ELSI: A Unified Software Interface for Kohn-Sham Electronic Structure Solvers

Solving the electronic structure from a generalized or standard eigenproblem is often the bottleneck in large scale calculations based on Kohn-Sham density-functional theory. This problem must be addressed by essentially all current electronic structure codes, based on similar matrix expressions, and by high-performance computation. We here present a unified software interface, ELSI, to access different strategies that address the Kohn-Sham eigenvalue problem. Currently supported algorithms include the dense generalized eigensolver library ELPA, the orbital minimization method implemented in libOMM, and the pole expansion and selected inversion (PEXSI) approach with lower computational complexity for semilocal density functionals. The ELSI interface aims to simplify the implementation and optimal use of the different strategies, by offering (a) a unified software framework designed for the electronic structure solvers in Kohn-Sham density-functional theory; (b) reasonable default parameters for a chosen solver; (c) automatic conversion between input and internal working matrix formats, and in the future (d) recommendation of the optimal solver depending on the specific problem. Comparative benchmarks are shown for system sizes up to 11,520 atoms (172,800 basis functions) on distributed memory supercomputing architectures.Comment: 55 pages, 14 figures, 2 table

Analysis of hybrid parallelization strategies: simulation of Anderson localization and Kalman Filter for LHCb triggers

This thesis presents two experiences of hybrid programming applied to condensed matter and high energy physics. The two projects differ in various aspects, but both of them aim to analyse the benefits of using accelerated hardware to speedup the calculations in current science-research scenarios. The first project enables massively parallelism in a simulation of the Anderson localisation phenomenon in a disordered quantum system. The code represents a Hamiltonian in momentum space, then it executes a diagonalization of the corresponding matrix using linear algebra libraries, and finally it analyses the energy-levels spacing statistics averaged over several realisations of the disorder. The implementation combines different parallelization approaches in an hybrid scheme. The averaging over the ensemble of disorder realisations exploits massively parallelism with a master-slave configuration based on both multi-threading and message passing interface (MPI). This framework is designed and implemented to easily interface similar application commonly adopted in scientific research, for example in Monte Carlo simulations. The diagonalization uses multi-core and GPU hardware interfacing with MAGMA, PLASMA or MKL libraries. The access to the libraries is modular to guarantee portability, maintainability and the extension in a near future. The second project is the development of a Kalman Filter, including the porting on GPU architectures and autovectorization for online LHCb triggers. The developed codes provide information about the viability and advantages for the application of GPU technologies in the first triggering step for Large Hadron Collider beauty experiment (LHCb). The optimisation introduced on both codes for CPU and GPU delivered a relevant speedup on the Kalman Filter. The two GPU versions in CUDA R and OpenCLTM have similar performances and are adequate to be considered in the upgrade and in the corresponding implementations of the Gaudi framework. In both projects we implement optimisation techniques in the CPU code. This report presents extensive benchmark analyses of the correctness and of the performances for both projects

PYDAC: A DISTRIBUTED RUNTIME SYSTEM AND PROGRAMMING MODEL FOR A HETEROGENEOUS MANY-CORE ARCHITECTURE

Heterogeneous many-core architectures that consist of big, fast cores and small, energy-efficient cores are very promising for future high-performance computing (HPC) systems. These architectures offer a good balance between single-threaded perfor- mance and multithreaded throughput. Such systems impose challenges on the design of programming model and runtime system. Specifically, these challenges include (a) how to fully utilize the chip’s performance, (b) how to manage heterogeneous, un- reliable hardware resources, and (c) how to generate and manage a large amount of parallel tasks. This dissertation proposes and evaluates a Python-based programming framework called PyDac. PyDac supports a two-level programming model. At the high level, a programmer creates a very large number of tasks, using the divide-and-conquer strategy. At the low level, tasks are written in imperative programming style. The runtime system seamlessly manages the parallel tasks, system resilience, and inter- task communication with architecture support. PyDac has been implemented on both an field-programmable gate array (FPGA) emulation of an unconventional het- erogeneous architecture and a conventional multicore microprocessor. To evaluate the performance, resilience, and programmability of the proposed system, several micro-benchmarks were developed. We found that (a) the PyDac abstracts away task communication and achieves programmability, (b) the micro-benchmarks are scalable on the hardware prototype, but (predictably) serial operation limits some micro-benchmarks, and (c) the degree of protection versus speed could be varied in redundant threading that is transparent to programmers

ELPA: A parallel solver for the generalized eigenvalue problem

For symmetric (hermitian) (dense or banded) matrices the computation of eigenvalues and eigenvectors Ax = λBx is an important task, e.g. in electronic structure calculations. If a larger number of eigenvectors are needed, often direct solvers are applied. On parallel architectures the ELPA implementation has proven to be very efficient, also compared to other parallel solvers like EigenExa or MAGMA. The main improvement that allows better parallel efficiency in ELPA is the two-step transformation of dense to band to tridiagonal form. This was the achievement of the ELPA project. The continuation of this project has been targeting at additional improvements like allowing monitoring and autotuning of the ELPA code, optimizing the code for different architectures, developing curtailed algorithms for banded A and B, and applying the improved code to solve typical examples in electronic structure calculations. In this paper we will present the outcome of this project

Towards efficient exploitation of GPUs : a methodology for mapping index-digit algorithms

