一个实用的本性并行差分格式

１．引 言 近些年来随着计算机尺度和复杂性的扩大，人们对计算机和计算方法提出了更高的要求．这表明了人们不仅需要高速的，大内存的并行计算机，而且需要有效的并行算法．为了适合并行计算。一些数值格式需要重新改造．然而确有一些数值格式本身具有并行特征，能够直接用于并行计算． 事实上，关于偏微分方程的有限差分格式都有这种情况，现在人们正尽力研究它们，并给出了一些方法［１，２］．在这篇文章里，我们以如下问题为例：给出了一个实用的本性并行差分格式．此格式基于在子区域边界上用显式格式，内部用隐式格式，此格式有许多…     

0-1规划中并行隐枚举法的实现方式

0-1规划中,当变量较大时,状态数过多、时间耗费较大,隐枚举法是目前解决0-1规划问题最有效的方法,并行计算的特点是快速解决大型且复杂的计算问题。结合并行计算和隐枚举法来解决这个问题,并且对隐枚举法做了一定的改进,使得在串行计算中难以实现的问题在并行计算机上得到了解决,并用实例验证了算法的可行性和优越性。     

自相关函数特征提取的向量并行算法研究

本文针对图象模块化的计算特点,研究了具有平移、相似不变性质的自相关函数特征的向量化并行算法。通过微机模拟给出了飞机图象的处理结果。此并行计算方法适用于一般图象模块卷积计算,可用于计算机视觉并行计算及大型并行机的图象处理,已移值于某并行机。     

二维热传导方程有限差分区域分解算法

本文讨论了一类数值求解二维热传导方程的并行差分格式．在这个算法中,通过引进内界点将求解区域分裂成若干子区域．在子区域间内界点上采用非对称格式计算,一旦这些点的值被计算出来,各子区域间的计算可完全并行．本文得到了稳定性条件和最大模误差估计．它表明我们的格式有令人满意的稳定性,并且有着较高的收敛阶．     

一种求解线性方程组的SOR并行算法

逐次松弛迭代算法(SOR)是求解线性方程组的一种常用迭代算法,当系数矩阵正定时,它具有较快的收敛速度。但是,由于每个迭代步内存在数据相关,它难以实现并行计算。目前的SOR并行算法采用数据分解的方法,但由于该法并行区域过小,同步通讯代价大,并行效率低。本文提出了SOR的一种新型并行算法,该算法与传统SOR方法等价,具有相同的收敛性和迭代结果。该并行算法通过矩阵分块增大了可并行计算的区域,并引入流水线技术,利用各处理器间通讯与计算时间的重叠,获得较理想的并行加速效率。通过多核微机以及小规模集群上的数值实验证明,本文提出的SOR并行算法在求解大型稠密线性方程组时具有较好的并行效率。     

求解对流-扩散方程的交替分段显-隐式方法

1．引言求解下述对流一扩散方程已有很多显式和隐式差分方法．显式方法很适合于并行计算，但由于稳定性条件的限制，必须采用非常小的时间步长进行计算．隐式格式一般无稳定性条件，但在每一时间层上要求解线性代数方程组，实现并行计算有一定困难．Evans和Abdullah［‘，‘l巧妙地利用Saul’yev非对称格式构造了交替分组显式（xGz）方法来求解扩散方程，其后又将方法推广到求解对流一扩散方程＊．张宝琳＊提出了求解扩散方程的交替分段显一隐式方法，在一定意义上推广了Evans和Abdullah的方法，并在计算上更为精确．本文根据＊中关于扩…     

共享存储并行机的算法设计

1．引言目前在工业设计部门使用的并行处理计算机中，共享存储计算机占有很大的比重，在今后一定的时间内仍然是一个比较重要的机型，怎样用好这些计算机是一个不可忽视的问题．在工业设计部门的计算中，大型稀疏线性代数方程组迭代求解占有不可忽视的比重．因此，研究大型稀疏线性代数方程组在共享存储计算机上的并行迭代算法就显得非常重要．我们所遇到的共享存储计算机主要有两大类：一是流水线式向量机或多CPU向量机；另一类是多CPU的并行机．这两类机型的算法设计思想是有比较大的差异的．就是同一种类型的机器，它的结构不一样，算…     

