粘性Burgers方程的高阶隐式WCNS格式

JFNK (Jacobian-free Newton-Krylov)方法是由外层Newton迭代法和内层Krylov子空间迭代法构成的嵌套迭代方法.本文提出了一种基于JFNK方法的高阶隐式WCNS (weighted compact nonlinear scheme)格式,并用于求解一维、二维粘性Burgers方程.外层迭代法采用含参数的多步Newton迭代法,给出了收敛性分析,内层迭代法采用无矩阵GMRES迭代法.粘性Burgers方程的非线性对流项采用五阶WCNS格式计算.为提高方法精度和计算效率,时间离散采用三阶隐式的DIRK (diagonal implicit Runge-Kutta)方法.数值结果表明基于JFNK方法的隐式WCNS格式在时间上能达到三阶精度,与显式TVD Runge-Kutta WCNS方法相比,计算效率更高.此外,基于JFNK方法的隐式WCNS格式稳定性好,且具有良好的激波捕捉能力.     

应用预处理AWE技术快速计算导体目标宽带RCS

应用渐近波形估计技术计算目标宽带雷达散射截面（RCS）,可有效提高计算效率。然而当目标为电大尺寸时,阻抗矩阵求逆运算将十分耗时,甚至无法计算。提出使用Krylov子空间迭代法取代矩阵逆来求解大型矩阵方程,应用双门槛不完全LU分解预处理技术降低迭代求解所需的迭代次数。数值计算表明,该方法结果与矩量法逐点求解结果吻合良好,并且计算效率大大提高。     

ILU预处理Newton-Krylov方法的潮流计算

由于电力系统修正方程组具有高维、稀疏的特点,本文提出将预处理Krylov子空间方法应用于潮流修正方程组的求解,形成预处理Newton-Krylov的潮流计算方法。结合ILU预处理方法,比较了最常用的3类Newton-Krylov方法求解潮流方程的计算效果。通过对 IEEE30、IEEE118、IEEE300 和3个Poland大规模电力系统进行潮流计算,结果表明：3类Newton-Krylov方法是电力系统潮流计算的有效方法,呈现出良好的收敛特性和计算效率。     

并行求解多维递归方程组的三种Krylov子空间迭代方法

多维递归方程组在并行求解时存在串并行不一致问题,提供三种Krylov子空间迭代求解方法———PCG/ATCG和GMRES来解决这一问题,并采用典型算例对这三种Krylov子空间迭代方法进行正确性验证和加速比测试.试验表明这三种Krylov子空间迭代法在并行规模较大的情况下,均能够正确求解多维递归方程组,并且加速特性良好.     

Krylov子空间方法及其并行计算

Krylov子空间方法在提高大型科学和工程计算效率上起着重要作用。本文阐述了Krylov子空间方．法产生的背景、Krylov子空间方法的分类,在此基础上,研完了分布式并行计算环境下Krylov子空间方法的并行计算方法,给出了Krylov子空间方法的并行化策略。     

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Matlab的解非线性方程算法设计

研究基于Matlab的非线性方程数值解法的算法设计。通过对Newton迭代法、二分法、弦位法分析研究得到其一般算法流程图,给出相应的Matlab程序和计算实例,比较它们的计算效率。这对在工程计算等领域广泛运用的非线性方程数值求解问题有一定启发意义。     

基于GaBP算法的快速潮流计算方法

研究在潮流迭代求解过程中雅可比矩阵方程组的迭代求解方法及其收敛性。首先利用PQ分解法进行潮流迭代求解,并针对求解过程中雅可比矩阵对称且对角占优的特性,对雅可比矩阵方程组采用高斯置信传播算法(GaBP)进行求解,再结合Steffensen加速迭代法以提高GaBP算法的收敛性。对IEEE118、IEEE300节点标准系统和两个波兰互联大规模电力系统进行仿真计算后结果表明：随着系统规模的增长,使用Steffensen加速迭代法进行加速的GaBP算法相对于基于不完全LU的预处理广义极小残余方法(GMRES)具有更好的收敛性,为大规模电力系统潮流计算的快速求解提供了一种新思路。     

