预条件处理CG法大规模电力系统潮流计算

研究了预条件处理的CG(ConjugateGradient)法求解大规模电力系统潮流方程的问题。采用预处理CG法代替传统的LU直接法对高维稀疏潮流方程进行求解,详细比较各种预条件处理技术对CG法潮流方程求解的效果,提出一种新的节点优化排序的IncompleteCholesky预处理方法,实验分析证明它是CG法快速求解潮流的一种十分有效的预处理方法。对IEEE-30、IEEE-118和多个合成的大规模电力系统进行潮流计算,结果表明:这种预处理方法比其它预处理方法需要更少的迭代次数和浮点运算次数,对超大规模电力系统潮流问题也比传统LU直接法更具速度和存储优势。在电力系统互联程度不断增加使其潮流计算面临大规模甚至超大规模计算压力时,该方法能够成为传统方法的一个替代。     

邻居预条件加速的多层快速非均匀平面波算法

采用邻居预条件加速的多层快速非均匀平面波算法求解三维导电目标的电磁散射.通过分组,将耦合划分为附近和非附近区,对于非附近区采用索末菲恒等式对格林函数展开,用修正最陡下降路径代替索末菲积分路径进行数值积分.采用内插与外推技术将复角谱序列转换成均匀实角谱序列,以便于算法的高效实施.该算法的计算复杂度与多层快速多极子相当,且更具潜在优势.为改善迭代特性,本文研究了一种邻居预条件方法,加速迭代收敛,数值结果验证了算法的准确和高效.     

Block constrained versus generalized Jacobi preconditioners for iterative solution of large-scale Biot’s FEM equations

Generalized Jacobi (GJ) diagonal preconditioner coupled with symmetric quasi-minimal residual (SQMR) method has been demonstrated to be efficient for solving the 2 × 2 block linear system of equations arising from discretized Biot’s consolidation equations. However, one may further improve the performance by employing a more sophisticated non-diagonal preconditioner. This paper proposes to employ a block constrained preconditioner P_c that uses the same 2 × 2 block matrix but its (1, 1) block is replaced by a diagonal approximation. Numerical results on a series of 3-D footing problems show that the SQMR method preconditioned by P_c is about 55% more efficient time-wise than the counterpart preconditioned by GJ when the problem size increases to about 180,000 degrees of freedom. Over the range of problem sizes studied, the P_c-preconditioned SQMR method incurs about 20% more memory than the GJ-preconditioned counterpart. The paper also addresses crucial computational and storage issues in constructing and storing P_c efficiently to achieve superior performance over GJ on the commonly available PC platforms.     

Two augmentation preconditioners for nonsymmetric and indefinite saddle point linear systems with singular (1, 1) blocks

In this paper, two preconditioners based on augmentation are introduced for the solution of large saddle point-type systems with singular (1, 1) blocks. We study the spectral characteristics of the preconditioners, show that all eigenvalues of preconditioned matrices are strongly clustered. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.     

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared with preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this article, several techniques are applied to monolithic two-level generalized Dryja-Smith-Widlund (GDSW) preconditioners to further improve the convergence speed and the computing time. In particular, reduced dimension GDSW coarse spaces, restricted and scaled versions of the first level, hybrid, and parallel coupling of the levels, and recycling strategies are investigated. Using a combination of all these improvements, for a small time-dependent Navier-Stokes problem on 240 message passing interface (MPI) ranks, a reduction of 86% of the time-to-solution can be obtained. Even without applying recycling strategies, the time-to-solution can be reduced by more than 50% for a larger steady Stokes problem on 4608 MPI ranks. For the largest problems with 11 979 MPI ranks, the scalability deteriorates drastically for the monolithic GDSW coarse space. On the other hand, using the reduced dimension coarse spaces, good scalability up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, could be achieved.     

A symmetric positive definite preconditioner for saddle-point problems

For the classical saddle-point problem, we present precisely two intervals containing the positive and the negative eigenvalues of the preconditioned matrix, respectively, when the inexact version of the symmetric positive definite preconditioner introduced in Section 2.1 of Gill et al. [Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl. 13 (1992), pp. 292–311] is employed. The model of Stokes problem is used to test the effectiveness of the presented bounds as well as the quality of the symmetric positive definite preconditioner.     

Computing the null space of finite element problems

We present a method for computing the null space of finite element models, including models with equality constraints. The method is purely algebraic; it requires access to the element matrices, but not to the geometry or material properties of the model.Theoretical considerations show that under certain conditions, both the amount of computation and the amount of memory required by our method scale linearly with model size; memory scales linearly but computation scales quadratically with the dimension of the null space. Our experiments confirm this: the method scales extremely well on 3-dimensional model problems. In general, large industrial models do not satisfy all the conditions that the theoretical results assume; however, experimentally the method performs well and outperforms an established method on industrial models, including models with many equality constraints.The accuracy of the computed null vectors is acceptable, but the method is usually less accurate than a more naive (and computationally much more expensive) method.     

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 efficient solver for the RANS equations and a one-equation turbulence model

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, this scheme has been used to solve the compressible Reynolds-averaged Navier–Stokes (RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equations and the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the turbulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational results are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.     

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。