An overview of block Gram-Schmidt methods and their stability properties

Block Gram-Schmidt algorithms serve as essential kernels in many scientific computing applications, but for many commonly used variants, a rigorous treatment of their stability properties remains open. This survey provides a comprehensive categorization of block Gram-Schmidt algorithms, particularly those used in Krylov subspace methods to build orthonormal bases one block vector at a time. All known stability results are assembled, and new results are summarized or conjectured for important communication-reducing variants. Additionally, new block versions of low-synchronization variants are derived, and their efficacy and stability are demonstrated for a wide range of challenging examples. Low-synchronization variants appear remarkably stable for s-step-like matrices built with Newton polynomials, pointing towards a new stable and efficient backbone for Krylov subspace methods. Numerical examples are computed with a versatile MATLAB package hosted at https://github.com/katlund/BlockStab, and scripts for reproducing all results in the paper are provided. Block Gram-Schmidt implementations in popular software packages are discussed, along with a number of open problems. An appendix containing all algorithms type-set in a uniform fashion is provided.Comment: 42 pages, 5 tables, 17 figures, 20 algorithm

Parallel unstructured solvers for linear partial differential equations

This thesis presents the development of a parallel algorithm to solve symmetric systems of linear equations and the computational implementation of a parallel partial differential equations solver for unstructured meshes. The proposed method, called distributive conjugate gradient - DCG, is based on a single-level domain decomposition method and the conjugate gradient method to obtain a highly scalable parallel algorithm. An overview on methods for the discretization of domains and partial differential equations is given. The partition and refinement of meshes is discussed and the formulation of the weighted residual method for two- and three-dimensions presented. Some of the methods to solve systems of linear equations are introduced, highlighting the conjugate gradient method and domain decomposition methods. A parallel unstructured PDE solver is proposed and its actual implementation presented. Emphasis is given to the data partition adopted and the scheme used for communication among adjacent subdomains is explained. A series of experiments in processor scalability is also reported. The derivation and parallelization of DCG are presented and the method validated throughout numerical experiments. The method capabilities and limitations were investigated by the solution of the Poisson equation with various source terms. The experimental results obtained using the parallel solver developed as part of this work show that the algorithm presented is accurate and highly scalable, achieving roughly linear parallel speed-up in many of the cases tested

Accelerating advanced preconditioning methods on hybrid architectures

Un gran número de problemas, en diversas áreas de la ciencia y la ingeniería, involucran la solución de sistemas dispersos de ecuaciones lineales de gran escala. En muchos de estos escenarios, son además un cuello de botella desde el punto de vista computacional, y por esa razón, su implementación eficiente ha motivado una cantidad enorme de trabajos científicos. Por muchos años, los métodos directos basados en el proceso de la Eliminación Gaussiana han sido la herramienta de referencia para resolver dichos sistemas, pero la dimensión de los problemas abordados actualmente impone serios desafíos a la mayoría de estos algoritmos, considerando sus requerimientos de memoria, su tiempo de cómputo y la complejidad de su implementación. Propulsados por los avances en las técnicas de precondicionado, los métodos iterativos se han vuelto más confiables, y por lo tanto emergen como alternativas a los métodos directos, ofreciendo soluciones de alta calidad a un menor costo computacional. Sin embargo, estos avances muchas veces son relativos a un problema específico, o dotan a los precondicionadores de una complejidad tal, que su aplicación en diversos problemas se vuelve poco práctica en términos de tiempo de ejecución y consumo de memoria. Como respuesta a esta situación, es común la utilización de estrategias de Computación de Alto Desempeño, ya que el desarrollo sostenido de las plataformas de hardware permite la ejecución simultánea de cada vez más operaciones. Un claro ejemplo de esta evolución son las plataformas compuestas por procesadores multi-núcleo y aceleradoras de hardware como las Unidades de Procesamiento Gráfico (GPU). Particularmente, las GPU se han convertido en poderosos procesadores paralelos, capaces de integrar miles de núcleos a precios y consumo energético razonables.Por estas razones, las GPU son ahora una plataforma de hardware de gran importancia para la ciencia y la ingeniería, y su uso eficiente es crucial para alcanzar un buen desempeño en la mayoría de las aplicaciones. Esta tesis se centra en el uso de GPUs para acelerar la solución de sistemas dispersos de ecuaciones lineales usando métodos iterativos precondicionados con técnicas modernas. En particular, se trabaja sobre ILUPACK, que ofrece implementaciones de los métodos iterativos más importantes, y presenta un interesante y moderno precondicionador de tipo ILU multinivel. En este trabajo, se desarrollan versiones del precondicionador y de los métodos incluidos en el paquete, capaces de explotar el paralelismo de datos mediante el uso de GPUs sin afectar las propiedades numéricas del precondicionador. Además, se habilita y analiza el uso de las GPU en versiones paralelas existentes, basadas en paralelismo de tareas para plataformas de memoria compartida y distribuida. Los resultados obtenidos muestran una sensible mejora en el tiempo de ejecución de los métodos abordados, así como la posibilidad de resolver problemas de gran escala de forma eficiente

