1,010 research outputs found
Dense and sparse parallel linear algebra algorithms on graphics processing units
Una lÃnea de desarrollo seguida en el campo de la supercomputación es el uso de procesadores de propósito especÃfico para acelerar determinados tipos de cálculo. En esta tesis estudiamos el uso de tarjetas gráficas como aceleradores de la computación y lo aplicamos al ámbito del álgebra lineal. En particular trabajamos con la biblioteca SLEPc para resolver problemas de cálculo de autovalores en matrices de gran dimensión, y para aplicar funciones de matrices en los cálculos de aplicaciones cientÃficas. SLEPc es una biblioteca paralela que se basa en el estándar MPI y está desarrollada con la premisa de ser escalable, esto es, de permitir resolver problemas más grandes al aumentar las unidades de procesado.
El problema lineal de autovalores, Ax = lambda x en su forma estándar, lo abordamos con el uso de técnicas iterativas, en concreto con métodos de Krylov, con los que calculamos una pequeña porción del espectro de autovalores. Este tipo de algoritmos se basa en generar un subespacio de tamaño reducido (m) en el que proyectar el problema de gran dimensión (n), siendo m << n. Una vez se ha proyectado el problema, se resuelve este mediante métodos directos, que nos proporcionan aproximaciones a los autovalores del problema inicial que querÃamos resolver. Las operaciones que se utilizan en la expansión del subespacio varÃan en función de si los autovalores deseados están en el exterior o en el interior del espectro. En caso de buscar autovalores en el exterior del espectro, la expansión se hace mediante multiplicaciones matriz-vector. Esta operación la realizamos en la GPU, bien mediante el uso de bibliotecas o mediante la creación de funciones que aprovechan la estructura de la matriz. En caso de autovalores en el interior del espectro, la expansión requiere resolver sistemas de ecuaciones lineales. En esta tesis implementamos varios algoritmos para la resolución de sistemas de ecuaciones lineales para el caso especÃfico de matrices con estructura tridiagonal a bloques, que se ejecutan en GPU.
En el cálculo de las funciones de matrices hemos de diferenciar entre la aplicación directa de una función sobre una matriz, f(A), y la aplicación de la acción de una función de matriz sobre un vector, f(A)b. El primer caso implica un cálculo denso que limita el tamaño del problema. El segundo permite trabajar con matrices dispersas grandes, y para resolverlo también hacemos uso de métodos de Krylov. La expansión del subespacio se hace mediante multiplicaciones matriz-vector, y hacemos uso de GPUs de la misma forma que al resolver autovalores. En este caso el problema proyectado comienza siendo de tamaño m, pero se incrementa en m en cada reinicio del método. La resolución del problema proyectado se hace aplicando una función de matriz de forma directa. Nosotros hemos implementado varios algoritmos para calcular las funciones de matrices raÃz cuadrada y exponencial, en las que el uso de GPUs permite acelerar el cálculo.One line of development followed in the field of supercomputing is the use of specific purpose processors to speed up certain types of computations. In this thesis we study the use of graphics processing units as computer accelerators and apply it to the field of linear algebra. In particular, we work with the SLEPc library to solve large scale eigenvalue problems, and to apply matrix functions in scientific applications. SLEPc is a parallel library based on the MPI standard and is developed with the premise of being scalable, i.e. to allow solving larger problems by increasing the processing units.
We address the linear eigenvalue problem, Ax = lambda x in its standard form, using iterative techniques, in particular with Krylov's methods, with which we calculate a small portion of the eigenvalue spectrum. This type of algorithms is based on generating a subspace of reduced size (m) in which to project the large dimension problem (n), being m << n. Once the problem has been projected, it is solved by direct methods, which provide us with approximations of the eigenvalues of the initial problem we wanted to solve. The operations used in the expansion of the subspace vary depending on whether the desired eigenvalues are from the exterior or from the interior of the spectrum. In the case of searching for exterior eigenvalues, the expansion is done by matrix-vector multiplications. We do this on the GPU, either by using libraries or by creating functions that take advantage of the structure of the matrix. In the case of eigenvalues from the interior of the spectrum, the expansion requires solving linear systems of equations. In this thesis we implemented several algorithms to solve linear systems of equations for the specific case of matrices with a block-tridiagonal structure, that are run on GPU.
