Parallel and Systolic Solution of Normalized Explicit Approximate Inverse Preconditioning

A new class of normalized approximate inverse matrix techniques, based on the concept of sparse normalized approximate factorization procedures are introduced for solving sparse linear systems derived from the finite difference discretization of partial differential equations. Normalized explicit preconditioned conjugate gradient type methods in conjunction with normalized approximate inverse matrix techniques are presented for the efficient solution of sparse linear systems. Theoretical results on the rate of convergence of the normalized explicit preconditioned conjugate gradient scheme and estimates of the required computational work are presented. Application of the new proposed methods on two dimensional initial/boundary value problems is discussed and numerical results are given. The parallel and systolic implementation of the dominant computational part is also investigated.     

Explicit approximate inverse preconditioning techniques

Summary  The numerical treatment and the production of related software for solving large sparse linear systems of algebraic equations, derived mainly from the discretization of partial differential equation, by preconditioning techniques has attracted the attention of many researchers. In this paper we give an overview of explicit approximate inverse matrix techniques for computing explicitly various families of approximate inverses based on Choleski and LU—type approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference, finite element and the domain decomposition discretization of elliptic and parabolic partial differential equations. Composite iterative schemes, using inner-outer schemes in conjunction with Picard and Newton method, based on approximate inverse matrix techniques for solving non-linear boundary value problems, are presented. Additionally, isomorphic iterative methods are introduced for the efficient solution of non-linear systems. Explicit preconditioned conjugate gradient—type schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear system of algebraic equations. Theoretical estimates on the rate of convergence and computational complexity of the explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.     

Normalized explicit finite element approximate inverse preconditioning

Normalized explicit approximate inverse matrix techniques for computing explicitly various families of normalized approximate inverses based on normalized approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference and finite element discretization of partial differential equations are presented. Normalized explicit preconditioned conjugate gradient-type schemes in conjunction with normalized approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear systems. Theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.     

High Performance Inverse Preconditioning

The derivation of parallel numerical algorithms for solving sparse linear systems on modern computer systems and software platforms has attracted the attention of many researchers over the years. In this paper we present an overview on the design issues of parallel approximate inverse matrix algorithms, based on an anti-diagonal “wave pattern” approach and a “fish-bone” computational procedure, for computing explicitly various families of exact and approximate inverses for solving sparse linear systems. Parallel preconditioned conjugate gradient-type schemes in conjunction with parallel approximate inverses are presented for the efficient solution of sparse linear systems. Applications of the proposed parallel methods by solving characteristic sparse linear systems on symmetric multiprocessor systems and distributed systems are discussed and the parallel performance of the proposed schemes is given, using MPI, OpenMP and Java multithreading.     

Sparse iterative algorithm software for large-scale MIMD machines: An initial discussion and implementation

The parallelization of sophisticated applications has dramatically increased in recent years. As machine capabilities rise, greater emphasis on modeling complex phenomena can be expected. Many of these applications require the solution of large sparse matrix equations which approximate systems of partial differential equations (PDEs). Therefore we consider parallel iterative solvers for large sparse non-symmetric systems and issues related to parallel sparse matrix software. We describe a collection of parallel iterative solvers which use a distributed sparse matrix format that facilitates the interface between specific applications and a variety of Krylov subspace techniques and multigrid methods. These methods have been used to solve a number of linear and non-linear PDE problems on a 1024-processor NCUBE 2 hypercube. Over 1 Gflop sustained computation rates are achieved with many of these solvers, demonstrating that high performance can be attained even when using sparse matrix data structures.     

Parallel Solutions for Large-Scale General Sparse Nonlinear Systems of Equations

In solving application problems,many large-scale nonlinear systems of equaions result in sparse Jacobian matrices.Such nonlinear systems are called sparse nonlinear systems.The irregularity of the locations of nonzrero elements of a general sparse matrix makes it very difficult to generally map sparse matrix computations to multiprocessors for parallel processing in a well balanced manner.To overcome this difficulty,we define a new storage scheme for general sparse matrices in this paper,With the new storage scheme,we develop parallel algorithms to solve large-scale general sparse systems of equations by interval Newton/Generalized bisection methods which reliably find all numerical solutions within a given domain.I n Section 1,we provide an introduction to the addressed problem and the interval Newton‘s methods.In Section 2,some currently used storage schemes for sparse systems are reviewed.In Section 3,new index schemes to store general sparse matrices are reported.In Section 4,we present a parallel algorithm to evaluate a general sparse Jacobian matrix.In Section 5,we present a parallel algorithm to solve the corresponding interval linear system by the all-row preconditioned scheme.Conclusions and future work are discussed in Section 6.     

