A new family of global methods for linear systems with multiple right-hand sides

The global bi-conjugate gradient (Gl-BCG) method is an attractive matrix Krylov subspace method for solving nonsymmetric linear systems with multiple right-hand sides, but it often show irregular convergence behavior in many applications. In this paper, we present a new family of global A-biorthogonal methods by using short two-term recurrences and formal orthogonal polynomials, which contain the global bi-conjugate residual (Gl-BCR) algorithm and its improved version. Finally, numerical experiments illustrate that the proposed methods are highly competitive and often superior to originals.     

A new family of global methods for linear systems with multiple right-hand sides

The global bi-conjugate gradient (Gl-BCG) method is an attractive matrix Krylov subspace method for solving nonsymmetric linear systems with multiple right-hand sides, but it often show irregular convergence behavior in many applications. In this paper, we present a new family of global $A$-biorthogonal methods by using short two-term recurrences and formal orthogonal polynomials, which contain the global bi-conjugate residual (Gl-BCR) algorithm and its improved version. Finally, numerical experiments illustrate that the proposed methods are highly competitive and often superior to originals.     

Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems

We prove local a posteriori error estimates for pointwise gradient errors in finite element methods for a second-order linear elliptic model problem. First we split the local gradient error into a computable local residual term and a weaker global norm of the finite element error (the ``pollution term'). Using a mesh-dependent weight, the residual term is bounded in a sharply localized fashion. In specific situations the pollution term may also be bounded by computable residual estimators. On nonconvex polygonal and polyhedral domains in two and three space dimensions, we may choose estimators for the pollution term which do not employ specific knowledge of corner singularities and which are valid on domains with cracks. The finite element mesh is only required to be simplicial and shape-regular, so that highly graded and unstructured meshes are allowed.

     

New perturbation bounds for denumerable Markov chains

This paper is devoted to perturbation analysis of denumerable Markov chains. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where the bounds are given by the weighted supremum norm. In addition, bounds for the perturbed stationary probabilities are established. Furthermore, bounds on the norm of the asymptotic decomposition of the perturbed stationary distribution are provided, where the bounds are expressed in terms of the norm of the ergodicity coefficient, or the norm of a special residual matrix. Refinements of our bounds for Doeblin Markov chains are considered as well. Our results are illustrated with a number of examples.     

几乎各向同性的高维空间分数阶扩散方程的分块快速正则Hermite分裂预处理方法

一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度.     

模糊线性系统的扰动分析

使用谱范数分析了模糊线性系统在三种情形下的扰动: (1)右端模糊向量有扰动, 系数矩阵不变; (2)系数矩阵有扰动,右端模糊向量不变; (3)系数矩阵和右端模糊向量都有扰动,并通过数值实例验证给出的扰动界的估计.     

The Worst-Case GMRES for Normal Matrices

We study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. This bound is sharp, i.e. it is attainable by the GMRES residual norm. The question is how to evaluate or estimate the standard bound, and if it is possible to characterize the GMRES-related quantities for which this bound is attained (worst-case GMRES). In this paper we completely characterize the worst-case GMRES-related quantities in the next-to-last iteration step and evaluate the standard bound in terms of explicit polynomials involving the matrix eigenvalues. For a general iteration step, we develop a computable lower and upper bound on the standard bound. Our bounds allow us to study the worst-case GMRES residual norm as a function of the eigenvalue distribution. For hermitian matrices the lower bound is equal to the worst-case residual norm. In addition, numerical experiments show that the lower bound is generally very tight, and support our conjecture that it is to within a factor of 4/π of the actual worst-case residual norm. Since the worst-case residual norm in each step is to within a factor of the square root of the matrix size to what is considered an “average” residual norm, our results are of relevance beyond the worst case. This revised version was published online in July 2006 with corrections to the Cover Date.     

Variants of the groupwise update strategy for short-recurrence Krylov subspace methods

Krylov subspace methods often use short-recurrences for updating approximations and the corresponding residuals. In the bi-conjugate gradient (Bi-CG) type methods, rounding errors arising from the matrix–vector multiplications used in the recursion formulas influence the convergence speed and the maximum attainable accuracy of the approximate solutions. The strategy of a groupwise update has been proposed for improving the convergence of the Bi-CG type methods in finite-precision arithmetic. In the present paper, we analyze the influence of rounding errors on the convergence properties when using alternative recursion formulas, such as those used in the bi-conjugate residual (Bi-CR) method, which are different from those used in the Bi-CG type methods. We also propose variants of a groupwise update strategy for improving the convergence speed and the accuracy of the approximate solutions. Numerical experiments demonstrate the effectiveness of the proposed method.     