In the computation of matrix functions we have to distinguish between the direct application of a matrix function, f(A), and the action of a matrix function on a vector, f(A)b. The first case involves a dense computation that limits the size of the problem. The second allows us to work with large sparse matrices, and to solve it we also make use of Krylov's methods. The expansion of subspace is done by matrix-vector multiplication, and we use GPUs in the same way as when solving eigenvalues. In this case the projected problem starts being of size m, but it is increased by m on each restart of the method. The solution of the projected problem is done by directly applying a matrix function. We have implemented several algorithms to compute the square root and the exponential matrix functions, in which the use of GPUs allows us to speed up the computation.Una lÃnia de desenvolupament seguida en el camp de la supercomputació és l'ús de processadors de propòsit especÃfic per a accelerar determinats tipus de cà lcul. En aquesta tesi estudiem l'ús de targetes grà fiques com a acceleradors de la computació i ho apliquem a l'à mbit de l'à lgebra lineal. En particular treballem amb la biblioteca SLEPc per a resoldre problemes de cà lcul d'autovalors en matrius de gran dimensió, i per a aplicar funcions de matrius en els cà lculs d'aplicacions cientÃfiques. SLEPc és una biblioteca paral·lela que es basa en l'està ndard MPI i està desenvolupada amb la premissa de ser escalable, açò és, de permetre resoldre problemes més grans en augmentar les unitats de processament.
El problema lineal d'autovalors, Ax = lambda x en la seua forma està ndard, ho abordem amb l'ús de tècniques iteratives, en concret amb mètodes de Krylov, amb els quals calculem una xicoteta porció de l'espectre d'autovalors. Aquest tipus d'algorismes es basa a generar un subespai de grandà ria reduïda (m) en el qual projectar el problema de gran dimensió (n), sent m << n. Una vegada s'ha projectat el problema, es resol aquest mitjançant mètodes directes, que ens proporcionen aproximacions als autovalors del problema inicial que volÃem resoldre. Les operacions que s'utilitzen en l'expansió del subespai varien en funció de si els autovalors desitjats estan en l'exterior o a l'interior de l'espectre. En cas de cercar autovalors en l'exterior de l'espectre, l'expansió es fa mitjançant multiplicacions matriu-vector. Aquesta operació la realitzem en la GPU, bé mitjançant l'ús de biblioteques o mitjançant la creació de funcions que aprofiten l'estructura de la matriu. En cas d'autovalors a l'interior de l'espectre, l'expansió requereix resoldre sistemes d'equacions lineals. En aquesta tesi implementem diversos algorismes per a la resolució de sistemes d'equacions lineals per al cas especÃfic de matrius amb estructura tridiagonal a blocs, que s'executen en GPU.
En el cà lcul de les funcions de matrius hem de diferenciar entre l'aplicació directa d'una funció sobre una matriu, f(A), i l'aplicació de l'acció d'una funció de matriu sobre un vector, f(A)b. El primer cas implica un cà lcul dens que limita la grandà ria del problema. El segon permet treballar amb matrius disperses grans, i per a resoldre-ho també fem ús de mètodes de Krylov. L'expansió del subespai es fa mitjançant multiplicacions matriu-vector, i fem ús de GPUs de la mateixa forma que en resoldre autovalors. En aquest cas el problema projectat comença sent de grandà ria m, però s'incrementa en m en cada reinici del mètode. La resolució del problema projectat es fa aplicant una funció de matriu de forma directa. Nosaltres hem implementat diversos algorismes per a calcular les funcions de matrius arrel quadrada i exponencial, en les quals l'ús de GPUs permet accelerar el cà lcul.Lamas Daviña, A. (2018). Dense and sparse parallel linear algebra algorithms on graphics processing units [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/112425TESI
Real-Time Bearing Estimation in a Multi-Source Environment Using Multi-Processor, Multi-Algorithmic Acceleration
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sources, the amount of content made available to users has become overwhelming and in need to be managed. Recommender systems emerged with the purpose of providing
personalized and meaningful content recommendations based on users’ preferences and usage history. Due to their utility and commercial potential, recommender systems integrate many audiovisual content providers and represent one of their most important and
valuable services. The goal of this thesis is to develop a recommender system based on
matrix factorization methods, capable of providing meaningful and personalized product
recommendations to individual users and groups of users, by taking into account
users’ rating patterns and biased tendencies, as well as their fluctuations throughout time
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