Explicit isomorphic iterative methods for solving arrow-type linear systems

A new class of approximate inverse arrow-type matrix techniques based on the concept of sparse approximate LU-type factorization procedures is introduced for computing explicitly approximate inverses without inverting the decomposition factors. Isomorphic methods in conjunction with explicit preconditioned schemes based on approximate inverse matrix techniques are presented for the efficient solution of arrow-type linear systems. Applications of the proposed method on linear systems is discussed and numerical results are given     

Design and implementation of parallel approximate inverse classes using OpenMP

A new parallel normalized optimized approximate inverse algorithm, based on the concept of antidiagonal wave pattern, for computing classes of explicitly approximate inverses, is introduced for symmetric multiprocessor systems. The parallel normalized explicit approximate inverses are used in conjunction with parallel normalized explicit preconditioned conjugate gradient schemes for the efficient solution of finite element sparse linear systems. The parallel design and implementation issues of the new algorithm are discussed and the parallel performance is presented using OpenMP. Copyright © 2008 John Wiley & Sons, Ltd.     

稀疏近似逆预条件子及其并行计算

文中使用范数极小技术,提出一种构造稀疏矩阵并行近似逆预条件子的方法,所构造的稀疏矩阵近似逆的稀疏结构和数据矩阵的转置矩阵相同,计算量和存储量上,其求解过程易于并行。且并行计算不影响其收敛效果。通过试算表明,该方法对很多问题的求解具有明显的加速效果。文中给出了该方法的并行算法,并提出了一种自适应分配算法来解决负载平衡问题。     

A parallel Self Mesh-Adaptive N-body method based on approximate inverses

A new parallel Self Mesh-Adaptive N-body method based on approximate inverses is proposed. The scheme is a three-dimensional Cartesian-based method that solves the Poisson equation directly in physical space, using modified multipole expansion formulas for the boundary conditions. Moreover, adaptive-mesh techniques are utilized to form a class of separate smaller n-body problems that can be solved in parallel and increase the total resolution of the system. The solution method is based on multigrid method in conjunction with the symmetric factored approximate sparse inverse matrix as smoother. The design of the parallel Self Mesh-Adaptive method along with discussion on implementation issues for shared memory computer systems is presented. The new parallel method is evaluated through a series of benchmark simulations using N-body models of isolated galaxies or galaxies interacting with dwarf companions. Furthermore, numerical results on the performance and the speedups of the scheme are presented.     

Efficient parallel implementation of Bose Hubbard model: Exact numerical ground states and dynamics of gaseous Bose-Einstein condensates

We present a parallel implementation of the Bose Hubbard model, using imaginary time propagation to find the lowest quantum eigenstate and real time propagation for simulation of quantum dynamics. Scaling issues, performance of sparse matrix-vector multiplication, and a parallel algorithm for determining nonzero matrix elements are described. Implementation of imaginary time propagation yields an O(N) linear convergence on a single processor and slightly better than ideal performance on up to 160 processors for a particular problem size. The determination of the nonzero matrix elements is intractable using sequential non-optimized techniques for large problem sizes. Thus, we discuss a parallel algorithm that takes advantage of the intrinsic structural characteristics of the Fock-space matrix representation of the Bose Hubbard Hamiltonian and utilizes a parallel implementation of a Fock state look up table to make this task solvable within reasonable timeframes. Our parallel algorithm demonstrates near ideal scaling on thousand of processors. We include results for a matrix 22.6 million square, with 202 million nonzero elements, utilizing 2048 processors.     

Explicit preconditioned domain decomposition schemes for solving nonlinear boundary value problems

A new class of inner-outer iterative procedures in conjunction with Picard-Newton methods based on explicit preconditioning iterative methods for solving nonlinear systems is presented. Explicit preconditioned iterative schemes, based on the explicit computation of a class of domain decomposition generalized approximate inverse matrix techniques are presented for the efficient solution of nonlinear boundary value problems on multiprocessor systems. Applications of the new composite scheme on characteristic nonlinear boundary value problems are discussed and numerical results are given.     

稀疏有限元线性系统的并行算法实现

在对称多处理机系统上,提出了一种求解稀疏对称有限元线性系统的正规化精确并行逆算法。该算法以一种避免数据依赖的反对角运动方法为基础,使用OpenMP编译指导来实现。诸如加速比和效率等数值实验结果的推出,说明在一个对称多处理机系统上,所提出的算法求解方法能更好地提高性能,获得更大的加速。     

Parallel,iterative solution of sparse linear systems: Models and architectures

Solving large, sparse, linear systems of equations is a 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. A data transfer model predicting both the probability that data must be transferred between two tasks and the amount of data to be transferred is presented. This model is used to derive an execution time model for predicting parallel execution time and an optimal number of tasks given the dimension and sparsity of the coefficient matrix and the costs of computation, synchronization, and communication.The suitability of different parallel architectures for solving randomly sparse linear systems is discussed. Based on the complexity of task scheduling, one parallel architecture, based on a broadcast bus, is presented and analyzed.     