求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法

该文建立了求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法.使用该算法不仅可以判断该矩阵方程的中心对称解的存在性,而且无论中心对称解是否存在,都能够在有限步迭代计算之后得到中心对称最小二乘解.选取特殊的初始矩阵时,可求得极小范数中心对称最小二乘解.同时,也能给出指定矩阵的最佳逼近中心对称矩阵.     

Perturbations of Jordan matrices

We consider perturbations of a large Jordan matrix, either random and small in norm or of small rank. In the case of random perturbations we obtain explicit estimates which show that as the size of the matrix increases, most of the eigenvalues of the perturbed matrix converge to a certain circle with centre at the origin. In the case of finite rank perturbations we completely determine the spectral asymptotics as the size of the matrix increases.     

Minimizing Quadratic Functions Subject to Bound Constraints with the Rate of Convergence and Finite Termination

A new active set based algorithm is proposed that uses the conjugate gradient method to explore the face of the feasible region defined by the current iterate and the reduced gradient projection with the fixed steplength to expand the active set. The precision of approximate solutions of the auxiliary unconstrained problems is controlled by the norm of violation of the Karush-Kuhn-Tucker conditions at active constraints and the scalar product of the reduced gradient with the reduced gradient projection. The modifications were exploited to find the rate of convergence in terms of the spectral condition number of the Hessian matrix, to prove its finite termination property even for problems whose solution does not satisfy the strict complementarity condition, and to avoid any backtracking at the cost of evaluation of an upper bound for the spectral radius of the Hessian matrix. The performance of the algorithm is illustrated on solution of the inner obstacle problems. The result is an important ingredient in development of scalable algorithms for numerical solution of elliptic variational inequalities.     

特征值问题协调/非协调有限元后验误差分析

本文研究对称椭圆特征值问题的有限元后验误差估计,包括协调元和非协调元,具有下列特色:(1)对协调/非协调元建立了有限元特征函数uh的误差与相应的边值问题有限元解的误差在局部能量模意义下的恒等关系式,该边值问题的右端为有限元特征值λh与uh的乘积,有限元解恰好为uh.从而边值问题有限元解在能量模意义下的局部后验误差指示子,包括残差型和重构型后验误差指示子,成为有限元特征函数在能量模意义下的局部后验误差指示子.(2)讨论了协调有限元特征函数的基于插值后处理的梯度重构型后验误差估计,对有限元特征函数的导数得到了最大模意义下的渐近准确局部后验误差指示子.     

Convection-diffusion-reaction problems on a B-type mesh

One-dimensional singularly perturbed problems with two small parameters are considered. Numerical methods for such problems are discussed in several papers, but on a Shishkin-type mesh. The first optimal result of convergence in an energy norm on a Bakhvalov-type mesh for one-dimensional convection-diffusion problem was given by Roos in [9]. In this paper we analyze Galerkin finite element method on a Bakhvalov-type mesh for two-parameter convection-diffusion-reaction problems. In the interpolation error analysis, instead of the usual interpolation operator in the finite element space we use a quasi-interpolant with improved stability properties. We prove that the finite element method for these problems is uniformly convergent in the energy norm. Numerical results confirm our theoretical analysis and show first-order convergence rate. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

冲击接触问题的一种双共振投影梯度算法

根据冲击接触计算模型所需满足的基本控制方程和非线性互补条件,应用非线性互补问题与约束优化的等价关系将非线性互补接触问题转变成一个非线性规划问题,系统地推导建立了冲击接触问题的一种双共轭投影梯度计算方法．增广Lagrange乘子法克服了罚函数要求减小迭代步长以达到计算稳定的限制,即使对于冲击接触问题亦可以采用较大迭代步长,在形成的与原互补问题等价的无约束规划模式下,应用双共轭投影梯度算法提高非线性搜索速度和计算效率．算法模型计算结果表明,所建立的双共轭投影梯度计算理论及方法是正确有效的．     

