基于MPI的Jacobi迭代算法的并行化

Jacobi迭代算法是解线性方程组的最常用的方法,具有广泛的应用。Jacobi迭代属于计算密集型[1],将并行计算技术应用到Jacobi迭代中,具有重要的意义。通过使用消息传递编程模型mpi提供的向量数据类型和虚拟进程拓扑来实现Jacobi迭代的并行化。     

基于FPGA的Jacobi迭代求解器研究

针对特定的数值算法进行硬件加速是当前体系结构的趋势之一。Jacobi迭代是典型的数值迭代算法,针对软件Jacobi迭代求解器性能慢,实时性差的缺点,在FPGA硬件平台上设计和实现了硬件Jacobi迭代求解器。求解器采用高度并行、流水的数据通路和优化的归约电路设计,充分利用了Jacobi迭代本身固有的并行性和FPGA的并发式结构,有效地提升求解器的性能。实验结果表明,Jacobi求解器具有良好的可扩展性和较高的计算性能。     

一种用于模拟电路仿真的改进型迭代算法

模拟电路的仿真问题最终归结为对线性代数方程组的求解。利用分块化方法可以降低求解过程中Jacobi矩阵的维数,从而有效降低求解时间。如何降低求解线性方程组的迭代次数,是有效降低求解时间的另一重要问题。首先详细分析了用于求解模拟电路代数方程中Jacobi矩阵的划分问题,然后提出一种改进的隐式迭代方法。最后,通过实验分析了算法中内迭代次数Iin对总迭代次数的影响,该结论对提高整体加速比具有指导意义。     

基于Jacobi方法的线性离散时间系统的迭代学习控制

提出线性离散时间系统基于Jacobi方法的迭代学习控制问题.通过构建线性迭代学习控制问题与线性方程组之间的联系,将Jacobi方法引入到迭代学习控制中,并由此构建得到迭代学习控制律.借助于矩阵运算,证明这种学习律能使得系统的输出跟踪误差经有限次迭代后为零.数值例子说明了算法的可适用性.     

线性方程组迭代法在流处理器上的映射与分析

斯坦福大学的Imagine流处理器具有很强的计算能力，如何将该体系结构应用在科学计算领域是当前研究的热点。解线性方程组的迭代法在工程和科学计算的各个领域中有着十分广泛的应用，该算法具有较好的计算密集性和并行性，十分适合流处理器的计算模型。本文分别针对系数矩阵的规模大小和稠密程度，介绍了Jacobi和Seidel迭代在流处
理器上的映射。实验结果表明，迭代算法能高效地开发Imagine的计算能力，取得较高的性能加速。     

拟牛顿法在航空发动机特性仿真中的应用

航空发动机特性仿真中常用牛顿迭代法求解非线性方程组,牛顿法每一步迭代计算都需要计算Jacobi矩阵,这需要多次发动机气动热力过程计算.因此避免大量重复计算Jacobi矩阵可以减少发动机计算整机的气动热力计算次数,从而提高发动机特性计算的速度.文中采用基于Broyden原理的拟牛顿法求解发动机非线性方程组,这种方法可以直接求出第一步迭代后的Jacobi矩阵,从而大幅度提高计算速度.将拟牛顿法应用于某型涡喷和涡扇发动机特性计算,通过分析计算结果,证明了采用拟牛顿法可以提高发动机特性模拟的计算速度.     

预条件的平方Smith法求解大型Sylvester矩阵方程

提出了一种预条件的平方Smith算法求解大型连续Sylvester矩阵方程,该算法利用交替方向隐式迭代(ADI)来构造预条件算子,将原方程转换为非对称Stein方程,并在Krylov子空间中应用平方Smith法迭代产生低秩逼近解。数值实验表明,与已知的Jacobi迭代法等算法相比,该算法有更好的迭代效率和收敛精度。     

基于PVM的Jacobi迭代算法的设计与实现

本文设计并实现了基于PVM的Jacobi迭代算法,在Linux操作系统上构建PVM系统进行验证,获得了较理想的加速比。     

在Excel中应用迭代法求解线性方程组——雅可比（Jacobi）和塞德尔（Seidel）迭代法

本文通用Excel软件计算《实用数值分析》中的线性方程组的迭代解法中的雅可比(Jacobi)迭代法和塞德尔(Seidel)迭代法,说明用Excel计算线性方程组的迭代解法是完全可行的     

求解一类边界问题的最小曲面的数值并行算法

本文给出了具有最小面积约束的一类边界问题的数值求法，同时实现了该算法的并行化，在算例中，介绍了利用Jacobi迭代求解曲顶柱体顶面面积最小值的一种并行算法，并阐述了解决这一问题的实际意义，算例结果表明，该并行算法的并行效率令人满意。     

A stencil of the finite-difference method for the 2D convection diffusion equation and its new iterative scheme

The paper gives the numerical stencil for the two-dimensional convection diffusion equation and the technique of elimination, and builds up the new iterative scheme to solve the implicit difference equation. The scheme's convergence and its higher rate of convergence than the Jacobi iteration are proved. And the numerical example indicates that the new scheme has the same parallelism and a higher rate of convergence than the Jacobi iteration.     

