共查询到19条相似文献，搜索用时 296 毫秒
1.
Lagrangian乘子区域分解法的一类预条件子 总被引：3，自引：2，他引：1
1．引言非重叠区域分解的Lagrangian乘子法已被许多作者讨论［1今它是一类非协调区域分解法（与通常的非协调元区域分解不同）,特别适合于非匹配网格的情形（即相邻子域在公共边或公共面上的结点不重合,参见14］［6］）．这种方法的一个最大优点是不要求界面变量在内交点（或内交边）上的连续性,从而界面方程易于建立,程序易于实现,而又正因为这个特点,使得界面矩阵的预条件子不能按通常的方法构造,故目前还未见到理想的预条件子（或者条件数差,或者应用上不方便）．本文在很大程度上解决了这一问题．1）工作单位：湘潭大学数学系… 相似文献
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1 问题的提出一个n阶方阵Tn被称为Toeplitz矩阵,如果它满足 相似文献
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１．引言考虑线性方程组ＴＮｘ＝ｂ（１．１）其中ＴＮ＝（ｔｉ,ｊ）是ＮｘＮ对称正定（ＳＰＤ）Ｔｏｅｐｌｉｔｚ矩阵,即ｔｉ,ｊ＝ｔ｜ｉ－ｊ｜（ｉ,ｊ＝０,１,...,Ｎ－１）且ＴＮ的所有特征值均为正数,并表为ＴＮ：＝Ｔ（ｔ。,ｔｉ,...,ｔＮ－１）．如果我们用预条件子共轭梯度法（ＰＣＧ）求解方程组（１．１）,最关健的任务是构造出高效的预条件子．而预条件子最自然的选择似乎其逆矩阵易求且构成矩阵ＴＮ的某种最优逼近．由于循环矩阵ＣＮ的逆矩阵ＣＲ'仍为循环矩阵,因此ＣＮ和ＣＨ'与向量的乘积可通ｉｓ速Ｆｏｕｒｉｅｒ… 相似文献
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广义鞍点问题的块三角预条件子 总被引：2，自引：2，他引：0
本文对Golub和Yuan(2002)中给出的ST分解推广到广义鞍点问题上,给出了三种块预条件子,并重点分析了其中两种预条件子应用到广义鞍点问题上所得到的对称正定阵,得出了其一般的性质并重点研究了预处理矩阵条件数的上界,最后给出了数值算例. 相似文献
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讨论位移方程组(A α_jI)x(α_j)=b(α_j,x(α_i)),(i相似文献
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椭圆离散方程并行预条件子局部构造算法Ⅱ：非自共轭型方程孙家昶,曹建文（中国科学院软件研究所并行软件研究开发中心）ＡＣＬＡＳＳＯＦＬＯＣＡＬＧＲＥＥＮ－ＬＩＫＥＰＡＲＡＬＬＥＬＰＲＥＣＯＮＤＩＴＩＯＮＥＲＡＬＧＯＲＩＴＨＭＦＯＲＥＬＬＩＰＴＩＣＤＩＳＣ... 相似文献
7.
针对由Galerkin有限元离散椭圆PDE-约束优化问题产生的具有特殊结构的3×3块线性鞍点系统,提出了一个预条件子并给出了预处理矩阵特征值及特征向量的具体表达形式.数值结果表明了该预条件子能够有效地加速Krylov子空间方法的收敛速率,同时也验证了理论结果. 相似文献
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１．引言 近年来,一类新的非重叠区域分解方法一非匹配网格区域分解法,日益引起人们的广泛兴趣,并已成为当今区域分解方法研究的热门课题。这类区域分解方法的特点是：相邻子区域在公共边（或面）上的结点可以不重合,从而能解决许多传统区域分解方法不便解决的问题（如变动网格问题）．目前主要有两类方法来处理这种区域分解的强非协调性：Ｍｏｒｔａｒ无法（见［１－２］和［９－１０］）和拉格朗日乘子法（见［５］,［８］,［１１］和［１２］）．拉格朗日乘子法比Ｍｏｒｔａｒ无法有明显的优点：（１）界面变量（即拉格朗日乘子）… 相似文献
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Factorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree. 相似文献
12.
Numerical solution of the elastic body-plate problem by nonoverlapping domain decomposition type techniques 总被引：2，自引：0，他引：2
Jianguo Huang. 《Mathematics of Computation》2004,73(245):19-34
The purpose of this paper is to provide two numerical methods for solving the elastic body-plate problem by nonoverlapping domain decomposition type techniques, based on the discretization method by Wang. The first one is similar to an older method, but here the corresponding Schur complement matrix is preconditioned by a specific preconditioner associated with the plate problem. The second one is a ``displacement-force' type Schwarz alternating method. At each iteration step of the two methods, either a pure body or a pure plate problem needs to be solved. It is shown that both methods have a convergence rate independent of the size of the finite element mesh.
