Block preconditioners with circulant blocks for general linear systems

Block preconditioner with circulant blocks (BPCB) has been used for solving linear systems with block Toeplitz structure since 1992 [R. Chan, X. Jin, A family of block preconditioners for block systems, SIAM J. Sci. Statist. Comput. (13) (1992) 1218–1235]. In this new paper, we use BPCBs to general linear systems (with no block structure usually). The BPCBs are constructed by partitioning a general matrix into a block matrix with blocks of the same size and then applying T. Chan’s optimal circulant preconditioner [T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput. (9) (1988) 766–771] to each block. These BPCBs can be viewed as a generalization of T. Chan’s preconditioner. It is well-known that the optimal circulant preconditioner works well for solving some structured systems such as Toeplitz systems by using the preconditioned conjugate gradient (PCG) method, but it is usually not efficient for solving general linear systems. Unlike T. Chan’s preconditioner, BPCBs used here are efficient for solving some general linear systems by the PCG method. Several basic properties of BPCBs are studied. The relations of the block partition with the cost per iteration and the convergence rate of the PCG method are discussed. Numerical tests are given to compare the cost of the PCG method with different BPCBs.     

An extended basis inexact shift–invert Lanczos for the efficient solution of large-scale generalized eigenproblems

This paper proposes a technique, based on the Inexact Shift–Invert Lanczos (ISIL) method with Inexact Jacobi Orthogonal Component Correction (IJOCC) refinement, and a preconditioned conjugate-gradient (PCG) linear solver with multilevel preconditioner, for finding several eigenvalues for generalized symmetric eigenproblems. Several eigenvalues are found by constructing (with the ISIL process) an extended projection basis. Presented results of numerical experiments confirm the technique can be effectively applied to challenging, large-scale problems characterized by very dense spectra, such as resonant cavities with spatial dimensions which are large with respect to wavelengths of the resonating electromagnetic fields. It is also shown that the proposed scheme based on inexact linear solves delivers superior performance, as compared to methods which rely on exact linear solves, indicating tremendous potential of the ‘inexact solve’ concept. Finally, the scheme which generates an extended projection basis is found to provide a cost-efficient alternative to classical deflation schemes when several eigenvalues are computed.     

Comparison of Two-Level Preconditioners Derived from Deflation,Domain Decomposition and Multigrid Methods

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.     

A multilevel hierarchical preconditioner for thin elastic solids

In this study, a multilevel, recursively defined preconditioner, for use with the Preconditioned Conjugate Gradient (PCG) algorithm in connection with the finite element analysis of elastostatics is developed. The preconditioner is constructed from a sequence of hierarchical vector spaces arising from the p-version of the finite element method. Results from parametric studies evaluating the effects of skewed elements, orthotropic material properties, and extreme span ratios, for p=2 and 3 are given. The results indicate that the preconditioner may be used to produce an efficient solver. The efficiency of the iterative procedure is illustrated using thin elastic solids. Results indicate that the preconditioner developed herein can be used to produce an efficient iterative solver for two- and three-dimensional problems in structural mechanics. © 1998 John Wiley & Sons, Ltd.     

Using recursion to compute the inverse of the genomic relationship matrix

Computing the inverse of the genomic relationship matrix using recursion was investigated. A traditional algorithm to invert the numerator relationship matrix is based on the observation that the conditional expectation for an additive effect of 1 animal given the effects of all other animals depends on the effects of its sire and dam only, each with a coefficient of 0.5. With genomic relationships, such an expectation depends on all other genotyped animals, and the coefficients do not have any set value. For each animal, the coefficients plus the conditional variance can be called a genomic recursion. If such recursions are known, the mixed model equations can be solved without explicitly creating the inverse of the genomic relationship matrix. Several algorithms were developed to create genomic recursions. In an algorithm with sequential updates, genomic recursions are created animal by animal. That algorithm can also be used to update a known inverse of a genomic relationship matrix for additional genotypes. In an algorithm with forward updates, a newly computed recursion is immediately applied to update recursions for remaining animals. The computing costs for both algorithms depend on the sparsity pattern of the genomic recursions, but are lower or equal than for regular inversion. An algorithm for proven and young animals assumes that the genomic recursions for young animals contain coefficients only for proven animals. Such an algorithm generates exact genomic EBV in genomic BLUP and is an approximation in single-step genomic BLUP. That algorithm has a cubic cost for the number of proven animals and a linear cost for the number of young animals. The genomic recursions can provide new insight into genomic evaluation and possibly reduce costs of genetic predictions with extremely large numbers of genotypes.     

基于Beowulf集群的大规模电力系统方程并行PCG求解

研究基于Beowulf集群的大规模电力系统方程并行PCG求解问题,采用常见的硬件设备和廉价且广为传播的软件构建出Beowulf分布式集群环境,基于这一环境,彻底摒弃矩阵的传统直接分解算法,采用多项式预处理的PCG法并行求解大型稀疏线性方程组.文中方法无需进行任何形式的电网络划分,也无需进行任何的矩阵三角分解和前推回代过程,适合各种类型的大规模电力系统方程的并行求解.本文分别在潮流计算、状态估计和静态安全分析中对大规模电力系统方程实现了并行求解,并获得一定的加速比和并行效率,为大规模电力系统快速、准确的仿真计算和在线分析提供一种可行的新途径.     

电磁场分析中大型稀疏方程组迭代解法的改进

针对电磁场数值分析中的大型稀疏对称线性方程组,尤其是求解棱边有限元法生成的奇异方程组,通过时谐涡流场实例计算,比较了目前文献中出现的各种预处理共轭梯度算法,提出了一种改进的预优处理的不完全乔列斯基分解共轭梯度算法,并得出了分别适用于节点有限元与棱边有限元离散方程组的最优预处理共轭梯度算法。最后对非对称方程组的求解进行了讨论。     

基于NetSolve的并行PCG实现及其性能分析

系统地介绍了基于Dell服务器集群的NetSolve系统的体系结构和工作原理。并在该系统上进行了并行实验,采用预条件共轭梯度法（PCG）来求解有限元结构分析中常常需要解的大规模对称正定的线性系统Ax=b,并利用阻塞调用和任务族调用实现了NetSolve并行的PCG,最后对应用的性能进行了分析。     

基于无线传输的心电心音监护系统设计

针对现有心电心音监测仪覆盖性不足、时效性差等问题,提出了一种基于无线传输的心电(ECG)心音(PCG)监护系统设计.本系统设计有64个监护节点,各节点由传感模块、调理采集模块、存储模块、主控管理模块、通信模块、电源管理模块组成,各节点可对12导联心电信号、HKY-06B心音信号进行同步采集传输,将信息汇总给监护中心并及...     

GridSolve系统中的任务迁移

描述了NetSolve系统的基本结构,及在此系统基础上发展出的网格系统GridSolve。重点讲述了在NetSolve代理中加入迁移者这个新的组件及适应性资源选择机制,讨论了怎样使用它来获取自动任务迁移到有更好资源的地方。描述的应用迁移对于NetSolve中代理的实现是非常有趣而具有挑战性的问题。