雅可比迭代法在图形处理器上实现的研究

雅可比迭代法是求解大型线性方程组的基本方法。利用GPU（Graphics Processing Unit,图形处理器）的并行处理能力,将雅可比迭代求解线性方程组过程中运算量较大的部分移植到GPU上执行,以提高运算速度。并分析了影响运算速度的两个因素：CPU-GPU数据交换和共享变量的访问;实验结果表明采用单个thread访问共享变量判断迭代是否收敛时,线性方程组的阶数为500,速度可以提高45倍以上。     

ISE-optimal nonminimum-phase compensation for nonlinear processes

This work concerns the optimal regulation of single-input–single-output nonminimum-phase nonlinear processes. The problem of calculation of an ISE-optimal, statically equivalent, minimum-phase output for nonminimum-phase compensation is formulated using Hamilton–Jacobi theory and the normal form representation of the nonlinear system. A Newton–Kantorovich iteration is developed for the solution of the pertinent Hamilton–Jacobi equations, which involves solving a Zubov equation at each step of the iteration. The method is applied to the problem of controlling a nonisothermal CSTR with Van de Vusse kinetics, which exhibits nonminimum-phase behaviour.     

Reducing the number of multiplikations in iterative processes

Summary In any iteration scheme, such as v ^k=f(Qv ^k–1), where a fixed matrix multiplies a vector that depends on the iteration number, Winograd's method for computing inner products can be used in a straightforward manner to reduce the number of multiplications required at the cost of more additions. The key observation is that certain quantities required by Winograd's method have to be computed only at the first iteration. In the Jacobi method for solving systems of linear equations, f is linear. Gauss-Seidel iteration often converges faster than Jacobi iteration, but it cannot be put in the above form. A simple trick is necessary to apply Winograd's method in an efficient recursive manner. Our proposed method is better than the naive method when it is faster to add than to multiply. Versions of Jacobi and Gauss-Seidel iteration appropriate for optimization (as in Markov decision problems) are presented. The analysis specializes easily to the linear equation case.This research was supported by NRC grant A8565.     

Iterative Path Integral Approach to Nonlinear Stochastic Optimal Control Under Compound Poisson Noise

Nonlinear stochastic optimal control theory has played an important role in many fields. In this theory, uncertainties of dynamics have usually been represented by Brownian motion, which is Gaussian white noise. However, there are many stochastic phenomena whose probability density has a long tail, which suggests the necessity to study the effect of non‐Gaussianity. This paper employs Lévy processes, which cause outliers with a significantly higher probability than Brownian motion, to describe such uncertainties. In general, the optimal control law is obtained by solving the Hamilton–Jacobi–Bellman equation. This paper shows that the path‐integral approach combined with the policy iteration method is efficiently applicable to solve the Hamilton–Jacobi–Bellman equation in the Lévy problem setting. Finally, numerical simulations illustrate the usefulness of this method.     

Matrix iterative methods for structural reanalysis

The theory of linear, stationary, norm-reducing type iterations for the solution of linear, simultaneous equations is briefly reviewed and the genesis of simple iterition, Jacobi iteration and Gauss-Seidel iteration is shown to be the consequence of ‘splitting’ the coefficient matrix in different ways. For positive definite, sparse matrices arising in structural applications, block Gauss-Seidel iteration is shown to be effective for both reanalysis and initial analysis, through its influence as a norm-reducing aid which results in more pronounced ‘diagonal dominance’ and a better initial choice of starting vector. A numerical example is used to show the effectiveness of the method.     

The finite-horizon optimal control for a class of time-delay affine nonlinear system

In this paper, a new iteration algorithm is proposed to solve the finite-horizon optimal control problem for a class of time-delay affine nonlinear systems with known system dynamic. First, we prove that the algorithm is convergent as the iteration step increases. Then, a theorem is presented to demonstrate that the limit of the iteration performance index function satisfies discrete-time Hamilton–Jacobi–Bellman (DTHJB) equation, and the finite-horizon iteration algorithm is presented with satisfactory accuracy error. At last, two neural networks are used to approximate the iteration performance index function and the corresponding control policy. In simulation part, an example is given to demonstrate the effectiveness of the proposed iteration algorithm.