ON AUGMENTED LAGRANGIAN METHODS FOR SADDLE-POINT LINEAR SYSTEMS WITH SINGULAR OR SEMIDEFINITE （1, 1） BLOCKS

An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite （1, 1） blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems.     

Restarted FOM Augmented with Ritz Vectors for Shifted Linear Systems

The restarted FOM method presented by Simoncini[7]according to the natural collinearity of all residuals is an efficient method for solving shifted systems,which generates the same Krylov subspace when the shifts are handled simultaneously.However,restarting slows down the convergence.We present a practical method for solving the shifted systems by adding some Ritz vectors into the Krylov subspace to form an augmented Krylov subspace. Numerical experiments illustrate that the augmented FOM approach(restarted version)can converge more quickly than the restarted FOM method.     

CGS/GMRES(k): AN ADAPTIVE PRECONDITIONED CGS ALGORITHM FOR NONSYMMETRIC LINEAR SYSTEMS

Recently Y. Saad proposed a flexible inner-outer preconditioned GMRES algorithm for nonsymmetric linear systems [4]. Following their ideas, we suggest an adaptive preconditioned CGS method, called CGS/GMRES (k), in which the preconditioner is constructed in the iteration step of CGS, by several steps of GMRES(k). Numerical experiments show that the residual of the outer iteration decreases rapidly. We also found the interesting residual behaviour of GMRES for the skewsymmetric linear system Ax = b, which gives a convergence result for restarted GMRES (k). For convenience, we discuss real systems.     

A New Class AOR Preconditioner for $L$-Matrices

Hadjidimos(1978) proposed a classical accelerated overrelaxation(AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant, L-matrices, and consistently orders matrices. Several preconditioned AOR methods have been proposed to solve system of linear equations Ax = b, where A ∈ R~(n×n) is an L-matrix. In this work, we introduce a new class preconditioners for solving linear systems and give a comparison result and some convergence result for this class of preconditioners. Numerical results for corresponding preconditioned GMRES methods are given to illustrate the theoretical results.     

关于解线性系统的一个有效的贝叶斯迭代方法

This paper concerns with the statistical methods for solving general linear systems. After a brief review of Bayesian perspective for inverse problems,a new and efficient iterative method for general linear systems from a Bayesian perspective is proposed.The convergence of this iterative method is proved,and the corresponding error analysis is studied.Finally, numerical experiments are given to support the efficiency of this iterative method,and some conclusions are obtained.     

A PRODUCT HYBRID GMRES ALGORITHM FOR NONSYMMETRIC LINEAR SYSTEMS

It has been observed that the residual polynomials resulted from successive restarting cycles of GMRES(m) may differ from one another meaningfully. In this paper, it is further shown that the polynomials can complement one another harmoniously in reducing the iterative residual. This characterization of GMRES(m) is exploited to formulate an efficient hybrid iterative scheme, which can be widely applied to existing hybrid algorithms for solving large nonsymmetric systems of linear equations. In particular, a variant of the hybrid GMRES algorithm of Nachtigal, Reichel and Trefethen (1992) is presented. It is described how the new algorithm may offer significant performance improvements over the original one.     

Inner iterations in the shift-invert residual Arnoldi method and the Jacobi-Davidson method

We establish a general convergence theory of the Shift-Invert Residual Arnoldi(SIRA)method for computing a simple eigenvalue nearest to a given targetσand the associated eigenvector.In SIRA,a subspace expansion vector at each step is obtained by solving a certain inner linear system.We prove that the inexact SIRA method mimics the exact SIRA well,i.e.,the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with low or modest accuracy during outer iterations.Based on the theory,we design practical stopping criteria for inner solves.Our analysis is on one step expansion of subspace and the approach applies to the Jacobi-Davidson(JD)method with the fixed targetσas well,and a similar general convergence theory is obtained for it.Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.     

ON THE OPTIMIZATION OF EXTRAPOLATION METHODS FOR SINGULAR LINEAR SYSTEMS

We discuss semiconvergence of the extrapolated iterative methods for solving singular linear systems. We obtain the upper bounds and the optimum convergence factor of the extrapolation method as well as its associated optimum extrapolation parameter. Numerical examples are given to illustrate the theoretical results.     

MODIFIED BERNOULLI ITERATION METHODS FOR QUADRATIC MATRIX EQUATION

We construct a modified Bernoulli iteration method for solving the quadratic matrix equation AX^2 ＋ BX ＋ C = 0, where A, B and C are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the ShermanMorrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.     

A SPARSE SUBSPACE TRUNCATED NEWTON METHOD FOR LARGE-SCALE BOUND CONSTRAINED NONLINEAR OPTIMIZATION

In this paper we report a sparse truncated Newton algorithm for handling large-scale simple bound nonlinear constrained minimixation problem. The truncated Newton method is used to update the variables with indices outside of the active set, while the projected gradient method is used to update the active variables. At each iterative level, the search direction consists of three parts, one of which is a subspace truncated Newton direction, the other two are subspace gradient and modified gradient directions. The subspace truncated Newton direction is obtained by solving a sparse system of linear equations. The global convergence and quadratic convergence rate of the algorithm are proved and some numerical tests are given.     