Java multithreading‐based parallel approximate arrow‐type inverses

A new parallel shared memory Java multithreaded design and implementation of the explicit approximate inverse preconditioning, for efficiently solving arrow‐type linear systems on symmetric multiprocessor systems (SMPs), is presented. A new parallel algorithm for computing a class of optimized approximate arrow‐type inverse matrix is introduced. The performance on an SMP, using Java multithreading, is investigated by solving arrow‐type linear systems and numerical results are given. The parallel performance of the construction of the optimized approximate inverse and the explicit preconditioned generalized conjugate gradient square scheme, using a dynamic workload scheduling, is also presented. Copyright © 2007 John Wiley & Sons, Ltd.     

On some new approximate factorization methods for block tridiagonal matrices suitable for vector and parallel processors

In this paper, to obtain an efficient parallel algorithm to solve sparse block-tridiagonal linear systems, stair matrices are used to construct some parallel polynomial approximate inverse preconditioners. These preconditioners are suitable when the desired goal is to maximize parallelism. Moreover, some theoretical results concerning these preconditioners are presented and how to construct preconditioners effectively for any nonsingular block tridiagonal H-matrices is also described. In addition, the validity of these preconditioners is illustrated with some numerical experiments arising from the second order elliptic partial differential equations and oil reservoir simulations.     

增量式约简最小二乘孪生支持向量回归机

为了解决增量式最小二乘孪生支持向量回归机存在构成的核矩阵无法很好地逼近原核矩阵的问题,提出了一种增量式约简最小二乘孪生支持向量回归机(IRLSTSVR)算法.该算法首先利用约简方法,判定核矩阵列向量之间的相关性,筛选出用于构成核矩阵列向量的样本作为支持向量以降低核矩阵中列向量的相关性,使得构成的核矩阵能够更好地逼近原核...     

Parallel solution of sparse linear least squares problems on distributed-memory multiprocessors

This paper studies the parallel solution of large-scale sparse linear least squares problems on distributed-memory multiprocessors. The key components required for solving a sparse linear least squares problem are sparse QR factorization and sparse triangular solution. A block-oriented parallel algorithm for sparse QR factorization has already been described in the literature. In this paper, new block-oriented parallel algorithms for sparse triangular solution are proposed. The arithmetic and communication complexities of the new algorithms applied to regular grid problems are analyzed. The proposed parallel sparse triangular solution algorithms together with the block-oriented parallel sparse QR factorization algorithm result in a highly efficient approach to the parallel solution of sparse linear least squares problems. Performance results obtained on an IBM Scalable POWERparallel system SP2 are presented. The largest least squares problem solved has over two million rows and more than a quarter million columns. The execution speed for the numerical factorization of this problem achieves over 3.7 gigaflops per second on an IBM SP2 machine with 128 processors.     

Dense Linear System: A Parallel Self-verified Solver

This article presents a parallel self-verified solver for dense linear systems of equations. This kind of solver is commonly used in many different kinds of real applications which deal with large matrices. Nevertheless, two key problems appear to limit the use of linear system solvers to a more extensive range of real applications: solution correctness and high computational cost. In order to solve the first one, verified computing would be an interesting choice. An algorithm that uses this concept is able to find a highly accurate and automatically verified result providing more reliability. However, the performance of these algorithms quickly becomes a drawback. Aiming at a better performance, parallel computing techniques were employed. Two main parts of this method were parallelized: the computation of the approximate inverse of matrix A and the preconditioning step. The results obtained show that these optimizations increase significantly the overall performance.     

稀疏近似逆并行预条件子

1．引言考虑求解线性方程组AX一b,X,bE＊”,山其中A二（a;小＿是大型稀疏非对称矩阵．通常使用迭代法求解式（1）,如GMRESBICGSTAB,CGSTFQMRCGSZ等Kryl0V子空间迭代法．直接使用迭代法的收敛速度有时特别慢,或根本不收敛,需使用预条件以加速迭代法的收敛速度．通常使用左或右预条件子M使式（1）变成易于求解的形式＊M9一6,X二M队或＊AX二＊6．由然后用迭代法求解式（2）,M的选择要使得AM（或M则近似等于单位矩阵．构造预条件子的方法有很多,如不完全分解方法、SSOR方法、多项式方法等,不完全分解方法和SSOR…