一种孔隙介质中地下水流并行计算方法

针对孔隙介质中地下水流动问题提出了一种并行数值计算方法,并基于此设计了一套专用于求解大规模三维地下水流动方程的并行计算模块。计算模块基于区域分解的方法实现对模型区域的并行求解,采用了分布式内存和压缩矩阵技术解决大规模稀疏矩阵的存储及其计算,整合多种并行Krylov子空间方法和预条件子技术迭代求解大规模线性方程组。在Linux集群系统上进行了数值模拟实验,性能测试结果表明,程序具有良好的加速比和可扩展性。     

独立自主发展分布式应用软件

国际上已推出每秒万亿次并行机,目前美国正在研制每秒千万亿次运算(PetaFlo~)s)并行机。国内已研制成功千亿次分布式并行机,预期到2000年我国也将推出万亿次计算机。随着一批国产高性能并行机的问世与国外计算机的进口,我国高性能并行计算的硬件环境已有了重大改善。我国很多高性能计算的应用部门都面临着把现有大型串行软件或共享并行软件移植到分布式并行系.统的技术转轨。这些并行机是否好用,如何用好,发挥应有的作用,从而推动我国计算科学的发展,使计算真正成为我国高科技研究技术人员除理论与实验之外的第三只眼睛,事关国家重大目标的…     

关于采用流水线方式进行一簇递推关系式的并行计算

１．引言在科学与工程计算中经常需要进行大量的递推形式的计算．例如,采用隐式或半隐式格式计算偏微分方程的数值解时问题通常转化为线性方程组的求解．在实际工程计算中目前依然大量采用Ｇａｕｓｓ－Ｓｅｉｄｅｌ或ＳＯＲ类型的选代法．又如,采用ＡＤＩ［１］方法对偏微分方程进行离散,则往往归结为一组带状线性方程组的求解．一般认为在分布式并行系统上,这种速推类型的计算较难实现且并行效率不理想．近年来为了解决这一问题,人们对算法从各种角度进行改造以避免递推形式的计算,如将Ｇａｕｓｓ－Ｓｅｉｄｅｌ类型迭代改为Ｊａｃｏ…     

Explicit group over-relaxation methods for solving elliptic partial differential equations

Previous block (or line) iterative methods have been implicit in nature where a group of equations (or points on the grid mesh) are treated implicitly [2] and solved directly by a specialised algorithm, this has become the standard technique for solving the sparse linear systems derived from the discretisation of self-adjoint elliptic partial differential equations by finite difference/element techniques.The aim of this paper is to show that if a small group of points (i.e. 2, 4, 9, 16 or 25 point group) is chosen then each group can easily be initially inverted leading to a new class of Group Explicit iterative methods. A comparison with the usual 1-line and 2-line block S.O.R. schemes for the model problem confirm the new techniques to be computationally superior.     

Parallel simulation of anisotropic diffusion with human brain DT-MRI Data

We conduct simulations for the 3D unsteady state anisotropic diffusion process with DT-MRI data in the human brain by discretizing the governing diffusion equation on Cartesian grid and adopting a high performance differential-algebraic equation (DAE) solver, the parallel version of implicit differential-algebraic (IDA) solver, to tackle the resulting large scale system of DAEs. Parallel preconditioning techniques including sparse approximate inverse and banded-block-diagonal preconditioners are used with the GMRES method to accelerate the convergence rate of the iterative solution. We then investigate and compare the efficiency and effectiveness of the two parallel preconditioners. The experimental results of the diffusion simulations on a parallel supercomputer show that the sparse approximate inverse preconditioning strategy, which is robust and efficient with good scalability, gives a much better overall performance than the banded-block-diagonal preconditioner.     

Improved solution methods for inelastic rate problems

The objective of the paper is an assessment of the incremental solution methods for the analysis of inelastic rate problems. In particular, the possibilities of the initial load method are explored to improve the accuracy and stability of the traditional explicit operators by higher-order time expansions and implicit weighting schemes.The convergence limitations are examined for different classes of inelastic growth laws (viscous flow, viscoelasticity, viscoplasticity) which restrict the time step because of the iterative solution of the implicit algorithm. The range and rate of convergence of the initial load method (constant stiffness predictor-corrector iteration) is enlarged by tangential gradient techniques which account for the inelastic response in the structural stiffness matrix. In this way the time step restriction disappears although at a considerable increase of computational expense because of the costly computation and decomposition of structural gradients within each iteration cycle (Newton-Raphson methods).As compared to the linear single-step methods, the cubic Hermitian time expansions furnish far better accuracy than the traditional linear expansions for very little increase of computational cost. Stability and convergence limits correspond to those of the lower-order operators, whereby the implicit midstep of backward weighting schemes are most advantageous. In this context it is worth noting that aging or strain-hardening effects in the inelastic growth law reduce dramatically the time step restrictions of the iterative initial load solution methods (predictor-corrector schemes), as compared to the simplest creep model in which the inelastic growth law depends only on stress, e.g. for viscous flow and viscoplasticity.     