Restarted Full Orthogonalization Method for Shifted Linear Systems

Restarted GMRES is known to be inefficient for solving shifted systems when the shifts are handled simultaneously. Variants have been proposed to enhance its performance. We show that another restarted method, restarted Full Orthogonalization Method (FOM), can effectively be employed. The total number of iterations of restarted FOM applied to all shifted systems simultaneously is the same as that obtained by applying restarted FOM to the shifted system with slowest convergence rate, while the computational cost grows only sub-linearly with the number of shifts. Numerical experiments are reported.     

Complementary cycles of restarted GMRES

Restarted GMRES is one of the most popular methods for solving large nonsymmetric linear systems. It is generally thought that the information of previous GMRES cycles is lost at the time of a restart; therefore, each cycle contributes to the global convergence individually. However, this is not the full story. In this paper, we shed light on the relationship between different GMRES cycles. It is shown that successive GMRES cycles can complement one another harmoniously. These groups of cycles, called complementary cycles, are defined and studied. Copyright © 2008 John Wiley & Sons, Ltd.     

GMRES with adaptively deflated restarting and its performance on an electromagnetic cavity problem

GMRES with deflated restarting, denoted by GMRES-DR(m,k), is a variant of restarted GMRES for solving nonsymmetric linear systems, which improves the convergence of restarted GMRES by weakening the influence of the eigenvalues incurring poor numerical performance. However, it is difficult to determine the deflation parameter k. In this paper, we present a new deflation strategy which determines k dynamically in different cycles. Numerical results for an electromagnetic cavity problem show that the deflation strategy is efficient.     

Deflated GMRES for systems with multiple shifts and multiple right-hand sides

We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple right-hand sides. First, for a single right-hand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including lattice quantum chromodynamics where the matrices are complex and non-Hermitian. Some Krylov iterative methods such as GMRES and BiCGStab have been used to solve multiply shifted systems for about the cost of solving just one system. Restarted GMRES can be improved by deflating eigenvalues for matrices that have a few small eigenvalues. We show that a particular deflated method, GMRES-DR, can be applied to multiply shifted systems.In quantum chromodynamics, it is common to have multiple right-hand sides with multiple shifts for each right-hand side. We develop a method that efficiently solves the multiple right-hand sides by using a deflated version of GMRES and yet keeps costs for all of the multiply shifted systems close to those for one shift. An example is given showing this can be extremely effective with a quantum chromodynamics matrix.     

Circulant-block preconditioners for solving ordinary differential equations

Boundary value methods for solving ordinary differential equations require the solution of non-symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A circulant-block preconditioner is proposed to speed up the convergence rate of the GMRES method. Theoretical and practical arguments are given to show that this preconditioner is more efficient than some other circulant-type preconditioners in some cases. A class of waveform relaxation methods is also proposed to solve the linear systems.     

On preconditioned MHSS iteration methods for complex symmetric linear systems

We propose a preconditioned variant of the modified HSS (MHSS) iteration method for solving a class of complex symmetric systems of linear equations. Under suitable conditions, we prove the convergence of the preconditioned MHSS (PMHSS) iteration method and discuss the spectral properties of the PMHSS-preconditioned matrix. Numerical implementations show that the resulting PMHSS preconditioner leads to fast convergence when it is used to precondition Krylov subspace iteration methods such as GMRES and its restarted variants. In particular, both the stationary PMHSS iteration and PMHSS-preconditioned GMRES show meshsize-independent and parameter-insensitive convergence behavior for the tested numerical examples.     

Shifted skew-symmetric iteration methods for nonsymmetric linear complementarity problems

We present a shifted skew-symmetric iteration method for solving the nonsymmetric positive definite or positive semidefinite linear complementarity problems. This method is based on the symmetric and skew-symmetric splitting of the system matrix, which has been adopted to establish efficient splitting iteration methods for solving the nonsymmetric systems of linear equations. Global convergence of the method is proved, and the corresponding inexact splitting iteration scheme is established and analyzed in detail. Numerical results show that the new methods are feasible and effective for solving large sparse and nonsymmetric linear complementarity problems.     

GMRES方法的收敛率

1 引 言 GMRES方法是目前求解大型稀疏非对称线性方程组 Ax=b,A∈R~(n×n);x,b∈R~n (1)最为流行的方法之一.设x~((0))是(1)解的初始估计,r~((0))=b-Ax~((0))是初始残量,K_k=span{r~((0)),Ar~((0)),…A~(k-1)r~((0))}为由r~((0))和A产生的Krylov子空间.GMRES方法的第k步     

Some observations on weighted GMRES

We investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present a new alternative implementation of the weighted Arnoldi algorithm which under known circumstances will be favourable in terms of computational complexity. These implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used.     

A simpler GMRES and its adaptive variant for shifted linear systems

A variant of the simpler GMRES method is developed for solving shifted linear systems (SGMRES‐Sh), exhibiting almost the same advantage of the simpler GMRES method over the regular GMRES method. Because the remedy adapted by GMRES‐Sh is no longer feasible for SGMRES‐Sh due to the differences between simpler GMRES and GMRES for constructing the residual vectors of linear systems, we take an alternative strategy to force the residual vectors of the add system also be orthogonal to the subspaces, to which the residual vectors of the seed system are orthogonal when the seed system is solved with the simpler GMRES method. In addition, a seed selection strategy is also employed for solving the rest non‐converged linear systems. Furthermore, an adaptive version of SGMRES‐Sh is presented for the purpose of improving the stability of SGMRES‐Sh based on the technique of the adaptive choice of the Krylov subspace basis developed for the adaptive simpler GMRES. Numerical experiments demonstrate the benefits of the presented methods.