一个拟极小化左共轭梯度算法及其数值表现

左共轭梯度法是求解大型稀疏线性方程组的一种新兴的Krylov子空间方法.为克服该算法数值表现不稳定、迭代中断的缺点,本文对原方法进行等价变形,得到左共轭梯度方向的另一迭代格式,给出一个拟极小化左共轭梯度算法.数值结果证实了该变形算法与原算法的相关性.     

Computable error bounds and estimates for the conjugate gradient method

The conjugate gradient method is one of the most popular iterative methods for computing approximate solutions of linear systems of equations with a symmetric positive definite matrix A. It is generally desirable to terminate the iterations as soon as a sufficiently accurate approximate solution has been computed. This paper discusses known and new methods for computing bounds or estimates of the A-norm of the error in the approximate solutions generated by the conjugate gradient method.     

Conjugate directions method for solving interval linear systems

We propose the interval version of the conjugate directions method, to solve the problem of linear systems, with symmetric and positive definite interval matrix A, and a right-hand side interval vector b. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Multilevel AINV Preconditioner

In this paper we describe an algebraic multilevel extension of the approximate inverse AINV preconditioner for solving symmetric positive definite linear systems Ax=b with the preconditioned conjugate gradient method. The smoother is the approximate inverse M and the coarse grids and the interpolation operator are constructed by looking at the entries of M. Numerical examples are given for problems arising from discretization of partial differential equations.     

Analysis of a novel preconditioner for a class of p‐level lower rank extracted systems

This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, Âx=b, which we call p‐level lower rank extracted systems (p‐level LRES), by the preconditioned conjugate gradient method. The study of these systems is motivated by the numerical approximation of integral equations with convolution kernels defined on arbitrary p‐dimensional domains. This is in contrast to p‐level Toeplitz systems which only apply to rectangular domains. The coefficient matrix, Â, is a principal submatrix of a p‐level Toeplitz matrix, A, and the preconditioner for the preconditioned conjugate gradient algorithm is provided in terms of the inverse of a p‐level circulant matrix constructed from the elements of A. The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix which leads to a substantial reduction in the computational cost of solving LRE systems. Copyright © 2006 John Wiley & Sons, Ltd.     

On parallel solution of linear elasticity problems: Part I: theory

The discretized linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorization pcg-method, i.e., we perform inner iterations. A finite element discretization and node numbering giving a high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorization requires an M-matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg-method is analysed for different preconditionings based on the sdc-part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects. © 1998 John Wiley & Sons, Ltd.     

GMRES, L-Curves, and Discrete Ill-Posed Problems

The GMRES method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix. This paper discusses application of the GMRES method to the solution of large linear systems of equations that arise from the discretization of linear ill-posed problems. These linear systems are severely ill-conditioned and are referred to as discrete ill-posed problems. We are concerned with the situation when the right-hand side vector is contaminated by measurement errors, and we discuss how a meaningful approximate solution of the discrete ill-posed problem can be determined by early termination of the iterations with the GMRES method. We propose a termination criterion based on the condition number of the projected matrices defined by the GMRES method. Under certain conditions on the linear system, the termination index corresponds to the vertex of an L-shaped curve.     

A note on the efficiency of the conjugate gradient method for a class of time‐dependent problems

We discuss the efficiency of the conjugate gradient (CG) method for solving a sequence of linear systems; Auⁿ⁺¹ = uⁿ, where A is assumed to be sparse, symmetric, and positive definite. We show that under certain conditions the Krylov subspace, which is generated when solving the first linear system Au¹ = u⁰, contains the solutions {uⁿ} for subsequent time steps. The solutions of these equations can therefore be computed by a straightforward projection of the right‐hand side onto the already computed Krylov subspace. Our theoretical considerations are illustrated by numerical experiments that compare this method with the order‐optimal scheme obtained by applying the multigrid method as a preconditioner for the CG‐method at each time step. Copyright © 2007 John Wiley & Sons, Ltd.     

Block two-stage preconditioners

Linear systems of the form Ax = b, where the matrix A is symmetric and positive definite, often arise from the discretization of elliptic partial differential equations. A very successful method for solving these linear systems is the preconditioned conjugate gradient method. In this paper, we study parallel preconditioners for the conjugate gradient method based on the block two-stage iterative methods. Sufficient conditions for the validity of these preconditioners are given. Computational results of these preconditioned conjugate gradient methods on two parallel computing systems are presented.     

Conjugate gradient method for dual-dual mixed formulations

We deal with the iterative solution of linear systems arising from so-called dual-dual mixed finite element formulations. The linear systems are of a two-fold saddle point structure; they are indefinite and ill-conditioned. We define a special inner product that makes matrices of the two-fold saddle point structure, after a specific transformation, symmetric and positive definite. Therefore, the conjugate gradient method with this special inner product can be used as iterative solver. For a model problem, we propose a preconditioner which leads to a bounded number of CG-iterations. Numerical experiments for our model problem confirming the theoretical results are also reported.

     

A COMPUTATIONAL APPROACH FOR OPTIMAL CONTROL SYSTEMS GOVERNED BY PARABOLIC VARIATIONAL INEQUALITIES

Iterative techniques for solving optimal control systems governed by parabolic variational inequalities are presented. The techniques we use are based on linear finite elements method to approximate the state equations and nonlinear conjugate gradient methods to solve the discrete optimal control problem. Convergence results and numerical experiments are presented.     