基于牛顿-拉夫逊法和P-Q分解法的配网潮流计算联合迭策略

为提高配网线路潮流计算的快速性和收敛精度,本文通过对比分析牛顿-拉夫逊法和P-Q分解法在不同配网场景下的优点和局限性,提出结合两种算法共同提高计算效率和精度的潮流计算联合策略,以期提高大规模复杂配电网的潮流计算的准确性和实用性。以IEEE9和IEEE30节点为例,对比分析牛顿法、P-Q分解法及联合迭代策略法的迭代次数和计算时长,研究发现,对于大规模电网而言,采用联合迭代策略,潮流计算的效率较高。     

含分段下垂控制的柔性交直混联系统潮流计算统一表达研究

针对不同控制方式的柔性交直混联系统进行潮流计算的统一表达研究具有一定意义。为此,本文提出了一种考虑分段下垂控制方式的统一潮流表达式。首先,讨论了各类控制方式下的换流站潮流控制模型,其次,引入阶跃函数列写合成的分段下垂控制方式的潮流表达,然后,建立了直流系统针对多种控制方式的统一表达式;最后,基于潮流交替迭代法,借助MATLAB仿真平台,在一个修改后的含七端柔性直流电网的IEEE 57节点交直混联输电系统上,验证了所提表达式的有效性。     

Optimized Schwarz method without overlap for the gravitational potential equation on cluster of graphics processing unit

Many engineering and scientific problems need to solve boundary value problems for partial differential equations or systems of them. For most cases, to obtain the solution with desired precision and in acceptable time, the only practical way is to harness the power of parallel processing. In this paper, we present some effective applications of parallel processing based on hybrid CPU/GPU domain decomposition method. Within the family of domain decomposition methods, the so-called optimized Schwarz methods have proven to have good convergence behaviour compared to classical Schwarz methods. The price for this feature is the need to transfer more physical information between subdomain interfaces. For solving large systems of linear algebraic equations resulting from the finite element discretization of the subproblem for each subdomain, Krylov method is often a good choice. Since the overall efficiency of such methods depends on effective calculation of sparse matrix–vector product, approaches that use graphics processing unit (GPU) instead of central processing unit (CPU) for such task look very promising. In this paper, we discuss effective implementation of algebraic operations for iterative Krylov methods on GPU. In order to ensure good performance for the non-overlapping Schwarz method, we propose to use optimized conditions obtained by a stochastic technique based on the covariance matrix adaptation evolution strategy. The performance, robustness, and accuracy of the proposed approach are demonstrated for the solution of the gravitational potential equation for the data acquired from the geological survey of Chicxulub crater.     

Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi–Davidson eigensolver

In the numerical solution of large‐scale eigenvalue problems, Davidson‐type methods are an increasingly popular alternative to Krylov eigensolvers. The main motivation is to avoid the expensive factorizations that are often needed by Krylov solvers when the problem is generalized or interior eigenvalues are desired. In Davidson‐type methods, the factorization is replaced by iterative linear solvers that can be accelerated by a smart preconditioner. Jacobi–Davidson is one of the most effective variants. However, parallel implementations of this method are not widely available, particularly for non‐symmetric problems. We present a parallel implementation that has been included in SLEPc, the Scalable Library for Eigenvalue Problem Computations, and test it in the context of a highly scalable plasma turbulence simulation code. We analyze its parallel efficiency and compare it with a Krylov–Schur eigensolver. Copyright © 2011 John Wiley & Sons, Ltd.     

Modelling variable density flow problems in heterogeneous porous media using the method of lines and advanced spatial discretization methods

Modelling variable density flow problems under heterogeneous porous media conditions requires very long computation time and high performance equipments. In this work, the DASPK solver for temporal resolution is combined with advanced spatial discretization schemes in order to improve the computational efficiency while maintaining accuracy.The spatial discretization is based on a combination of Mixed Finite Element (MFE), Discontinuous Galerkin (DG) and Multi-point Flux Approximation methods (MPFA). The obtained non-linear ODE/DAE system is solved with the Method of Lines (MOL) using the DASPK time solver. DASPK uses the preconditioned Krylov iterative method to solve linear systems arising at each time step.Precise laboratory-scale 2D experiments were conducted in a heterogeneously packed porous medium flow tank and the measured concentration contour lines are used to evaluate the numerical model. Simulations show the high efficiency and accuracy of the code and the sensitivity analysis confirms the density dependence of dispersion.     

求解二维三温能量方程的半粗化代数多重网格法

§1.引言 二维三温辐射流体动力学方程组的求解是数值模拟的重要组成部分,而求解能量方程是一个十分重要的环节,而且在整个系统的计算中,能量方程求解所占的机时比重相当大(约80％以上)。因此,寻求一个收敛快、稳定性好的二维三温能量方程数值解法是一个值得探讨     

Updating preconditioner for iterative method in time domain simulation of power systems

The numerical solution of the differential-algebraic equations(DAEs) involved in time domain simulation(TDS) of power systems requires the solution of a sequence of large scale and sparse linear systems.The use of iterative methods such as the Krylov subspace method is imperative for the solution of these large and sparse linear systems.The motivation of the present work is to develop a new algorithm to efficiently precondition the whole sequence of linear systems involved in TDS.As an improvement of dishon...     

Auto-tuned Krylov methods on cluster of graphics processing unit

Exascale computers are expected to have highly hierarchical architectures with nodes composed by multiple core processors (CPU; central processing unit) and accelerators (GPU; graphics processing unit). The different programming levels generate new difficult algorithm issues. In particular when solving extremely large linear systems, new programming paradigms of Krylov methods should be defined and evaluated with respect to modern state of the art of scientific methods. Iterative Krylov methods involve linear algebra operations such as dot product, norm, addition of vectors and sparse matrix–vector multiplication. These operations are computationally expensive for large size matrices. In this paper, we aim to focus on the best way to perform effectively these operations, in double precision, on GPU in order to make iterative Krylov methods more robust and therefore reduce the computing time. The performance of our algorithms is evaluated on several matrices arising from engineering problems. Numerical experiments illustrate the robustness and accuracy of our implementation compared to the existing libraries. We deal with different preconditioned Krylov methods: Conjugate Gradient for symmetric positive-definite matrices, and Generalized Conjugate Residual, Bi-Conjugate Gradient Conjugate Residual, transpose-free Quasi Minimal Residual, Stabilized BiConjugate Gradient and Stabilized BiConjugate Gradient (L) for the solution of sparse linear systems with non symmetric matrices. We consider and compare several sparse compressed formats, and propose a way to implement effectively Krylov methods on GPU and on multicore CPU. Finally, we give strategies to faster algorithms by auto-tuning the threading design, upon the problem characteristics and the hardware changes. As a conclusion, we propose and analyse hybrid sub-structuring methods that should pave the way to exascale hybrid methods.     

On optimality of approximate low rank solutions of large-scale matrix equations

In this paper, we discuss some optimality results for the approximation of large-scale matrix equations. In particular, this includes the special case of Lyapunov and Sylvester equations, respectively. We show a relation between the iterative rational Krylov algorithm and a Riemannian optimization method which recently has been shown to locally minimize a certain energy norm of the underlying Lyapunov operator. Moreover, we extend the results for a more general setting leading to a slight modification of IRKA. By means of some numerical test examples, we show the efficiency of the proposed methods.     

GMRES implementations and residual smoothing techniques for solving ill-posed linear systems

There are verities of useful Krylov subspace methods to solve nonsymmetric linear system of equations. GMRES is one of the best Krylov solvers with several different variants to solve large sparse linear systems. Any GMRES implementation has some advantages. As the solution of ill-posed problems are important. In this paper, some GMRES variants are discussed and applied to solve these kinds of problems. Residual smoothing techniques are efficient ways to accelerate the convergence speed of some iterative methods like CG variants. At the end of this paper, some residual smoothing techniques are applied for different GMRES methods to test the influence of these techniques on GMRES implementations.