13.
Qiya Hu 《Advances in Computational Mathematics》2007,26(4):367-401
In this paper, we are concerned with the nonoverlapping domain decomposition method with Lagrange multiplier for three-dimensional
second-order elliptic problems with no zeroth-order term. It is known that the methods result in a singular subproblem on
each internal (floating) subdomain. To handle the singularity, we propose a regularization technique which transforms the
corresponding singular problems into approximate positive definite problems. For the regularized method, one can build the
interface equation of the multiplier directly. We first derive an optimal error estimate of the regularized approximation,
and then develop a cheap preconditioned iterative method for solving the interface equation. For the new method, the cost
of computation will not be increased comparing the case without any floating subdomain. The effectiveness of the new method
will be confirmed by both theoretical analyzes and numerical experiments.
The work is supported by Natural Science Foundation of China G10371129. 相似文献
14.
In this work we propose the use of alternating oblique projections (AOP) for the solution of the saddle points systems resulting from the discretization of domain decomposition problems. These systems are called coupled linear systems. The AOP method is a descent method in which the descent direction is defined by using alternating oblique projections onto the search subspaces. We prove that this method is a preconditioned simple gradient (Uzawa) method with a particular preconditioner. Finally, a preconditioned conjugate gradient based version of AOP is proposed.
AMS subject classification 65F10, 65N22, 65Y05 相似文献
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This paper introduces a robust preconditioner for general sparse matrices based on low‐rank approximations of the Schur complement in a Domain Decomposition framework. In this ‘Schur Low Rank’ preconditioning approach, the coefficient matrix is first decoupled by a graph partitioner, and then a low‐rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface unknowns. The method avoids explicit formation of the Schur complement. We show the feasibility of this strategy for a model problem and conduct a detailed spectral analysis for the relation between the low‐rank correction and the quality of the preconditioner. We first introduce the SLR preconditioner for symmetric positive definite matrices and symmetric indefinite matrices if the interface matrices are symmetric positive definite. Extensions to general symmetric indefinite matrices as well as to nonsymmetric matrices are also discussed. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
16.
非重叠区域分解算法在于建立和求解相关的界面方程．建立界面方程在理论上虽。然容易推导，例如某些问题可用Gauss块消去法，但在实际计算时并不可行，所以界面方程在一些算法中是陷式的．而求解界面方程一般要进行预处理，本提出一种区域分解算法，可得出界面方程的显式表达．算法是完全并行的，所得出的界面方程的系数矩阵的条件数已与网参数无关，事实上就是(Sh^（1）)^-1Sh，进而可直接用收敛速度较快的Chebyshev加速算法求解该界面方程，在充分应用并行计算方法的条件下，本算法与[4]中的算法相比计算效率提高． 相似文献
17.
Time harmonic Maxwell equations in lossless media lead to a second order differential equation for the electric field involving a differential operator that is neither elliptic nor definite. A Galerkin method using Nedelec spaces can be employed to get approximate solutions numerically. The problem of preconditioning the indefinite matrix arising from this method is discussed here. Specifically, two overlapping Schwarz methods will be shown to yield uniform preconditioners.
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In this article, we consider a class of unfitted finite element methods for scalar elliptic problems. These so-called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson interface problem. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The unfitted finite element space is split into two subspaces, where one subspace is the standard finite element space associated to the background mesh and the second subspace is spanned by all cut basis functions corresponding to nodes on the cut elements. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the interface in the triangulation. Based on this we introduce an efficient preconditioner that is uniformly spectrally equivalent to the stiffness matrix. Using a similar splitting, it is shown that the same preconditioning approach can also be applied to a fictitious domain CutFEM discretization of the Poisson equation. Results of numerical experiments are included that illustrate optimality of such preconditioners for the Poisson interface problem and the Poisson fictitious domain problem. 相似文献
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Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l2 norm, however, the optimal convergence result is available only in the energy norm (or the equivalent Sobolev H1 norm). Very little progress has been made in the theoretical understanding of the l2 behaviour of this very successful algorithm. To add to the difficulty in developing a full l2 theory, in this note, we construct explicit examples and show that the optimal convergence of additive Schwarz preconditioned GMRES in l2 cannot be obtained using the existing GMRES theory. More precisely speaking, we show that the symmetric part of the preconditioned matrix, which plays a role in the Eisenstat–Elman–Schultz theory, has at least one negative eigenvalue, and we show that the condition number of the best possible eigenmatrix that diagonalizes the preconditioned matrix, key to the Saad–Schultz theory, is bounded from both above and below by constants multiplied by h?1/2. Here h is the finite element mesh size. The results presented in this paper are mostly negative, but we believe that the techniques used in our proofs may have wide applications in the further development of the l2 convergence theory and in other areas of domain decomposition methods. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献