Fast Construction of Optimal Circulant Preconditioners for Matrices from the Fast Dense Matrix Method

In this paper, we consider solving non-convolution type integral equations by the preconditioned conjugate gradient method. The fast dense matrix method is a fast multiplication scheme that provides a dense discretization matrix A approximating a given integral equation. The dense matrix A can be constructed in O(n) operations and requires only O(n) storage where n is the size of the matrix. Moreover, the matrix-vector multiplication A xcan be done in O(n log n) operations. Thus if the conjugate gradient method is used to solve the discretized system, the cost per iteration is O(n log n) operations. However, for some integral equations, such as the Fredholm integral equations of the first kind, the system will be ill-conditioned and therefore the convergence rate of the method will be slow. In these cases, preconditioning is required to speed up the convergence rate of the method. A good choice of preconditioner is the optimal circulant preconditioner which is the minimizer of C – A _F in Frobenius norm over all circulant matrices C. It can be obtained by taking arithmetic averages of all the entries of A and therefore the cost of constructing the preconditioner is of O(n ²) operations for general dense matrices. In this paper, we develop an O(n log n) method of constructing the preconditioner for dense matrices A obtained from the fast dense matrix method. Application of these ideas to boundary integral equations from potential theory will be given. These equations are ill-conditioned whereas their optimal circulant preconditioned equations will be well-conditioned. The accuracy of the approximation A, the fast construction of the preconditioner and the fast convergence of the preconditioned systems will be illustrated by numerical examples.     

New results on the convergence of the conjugate gradient method

This paper is concerned with proving theoretical results related to the convergence of the conjugate gradient (CG) method for solving positive definite symmetric linear systems. Considering the inverse of the projection of the inverse of the matrix, new relations for ratios of the A‐norm of the error and the norm of the residual are provided, starting from some earlier results of Sadok (Numer. Algorithms 2005; 40 :201–216). The proofs of our results rely on the well‐known correspondence between the CG method and the Lanczos algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

双变量LMEs一种异类约束最小二乘解的MCG算法

借鉴求线性矩阵方程组(LMEs)同类约束最小二乘解的修正共轭梯度法,建立了求双变量LMEs的一种异类约束最小二乘解的修正共轭梯度法,并证明了该算法的收敛性.在不考虑舍入误差的情况下,利用该算法不仅可在有限步计算后得到LMEs的一组异类约束最小二乘解,而且选取特殊初始矩阵时,可求得LMEs的极小范数异类约束最小二乘解.另外,还可求得指定矩阵在该LMEs的异类约束最小二乘解集合中的最佳逼近.算例表明,该算法是有效的.     

A new globalization technique for nonlinear conjugate gradient methods for nonconvex minimization

It is well-known that the HS method and the PRP method may not converge for nonconvex optimization even with exact line search. Some globalization techniques have been proposed, for instance, the PRP+ globalization technique and the Grippo-Lucidi globalization technique for the PRP method. In this paper, we propose a new efficient globalization technique for general nonlinear conjugate gradient methods for nonconvex minimization. This new technique utilizes the information of the previous search direction sufficiently. Under suitable conditions, we prove that the nonlinear conjugate gradient methods with this new technique are globally convergent for nonconvex minimization if the line search satisfies Wolfe conditions or Armijo condition. Extensive numerical experiments are reported to show the efficiency of the proposed technique.     

On the semi-conjugate direction methods with dynamic preconditioning

The semi-conjugate residual methods and semi-conjugate gradient methods with dynamic preconditioners in Krylov subspaces are considered for solving the systems of linear algebraic equations whose matrices are not symmetric. Their orthogonal and variational properties are under study. New algorithms are proposed for choosing the inner iteration parameters in the preconditioning matrices corresponding to incomplete factorization methods. The efficiency of the resulting iterative processes is demonstrated by a set of numerical experiments for finite difference diffusion-convection equations.     

On modification of certain methods of the conjugate direction type for solving rectangular systems of linear algebraic equations

The use of modifications of certain well-known methods of the conjugate direction type for solving systems of linear algebraic equations with rectangular matrices is examined. The modified methods are shown to be superior to the original versions with respect to the round-off accumulation; the advantage is especially large for ill-conditioned matrices. Examples are given of the efficient use of the modified methods for solving certain fairly large ill-conditioned problems.     

SINE TRANSFORM PRECONDITIONERS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.     

Preconditioned Barzilai-Borwein method for the numerical solution of partial differential equations

The preconditioned Barzilai-Borwein method is derived and applied to the numerical solution of large, sparse, symmetric and positive definite linear systems that arise in the discretization of partial differential equations. A set of well-known preconditioning techniques are combined with this new method to take advantage of the special features of the Barzilai-Borwein method. Numerical results on some elliptic test problems are presented. These results indicate that the preconditioned Barzilai-Borwein method is competitive and sometimes preferable to the preconditioned conjugate gradient method.This author was partially supported by the Parallel and Distributed Computing Center at UCV.This author was partially supported by BID-CONICIT, project M-51940.     

A fast minimal residual algorithm for shifted unitary matrices

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρU + ζI, where ρ and ζ are constants, U is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended.