en la física, que consiste en la renumeración de la malla en la dirección de la convección. Esta renumeración se usa luego junto con el precondicionador Gauss-Seidel para propagar el resultado de los productos matriz-vector a lo largo de la convección. La robustez y eficiencia de este precondicionador se demuestra en varios ejemplos en los que se resuelve la ecuación de calor y las ecuaciones de Navier-Stokes tanto en secuencial como en paralelo utilizando la librería interfaz de paso de mensajes (MPI). Además, se propone la composición de precondicionadores para resolver casos donde diferentes comportamientos físicos locales coexisten en el mismo flujo. En particular, nos enfocamos en los casos donde un flujo altamente convectivo se encuentra un obstáculo. En este tipo de problemas nos encontramos dos zonas: una con alta convección lejos del obstáculo y otra donde se desarrolla una capa límite en los alrededores del obstáculo. En términos numéricos, estos comportamientos locales se traducen en estructuras matriciales específicas que aprovecharemos para adaptar localmente el precondicionador. Por un lado, sabemos que el linelet es un precondicionador eficiente para resolver problemas de capa límite donde la malla es altamente anisótropa. En particular resulta eficiente para resolver la ecuación de Poisson. Por otro lado, sabemos que el linelet aerodinámico, el precondicionador que proponemos en esta tesis (precondicionador Gauss-Seidel junto con una renumeración de malla en la dirección de la convección) está bien adaptado para flujos localmente hiperbólicos. Con todo esto, proponemos también una composición de los dos precondicionadores (combinación de ambos) de distintas formas para investigar su robustez en términos de convergencia, así como sus costes para resolver los problemas de transporte propuestos. Estudiaremos también el rendimiento en cuanto a la paralelización se refiere.Postprint (published version

    Inverse M-matrices, II

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    AbstractThis is an update of the 1981 survey by the first author. In the meantime, a considerable amount has been learned about the very special structure of the important class of inverse M-matrices. Developments since the earlier survey are emphasized, but we have tried to be somewhat complete; and, some results have not previously been published. Some proofs are given where appropriate and references are given for others. After some elementary preliminaries, results are grouped by certain natural categories

    Inverse M-matrices, II

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    AbstractThis is an update of the 1981 survey by the first author. In the meantime, a considerable amount has been learned about the very special structure of the important class of inverse M-matrices. Developments since the earlier survey are emphasized, but we have tried to be somewhat complete; and, some results have not previously been published. Some proofs are given where appropriate and references are given for others. After some elementary preliminaries, results are grouped by certain natural categories

    Computation of restricted maximum likelihood estimates of variance components

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    The method preferred by animal breeders for the estimation of variance components is restricted maximum likelihood (REML). Various iterative algorithms have been proposed for computing REML estimates. Five different computational strategies for implementing such an algorithm were compared in terms of flops (floating-point operations). These strategies were based respectively on the LDL\u27 decomposition, the W transformation, the SWEEP method, tridiagonalization and diagonalization of the coefficient matrix of the mixed-model equations;The computational requirements of the orthogonal transformations employed in tridiagonalization and diagonalization were found to be rather extensive. However, these transformations are performed prior to the initiation of the iterative estimation process and need not be repeated during the remainder of the process. Subsequent to either diagonalization or tridiagonalization, the flops required per iteration are very minimal. Thus, for most applications of mixed-effects linear models with a single set of random effects, the use of an orthogonal transformation prior to the initiation of the iterative process is recommended. For most animal breeding applications, tridiagonalization will generally be more efficient than diagonalization;In most animal breeding applications, the coefficient matrix of the mixed-model equations is extremely sparse and of very large order. The use of sparse-matrix techniques for the numerical evaluation of the log-likelihood function and its first- and second-order partial derivatives was investigated in the case of the simple sire and animal models. Instead of applying these techniques directly to the coefficient matrix of the mixed-model equations to obtain the Cholesky factor, they were used to obtain the Cholesky factor indirectly by carrying out a QR decomposition of an augmented model matrix;The feasibility of the computational method for the simple sire model was investigated by carrying out the most computationally intensive part of this method (which is the part consisting of the QR decomposition) for an animal breeding data set comprising 180,994 records and 1,264 sires. The total CPU time required for this part (using an NAS AS/9160 computer) was approximately 75,000 seconds

    Edge manipulation techniques for complex networks with applications to communicability and triadic closure.

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    Complex networks are ubiquitous in our everyday life and can be used to model a wide variety of phenomena. For this reason, they have captured the interest of researchers from a wide variety of fields. In this work, we describe how to tackle two problems that have their focus on the edges of networks. Our first goal is to develop mathematically inferred, efficient methods based on some newly introduced edge centrality measures for the manipulation of links in a network. We want to make a small number of changes to the edges in order to tune its overall ability to exchange information according to certain goals. Specifically, we consider the problem of adding a few links in order to increase as much as possible this ability and that of selecting a given number of connections to be removed from the graph in order to penalize it as little as possible. Techniques to tackle these problems are developed for both undirected and directed networks. Concerning the directed case, we further discuss how to approximate certain quantities that are used to measure the importance of edges. Secondly, we consider the problem of understanding the mechanism underlying triadic closure in networks and we describe how communicability distance functions play a role in this process. Extensive numerical tests are presented to validate our approaches
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