Three-dimensional reconstruction of cellular structures by electron microscope tomography and parallel computing

Electron microscope tomography has emerged as the leading technique for structure determination of cellular components with a resolution of a few nanometers, opening up exciting perspectives for visualizing the molecular architecture of the cytoplasm. This work describes and analyzes the parallelization of tomographic reconstruction algorithms for their application in electron microscope tomography of cellular structures. Efficient iterative algorithms that are characterized by a fast convergence rate have been used to tackle the image reconstruction problem. The use of smooth basis functions provides the reconstruction algorithms with an implicit regularization mechanism, very appropriate for highly noisy conditions such as those present in high-resolution electron tomographic studies. Parallel computing techniques have been applied so as to face the computational requirements demanded by the reconstruction of large volumes. An efficient domain decomposition scheme has been devised that leads to a parallel approach with capabilities of interprocessor communication latency hiding. The combination of efficient iterative algorithms and parallel computing techniques have proved to be well suited for the reconstruction of large biological specimens in electron tomography, yielding solutions in reasonable computational times. This work concludes that parallel computing will be the key to afford high-resolution structure determination of cells, so that the location of molecular signatures in their native cellular context can be made a reality.     

Computations with inverse Runge-Kutta schemes

A new subclass of schemes is considered formally reduced to the class of fully implicit Runge-Kutta schemes possessing outstanding accuracy and stability characteristics. The implementation details of the iterative algorithm for solving stiff systems of ODE and differential-algebraic systems of index 1 by means of the proposed schemes are given.     

Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton–Jacobi–Bellman Equations

We analyse two practical aspects that arise in the numerical solution of Hamilton–Jacobi–Bellman equations by a particular class of monotone approximation schemes known as semi-Lagrangian schemes. These schemes make use of a wide stencil to achieve convergence and result in discretization matrices that are less sparse and less local than those coming from standard finite difference schemes. This leads to computational difficulties not encountered there. In particular, we consider the overstepping of the domain boundary and analyse the accuracy and stability of stencil truncation. This truncation imposes a stricter CFL condition for explicit schemes in the vicinity of boundaries than in the interior, such that implicit schemes become attractive. We then study the use of geometric, algebraic and aggregation-based multigrid preconditioners to solve the resulting discretised systems from implicit time stepping schemes efficiently. Finally, we illustrate the performance of these techniques numerically for benchmark test cases from the literature.     

Parallel iterative methods for parabolic equations

For the problems of the parabolic equations in one- and two-dimensional space, the parallel iterative methods are presented to solve the fully implicit difference schemes. The methods presented are based on the idea of domain decomposition in which we divide the linear system of equations into some non-overlapping sub-systems, which are easy to solve in different processors at the same time. The iterative value is proved to be convergent to the difference solution resulted from the implicit difference schemes. Numerical experiments for both one- and two-dimensional problems show that the methods are convergent and may reach the linear speed-up.     

对抛物方程使用新显格式的区域分解算法

我们提出了两个具有改进稳定性限制条件的新显格式．与经典显格式相比,稳定性限制条 件分别对两维抛物问题放宽了4倍,对一维问题放宽了2倍,同时它的精度与经典全隐格式 的相同．然后,我们通过在内边界点使用大步长的这种新显格式,在内点使用全隐格式,设计 了一个有限差分区域分解算法,稳定性限制条件分别对一维抛物问题放宽了2m2倍,对二维 问题放宽了4m2倍．从而我们能使用一个大的时间步长,这使我们在并行求解抛物问题时能 节省大量的计算量．     

Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.     

Stabilizing large‐scale generalized systems on parallel computers using multithreading and message‐passing

We discuss the parallelization of an efficient algorithm for the partial stabilization of large‐scale linear control systems in generalized state‐space form. The algorithm is composed of highly parallel iterative schemes that appear in the computation of certain matrix functions. Here we evaluate different approaches to exploit parallelism at two levels, based on threads and processes. Our experimental results on a cluster of symmetric multiprocessors and a CC‐NUMA platform show that the efficiency of the matrix operations underlying the iterative schemes carry over to the parallel implementation of the stabilization algorithm. Copyright © 2006 John Wiley & Sons, Ltd.