Parallel unstructured solvers for linear partial differential equations

This thesis presents the development of a parallel algorithm to solve symmetric systems of linear equations and the computational implementation of a parallel partial differential equations solver for unstructured meshes. The proposed method, called distributive conjugate gradient - DCG, is based on a single-level domain decomposition method and the conjugate gradient method to obtain a highly scalable parallel algorithm. An overview on methods for the discretization of domains and partial differential equations is given. The partition and refinement of meshes is discussed and the formulation of the weighted residual method for two- and three-dimensions presented. Some of the methods to solve systems of linear equations are introduced, highlighting the conjugate gradient method and domain decomposition methods. A parallel unstructured PDE solver is proposed and its actual implementation presented. Emphasis is given to the data partition adopted and the scheme used for communication among adjacent subdomains is explained. A series of experiments in processor scalability is also reported. The derivation and parallelization of DCG are presented and the method validated throughout numerical experiments. The method capabilities and limitations were investigated by the solution of the Poisson equation with various source terms. The experimental results obtained using the parallel solver developed as part of this work show that the algorithm presented is accurate and highly scalable, achieving roughly linear parallel speed-up in many of the cases tested.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Contribution to the study of efficient iterative methods for the numerical solution of partial differential equations

Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modelling of many applications in Computational Science and Engineering. This manuscript covers certain aspects of modern iterative solution methods for the solution of large-scale problems issued from the discretization of partial differential equations. More specifically, we focus on geometric multigrid methods, non-overlapping substructuring methods and flexible Krylov subspace methods with a particular emphasis on their combination. Firstly, the combination of multigrid and Krylov subspace methods is investigated on a linear partial differential equation modelling wave propagation in heterogeneous media. Secondly, we focus on non-overlapping domain decomposition methods for a specific finite element discretization known as the hp finite element, where unrefinement/refinement is allowed both by decreasing/increasing the step size h or by decreasing/increasing the polynomial degree p of the approximation on each element. Results on condition number bounds for the domain decomposition preconditioned operators are given and illustrated by numerical results on academic problems in two and three dimensions. Thirdly, we review recent advances related to a class of Krylov subspace methods allowing variable preconditioning. We examine in detail flexible Krylov subspace methods including augmentation and/or spectral deflation, where deflation aims at capturing approximate invariant subspace information. We also present flexible Krylov subspace methods for the solution of linear systems with multiple right-hand sides given simultaneously. The efficiency of the numerical methods is demonstrated on challenging applications in seismics requiring the solution of huge linear systems of equations with multiple right-hand sides on parallel distributed memory computers. Finally, we expose current and future prospectives towards the design of efficient algorithms on extreme scale machines for the solution of problems coming from the discretization of partial differential equations

Parallel Algorithms for Time and Frequency Domain Circuit Simulation

As a most critical form of pre-silicon verification, transistor-level circuit simulation is an indispensable step before committing to an expensive manufacturing process. However, considering the nature of circuit simulation, it can be computationally expensive, especially for ever-larger transistor circuits with more complex device models. Therefore, it is becoming increasingly desirable to accelerate circuit simulation. On the other hand, the emergence of multi-core machines offers a promising solution to circuit simulation besides the known application of distributed-memory clustered computing platforms, which provides abundant hardware computing resources. This research addresses the limitations of traditional serial circuit simulations and proposes new techniques for both time-domain and frequency-domain parallel circuit simulations. For time-domain simulation, this dissertation presents a parallel transient simulation methodology. This new approach, called WavePipe, exploits coarse-grained application-level parallelism by simultaneously computing circuit solutions at multiple adjacent time points in a way resembling hardware pipelining. There are two embodiments in WavePipe: backward and forward pipelining schemes. While the former creates independent computing tasks that contribute to a larger future time step, the latter performs predictive computing along the forward direction. Unlike existing relaxation methods, WavePipe facilitates parallel circuit simulation without jeopardizing convergence and accuracy. As a coarse-grained parallel approach, it requires low parallel programming effort, furthermore it creates new avenues to have a full utilization of increasingly parallel hardware by going beyond conventional finer grained parallel device model evaluation and matrix solutions. This dissertation also exploits the recently developed explicit telescopic projective integration method for efficient parallel transient circuit simulation by addressing the stability limitation of explicit numerical integration. The new method allows the effective time step controlled by accuracy requirement instead of stability limitation. Therefore, it not only leads to noticeable efficiency improvement, but also lends itself to straightforward parallelization due to its explicit nature. For frequency-domain simulation, this dissertation presents a parallel harmonic balance approach, applicable to the steady-state and envelope-following analyses of both driven and autonomous circuits. The new approach is centered on a naturally-parallelizable preconditioning technique that speeds up the core computation in harmonic balance based analysis. The proposed method facilitates parallel computing via the use of domain knowledge and simplifies parallel programming compared with fine-grained strategies. As a result, favorable runtime speedups are achieved