[Resumen]La computación de propósito general en GPUs supuso un gran paso, llevando la computación de alto rendimiento a los equipos domésticos. Lenguajes de programación de alto nivel como OpenCL y CUDA redujeron en gran medida la complejidad de programación. Sin embargo, para poder explotar totalmente el poder computacional de las GPUs, se requieren algoritmos paralelos especializados. La complejidad en la jerarquía de memoria y su arquitectura masivamente paralela hace que la programación de GPUs sea una tarea compleja incluso para programadores experimentados. Debido a la novedad, las librerías de propósito general son escasas y las versiones paralelas de los algoritmos no siempre están disponibles. En lugar de centrarnos en la paralelización de algoritmos concretos, en esta tesis proponemos una metodología general aplicable a la mayoría de los problemas de tipo divide y vencerás con una estructura de mariposa que puedan formularse a través de la representación Indice-Dígito. En primer lugar, se analizan los diferentes factores que afectan al rendimiento de la arquitectura de las GPUs. A continuación, estudiamos varias técnicas de optimización y diseñamos una serie de bloques constructivos modulares y reutilizables, que se emplean para crear los diferentes algoritmos. Por último, estudiamos el equilibrio óptimo de los recursos, y usando vectores de mapeo y operadores algebraicos ajustamos los algoritmos para las configuraciones deseadas. A pesar del enfoque centrado en la exibilidad y la facilidad de programación, las implementaciones resultantes ofrecen un rendimiento muy competitivo, que llega a superar conocidas librerías recientes.[Resumo] A computación de propósito xeral en GPUs supuxo un gran paso, levando a computación de alto rendemento aos equipos domésticos. Linguaxes de programación de alto nivel como OpenCL e CUDA reduciron en boa medida a complexidade da programación. Con todo, para poder aproveitar totalmente o poder computacional das GPUs, requírense algoritmos paralelos especializados. A complexidade na xerarquía de memoria e a súa arquitectura masivamente paralela fai que a programación de GPUs sexa unha tarefa complexa mesmo para programadores experimentados. Debido á novidade, as librarías de propósito xeral son escasas e as versións paralelas dos algoritmos non sempre están dispoñibles. En lugar de centrarnos na paralelización de algoritmos concretos, nesta tese propoñemos unha metodoloxía xeral aplicable á maioría dos problemas de tipo divide e vencerás cunha estrutura de bolboreta que poidan formularse a través da representación Índice-Díxito. En primeiro lugar, analízanse os diferentes factores que afectan ao rendemento da arquitectura das GPUs. A continuación, estudamos varias técnicas de optimización e deseñamos unha serie de bloques construtivos modulares e reutilizables, que se empregan para crear os diferentes algoritmos. Por último, estudamos o equilibrio óptimo dos recursos, e usando vectores de mapeo e operadores alxbricos axustamos os algoritmos para as configuracións desexadas. A pesar do enfoque centrado na exibilidade e a facilidade de programación, as implementacións resultantes ofrecen un rendemento moi competitivo, que chega a superar coñecidas librarías recentes.[Abstract]GPU computing supposed a major step forward, bringing high performance computing to commodity hardware. Feature-rich parallel languages like CUDA and OpenCL reduced the programming complexity. However, to fully take advantage of their computing power, specialized parallel algorithms are required. Moreover, the complex GPU memory hierarchy and highly threaded architecture makes programming a difficult task even for experienced programmers. Due to the novelty of GPU programming, common general purpose libraries are scarce and parallel versions of the algorithms are not always readily available. Instead of focusing in the parallelization of particular algorithms, in this thesis we propose a general methodology applicable to most divide-and-conquer problems with a buttery structure which can be formulated through the Index-Digit representation. First, we analyze the different performance factors of the GPU architecture. Next, we study several optimization techniques and design a series of modular and reusable building blocks, which will be used to create the different algorithms. Finally, we study the optimal resource balance, and through a mapping vector representation and operator algebra, we tune the algorithms for the desired configurations. Despite the focus on programmability and exibility, the resulting implementations offer very competitive performance, being able to surpass other well-known state of the art libraries

A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

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

Algebraic, Block and Multiplicative Preconditioners based on Fast Tridiagonal Solves on GPUs

This thesis contributes to the field of sparse linear algebra, graph applications, and preconditioners for Krylov iterative solvers of sparse linear equation systems, by providing a (block) tridiagonal solver library, a generalized sparse matrix-vector implementation, a linear forest extraction, and a multiplicative preconditioner based on tridiagonal solves. The tridiagonal library, which supports (scaled) partial pivoting, outperforms cuSPARSE's tridiagonal solver by factor five while completely utilizing the available GPU memory bandwidth. For the performance optimized solving of multiple right-hand sides, the explicit factorization of the tridiagonal matrix can be computed. The extraction of a weighted linear forest (union of disjoint paths) from a general graph is used to build algebraic (block) tridiagonal preconditioners and deploys the generalized sparse-matrix vector implementation of this thesis for preconditioner construction. During linear forest extraction, a new parallel bidirectional scan pattern, which can operate on double-linked list structures, identifies the path ID and the position of a vertex. The algebraic preconditioner construction is also used to build more advanced preconditioners, which contain multiple tridiagonal factors, based on generalized ILU factorizations. Additionally, other preconditioners based on tridiagonal factors are presented and evaluated in comparison to ILU and ILU incomplete sparse approximate inverse preconditioners (ILU-ISAI) for the solution of large sparse linear equation systems from the Sparse Matrix Collection. For all presented problems of this thesis, an efficient parallel algorithm and its CUDA implementation for single GPU systems is provided