Out-of-core macromolecular simulations on multithreaded architectures

We address the solution of large-scale eigenvalue problems that appear in the motion simulation of complex macromolecules on multithreaded platforms, consisting of multicore processors and possibly a graphics processor (GPU). In particular, we compare specialized implementations of several high- performance eigensolvers that, by relying on disk storage and out-of-core (OOC) techniques, can in principle tackle the large memory requirements of these biological problems, which in general do not fit into the main memory of current desktop machines. All these OOC eigensolvers, except for one, are composed of compute-bound (i.e., arithmetically-intensive) operations, which we accelerate by exploiting the performance of current multicore processors and, in some cases, by additionally off-loading certain parts of the computation to a GPU accelerator. One of the eigensolvers is a memory-bound algorithm, which strongly constrains its performance when the data is on disk. However, this method exhibits a much lower arithmetic cost compared with its compute- bound alternatives for this particular application. Experimental results on a desktop platform, representative of current server technology, illustrate the potential of these methods to address the simulation of biological activity

Dense and sparse parallel linear algebra algorithms on graphics processing units

Una línea de desarrollo seguida en el campo de la supercomputación es el uso de procesadores de propósito específico 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 científicas. 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 queríamos resolver. Las operaciones que se utilizan en la expansión del subespacio varían 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 específico 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 raíz 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 línia de desenvolupament seguida en el camp de la supercomputació és l'ús de processadors de propòsit específic per a accelerar determinats tipus de càlcul. En aquesta tesi estudiem l'ús de targetes gràfiques com a acceleradors de la computació i ho apliquem a l'àmbit de l'àlgebra lineal. En particular treballem amb la biblioteca SLEPc per a resoldre problemes de càlcul d'autovalors en matrius de gran dimensió, i per a aplicar funcions de matrius en els càlculs d'aplicacions científiques. SLEPc és una biblioteca paral·lela que es basa en l'estàndard 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 estàndard, 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 grandària 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 volíem 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 específic de matrius amb estructura tridiagonal a blocs, que s'executen en GPU. En el càlcul 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 càlcul dens que limita la grandària 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 grandària 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 càlcul.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

Look-ahead in the two-sided reduction to compact band forms for symmetric eigenvalue problems and the SVD

We address the reduction to compact band forms, via unitary similarity transformations, for the solution of symmetric eigenvalue problems and the computation of the singular value decomposition (SVD). Concretely, in the first case, we revisit the reduction to symmetric band form, while, for the second case, we propose a similar alternative, which transforms the original matrix to (unsymmetric) band form, replacing the conventional reduction method that produces a triangular– band output. In both cases, we describe algorithmic variants of the standard Level 3 Basic Linear Algebra Subroutines (BLAS)-based procedures, enhanced with lookahead, to overcome the performance bottleneck imposed by the panel factorization. Furthermore, our solutions employ an algorithmic block size that differs from the target bandwidth, illustrating the important performance benefits of this decision. Finally, we show that our alternative compact band form for the SVD is key to introduce an effective look-ahead strategy into the corresponding reduction procedure

Divide and Conquer Symmetric Tridiagonal Eigensolver for Multicore Architectures

International audienceComputing eigenpairs of a symmetric matrix is a problem arising in many industrial applications, including quantum physics and finite-elements computation for automo-biles. A classical approach is to reduce the matrix to tridiagonal form before computing eigenpairs of the tridiagonal matrix. Then, a back-transformation allows one to obtain the final solution. Parallelism issues of the reduction stage have already been tackled in different shared-memory libraries. In this article, we focus on solving the tridiagonal eigenproblem, and we describe a novel implementation of the Divide and Conquer algorithm. The algorithm is expressed as a sequential task-flow, scheduled in an out-of-order fashion by a dynamic runtime which allows the programmer to play with tasks granularity. The resulting implementation is between two and five times faster than the equivalent routine from the INTEL MKL library, and outperforms the best MRRR implementation for many matrices

Analysis and Design of Communication Avoiding Algorithms for Out of Memory(OOM) SVD

Many applications — including big data analytics, information retrieval, gene expression analysis, and numerical weather prediction – require the solution of large, dense singular value decomposition (SVD). The size of matrices used in many of these applications is becoming too large to fit into into a computer’s main memory at one time, and the traditional SVD algorithms that require all the matrix components to be loaded into memory before computation starts cannot be used directly. Moving data (communication) between levels of memory hierarchy and the disk exposes extra challenges to design SVD for such big matrices because of the exponential growth in the gap between floating-point arithmetic rate and bandwidth for many different storage devices on modern high performance computers. In this dissertation, we have analyzed communication overhead on hierarchical memory systems and disks for SVD algorithms and designed communication-avoiding (CA) Out of Memory (OOM) SVD algorithms. By Out of Memory we mean that the matrix is too big to fit in the main memory and therefore must reside in external or internal storage. We have studied communication overhead for classical one-stage blocked SVD and two-stage tiled SVD algorithms and proposed our OOM SVD algorithm, which reduces the communication cost. We have presented theoretical analysis and strategies to design CA OOM SVD algorithms, developed optimized implementation of CA OOM SVD for multicore architecture, and presented its performance results. When matrices are tall, performance of OOM SVD can be improved significantly by carrying out QR decomposition on the original matrix in the first place. The upper triangular matrix generated by QR decomposition may fit in the main memory, and in-core SVD can be used efficiently. Even if the upper triangular matrix does not fit in the main memory, OOM SVD will work on a smaller matrix. That is why we have analyzed communication reduction for OOM QR algorithm, implemented optimized OOM tiled QR for multicore systems and showed performance improvement of OOM SVD algorithms for tall matrices

Structure Preserving Parallel Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem

The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized Bethe-Salpeter equation in the context of computing exciton energies and states. A computational challenge is that at least half of the eigenvalues and the associated eigenvectors are desired in practice. We establish the equivalence between Bethe-Salpeter eigenvalue problems and real Hamiltonian eigenvalue problems. Based on theoretical analysis, structure preserving algorithms for a class of Bethe-Salpeter eigenvalue problems are proposed. We also show that for this class of problems all eigenvalues obtained from the Tamm-Dancoff approximation are overestimated. In order to solve large scale problems of practical interest, we discuss parallel implementations of our algorithms targeting distributed memory systems. Several numerical examples are presented to demonstrate the efficiency and accuracy of our algorithms