Resilience for Asynchronous Iterative Methods for Sparse Linear Systems

Large scale simulations are used in a variety of application areas in science and engineering to help forward the progress of innovation. Many spend the vast majority of their computational time attempting to solve large systems of linear equations; typically arising from discretizations of partial differential equations that are used to mathematically model various phenomena. The algorithms used to solve these problems are typically iterative in nature, and making efficient use of computational time on High Performance Computing (HPC) clusters involves constantly improving these iterative algorithms. Future HPC platforms are expected to encounter three main problem areas: scalability of code, reliability of hardware, and energy efficiency of the platform. The HPC resources that are expected to run the large programs are planned to consist of billions of processing units that come from more traditional multicore processors as well as a variety of different hardware accelerators. This growth in parallelism leads to the presence of all three problems. Previously, work on algorithm development has focused primarily on creating fault tolerance mechanisms for traditional iterative solvers. Recent work has begun to revisit using asynchronous methods for solving large scale applications, and this dissertation presents research into fault tolerance for fine-grained methods that are asynchronous in nature. Classical convergence results for asynchronous methods are revisited and modified to account for the possible occurrence of a fault, and a variety of techniques for recovery from the effects of a fault are proposed. Examples of how these techniques can be used are shown for various algorithms, including an analysis of a fine-grained algorithm for computing incomplete factorizations. Lastly, numerous modeling and simulation tools for the further construction of iterative algorithms for HPC applications are developed, including numerical models for simulating faults and a simulation framework that can be used to extrapolate the performance of algorithms towards future HPC systems

Large sparse linear systems arising from mimetic discretization

AbstractIn this work we perform an experimental study of iterative methods for solving large sparse linear systems arising from a second-order 2D mimetic discretization. The model problem is the 2D Poisson equation with different boundary conditions. We use GMRES with the restarted parameter and BiCGstab as iterative methods. We also use various preconditioning techniques including the robust preconditioner ILUt. The numerical experiments consist of large sparse linear systems with up to 643200 degrees of freedom

Lecture 10: Preconditioned Iterative Methods for Linear Systems

Iterative methods for the solution of linear systems of equations – such as stationary, semi-iterative, and Krylov subspace methods – are classical methods taught in numerical analysis courses, but adapting these methods to run efficiently at large-scale on high-performance computers is challenging and a constantly evolving topic. Preconditioners – necessary to aid the convergence of iterative methods – come in many forms, from algebraic to physics-based, are regularly being developed for linear systems from different classes of problems, and similarly are evolving with high-performance computers. This lecture will cover the background and some recent developments on iterative methods and preconditioning in the context of high-performance parallel computers. Topics include asynchronous iterative methods that avoid the potentially high synchronization cost where there are very large numbers of computational threads, parallel sparse approximate inverse preconditioners, parallel incomplete factorization preconditioners and sparse triangular solvers, and preconditioning with hierarchical rank-structured matrices for kernel matrix equations

Serial and Parallel Krylov Methods for Implicit Finite Difference Schemes Arising in Multivariate Option Pricing

This paper investigates computational and implementation issues for the valuation of options on three underlying assets, focusing on the use of the finite difference methods. We demonstrate that implicit methods, which have good convergence and stability prooperties, can now be implemented efficiently due to the recent development of techniques that allow the efficient solution of large and sparse linear systems. In the trivariate option valuation problem, we use nonstationary iterative methods (also called Krylov methods) for the solution of the large and sparse linear systems arising while using implicit methods. Krylov methods are investigated both in serial and in parallel implementations. Computational results show that the parallel implementation is particularly efficient if a fine grid space is needed.Multivariate option pricing, finite difference methods; Krylov methods; parallel Krylov methods

A model of asynchronous iterative algorithms for solving large, sparse, linear systems

Solving large, sparse, linear systems of equations is one of the fundamental problems in large scale scientific and engineering computation. A model of a general class of asynchronous, iterative solution methods for linear systems is developed. In the model, the system is solved by creating several cooperating tasks that each compute a portion of the solution vector. This model is then analyzed to determine the expected intertask data transfer and task computational complexity as functions of the number of tasks. Based on the analysis, recommendations for task partitioning are made. These recommendations are a function of the sparseness of the linear system, its structure (i.e., randomly sparse or banded), and dimension

An efficient sparse conjugate gradient solver using a Beneš permutation network

© 2014 Technical University of Munich (TUM).The conjugate gradient (CG) is one of the most widely used iterative methods for solving systems of linear equations. However, parallelizing CG for large sparse systems is difficult due to the inherent irregularity in memory access pattern. We propose a novel processor architecture for the sparse conjugate gradient method. The architecture consists of multiple processing elements and memory banks, and is able to compute efficiently both sparse matrix-vector multiplication, and other dense vector operations. A Beneš permutation network with an optimised control scheme is introduced to reduce memory bank conflicts without expensive logic. We describe a heuristics for offline scheduling, the effect of which is captured in a parametric model for estimating the performance of designs generated from our approach