Backward errors for eigenvalue and singular value decompositions

Summary. We present bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two-norm and in the Frobenius norm. Through different orthogonal decompositions of the computed eigenvectors we can define different symmetric backward errors for the eigenvalue decomposition. When the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small. Consequently it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured. Analogous results hold for the singular vectors. We indicate the effect of our error bounds on implementations for eigenvector and singular vector computation. In a more general context we prove that the distance of an appropriately scaled matrix to its orthogonal QR factor is not much larger than its distance to the closest orthogonal matrix. Received July 19, 1993     

The penalty method for grid matching in mixed finite element methods

In this paper we prove the possibility of the use of the penalty method for grid matching in mixed finite element methods. We consider the Hermann-Johnson scheme for biharmonic equation. The main idea is to construct a perturbed problem with two parameters which play roles of penalties. The perturbed problem is built by the replacement of essential conditions on the interface in the mixed variational statement with natural conditions that contain parameters. The perturbed problem is discretized by the finite element method. We estimate the norm of the difference between a solution of the discrete perturbed problem and a solution of the initial problem; the obtained estimates depend on the step and the penalties. We give recommendations for the choice of penalties depending on the step.     

The generalized centro‐symmetric and least squares generalized centro‐symmetric solutions of the matrix equation AYB + CYTD = E

An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P^?1 = P^T. An n × n real matrix Y is called a generalized centro‐symmetric with respect to P, if Y = PYP. It is obvious that every matrix is also a generalized centro‐symmetric matrix with respect to I. In this work by extending the conjugate gradient approach, two iterative methods are proposed for solving the linear matrix equation and the minimum Frobenius norm residual problem over the generalized centro‐symmetric Y, respectively. By the first (second) algorithm for any initial generalized centro‐symmetric matrix, a generalized centro‐symmetric solution (least squares generalized centro‐symmetric solution) can be obtained within a finite number of iterations in the absence of round‐off errors, and the least Frobenius norm generalized centro‐symmetric solution (the minimal Frobenius norm least squares generalized centro‐symmetric solution) can be derived by choosing a special kind of initial generalized centro‐symmetric matrices. We also obtain the optimal approximation generalized centro‐symmetric solution to a given generalized centro‐symmetric matrix Y₀ in the solution set of the matrix equation (minimum Frobenius norm residual problem). Finally, some numerical examples are presented to support the theoretical results of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

矩阵方程AXA^T＋BYB^T=C的双对称最小二乘解及其最佳逼近

基于共轭梯度法的思想,通过特殊的变形,建立了一类求矩阵方程AXA^T＋BYB^T=C的双对称最小二乘解的迭代算法．对任意的初始双对称矩阵．在没有舍人误差的情况下,经过有限步迭代得到它的双对称最小二乘解;在选取特殊的初始双对称矩阵时,能得到它的的极小范数双对称最小二乘解．另外,给定任意矩阵,利用此方法可得到它的最佳逼近双对称解,数值例子表明,这种方法是有效的．     

Finite iterative method for solving coupled Sylvester-transpose matrix equations

This paper is concerned with solutions to the so-called coupled Sylvester-transpose matrix equations, which include the generalized Sylvester matrix equation and Lyapunov matrix equation as special cases. By extending the idea of conjugate gradient method, an iterative algorithm is constructed to solve this kind of coupled matrix equations. When the considered matrix equations are consistent, for any initial matrix group, a solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial matrix group is chosen. By applying the proposed algorithm, the optimal approximation solution group to a given matrix group can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, a numerical example is given to illustrate that the algorithm is effective.     

A penalty method for grid matching in the mixed Herrmann-Miyoshi scheme

A penalty method for mixed finite element methods is formulated and studied. The Herrmann-Miyoshi scheme for the biharmonic equation is considered. The main idea is to build a perturbed problem with two parameters playing the role of penalties. The perturbed problem is constructed by replacing principal conditions in the mixed variational formulation at the interface by natural conditions containing parameters. Discretization of the perturbed problem is effected by a finite element method. Estimates for the norm of the difference between the solutions of a discrete perturbed problem and of an initial value problem are derived depending on the mesh size and penalties. Recommendations are given as to how to choose penalties so as to fit a mesh size.