Parallel Factorizations in Numerical Analysis

In this paper we review the parallel solution of sparse linear systems, usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is based on the concept of parallel factorization of a (block) tridiagonal matrix. This allows to obtain efficient parallel extensions of many known matrix factorizations, and to derive, as a by-product, a unifying approach to the parallel solution of ODEs.Comment: 15 pages, 5 figure

Parallel Factorizations in Numerical Analysis

In this paper we review the parallel solution of sparse linear systems, usually deriving by the discretization of ODEIVPs or ODE-BVPs. The approach is based on the concept of parallel factorization of a (block) tridiagonal matrix. This allows to obtain efficient parallel extensions of many known matrix factorizations, and to derive, as a by-product, a unifying approach to the parallel solution of ODEs

Communication in task-parallel ILU-preconditioned CG solversusing MPI + OmpSs

We target the parallel solution of sparse linear systems via iterative Krylov subspace–based methods enhanced with incomplete LU (ILU)-type preconditioners on clusters of multicore processors. In order to tackle large-scale problems, we develop task-parallel implementations of the classical iteration for the CG method, accelerated via ILUPACK and ILU(0) preconditioners, using MPI + OmpSs. In addition, we integrate several communication-avoiding strategies into the codes, including the butterfly communication scheme and Eijkhout's formulation of the CG method. For all these implementations, we analyze the communication patterns and perform a comparative analysis of their performance and scalability on a cluster consisting of 16 nodes, with 16 cores each

MPI-based Parallel Solution of Sparse Linear Systems Using Chio\u27s Condensation Algorithm and Test Data from Power Flow Analysis

Solving sparse systems of linear equations permeates power system analysis. Newton-Raphson, decoupled, and fast decoupled algorithms all require the repeated solving of sparse systems of linear equations in order to capture the steady state operational conditions of the power system under test. Solving these systems of equations is usually done using LU Factorization which has an order of complexity O(n3) where n represents the number of equations in the system. The Chio’s condensation algorithm is an alternative approach, which in general has a complexity of O(n4). However, it has a straightforward formulation that can be easily implemented in a parallel computing architecture. Previous research has not investigated the application of the Chio’s algorithm under sparse matrix, which is typical for power system analysis. This thesis presents a MPI-based parallel solution of sparse linear systems using Chio’s condensation algorithm and realistic test data from power flow analysis. Different sparse matrix techniques are discussed, and a reordering scheme is applied to further improve the efficiency for solving the sparse linear system

Iteration-fusing conjugate gradient for sparse linear systems with MPI + OmpSs

In this paper, we target the parallel solution of sparse linear systems via iterative Krylov subspace-based method enhanced with a block-Jacobi preconditioner on a cluster of multicore processors. In order to tackle large-scale problems, we develop task-parallel implementations of the preconditioned conjugate gradient method that improve the interoperability between the message-passing interface and OmpSs programming models. Specifically, we progressively integrate several communication-reduction and iteration-fusing strategies into the initial code, obtaining more efficient versions of the method. For all these implementations, we analyze the communication patterns and perform a comparative analysis of their performance and scalability on a cluster consisting of 32 nodes with 24 cores each. The experimental analysis shows that the techniques described in the paper outperform the classical method by a margin that varies between 6 and 48%, depending on the evaluation

Iteration-fusing conjugate gradient for sparse linear systems with MPI + OmpSs

In this paper, we target the parallel solution of sparse linear systems via iterative Krylov subspace-based method enhanced with a block-Jacobi preconditioner on a cluster of multicore processors. In order to tackle large-scale problems, we develop task-parallel implementations of the preconditioned conjugate gradient method that improve the interoperability between the message-passing interface and OmpSs programming models. Specifically, we progressively integrate several communication-reduction and iteration-fusing strategies into the initial code, obtaining more efficient versions of the method. For all these implementations, we analyze the communication patterns and perform a comparative analysis of their performance and scalability on a cluster consisting of 32 nodes with 24 cores each. The experimental analysis shows that the techniques described in the paper outperform the classical method by a margin that varies between 6 and 48%, depending on the evaluation.This research was partially supported by the H2020 EU FETHPC Project 671602 “INTERTWinE.” The researchers from Universidad Jaume I were sponsored by Project TIN2017-82972-R of the Spanish Ministerio de Economía y Competitividad. Maria Barreda was supported by the POSDOC-A/2017/11 project from the Universitat Jaume I.Peer ReviewedPostprint (author's final draft

Solving Sparse Integer Linear Systems

We propose a new algorithm to solve sparse linear systems of equations over the integers. This algorithm is based on a $p$-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art

Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections

Block projections have been used, in [Eberly et al. 2006], to obtain an efficient algorithm to find solutions for sparse systems of linear equations. A bound of softO(n^(2.5)) machine operations is obtained assuming that the input matrix can be multiplied by a vector with constant-sized entries in softO(n) machine operations. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections, and this has been conjectured. In this paper we establish the correctness of the algorithm from [Eberly et al. 2006] by proving the existence of efficient block projections over sufficiently large fields. We demonstrate the usefulness of these projections by deriving improved bounds for the cost of several matrix problems, considering, in particular, ``sparse'' matrices that can be be multiplied by a vector using softO(n) field operations. We show how to compute the inverse of a sparse matrix over a field F using an expected number of softO(n^(2.27)) operations in F. A basis for the null space of a sparse matrix, and a certification of its rank, are obtained at the same cost. An application to Kaltofen and Villard's Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an integer matrix yields algorithms requiring softO(n^(2.66)) machine operations. The derived algorithms are all probabilistic of the Las Vegas type

Exact Sparse Matrix-Vector Multiplication on GPU's and Multicore Architectures

We propose different implementations of the sparse matrix--dense vector multiplication (\spmv{}) for finite fields and rings \Zb/m\Zb. We take advantage of graphic card processors (GPU) and multi-core architectures. Our aim is to improve the speed of \spmv{} in the \linbox library, and henceforth the speed of its black box algorithms. Besides, we use this and a new parallelization of the sigma-basis algorithm in a parallel block Wiedemann rank implementation over finite fields

Efficient Computation of the Characteristic Polynomial

This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices