Strang-Type Preconditioners for Solving Linear Systems from Delay Differential Equations

We consider the solution of delay differential equations (DDEs) by using boundary value methods (BVMs). These methods require the solution of one or more nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed for solving these linear systems. We show that if a P _{k 1},k ₂-stable BVM is used for solving an m-by-m system of DDEs, then our preconditioner is invertible and all the eigenvalues of the preconditioned system are clustered around 1. It follows that when the GMRES method is applied to solving the preconditioned systems, the method may converge fast. Numerical results are given to illustrate the effectiveness of our methods.     

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.     

Diagonal and Toeplitz splitting iteration methods for diagonal‐plus‐Toeplitz linear systems from spatial fractional diffusion equations

The finite difference discretization of the spatial fractional diffusion equations gives discretized linear systems whose coefficient matrices have a diagonal‐plus‐Toeplitz structure. For solving these diagonal‐plus‐Toeplitz linear systems, we construct a class of diagonal and Toeplitz splitting iteration methods and establish its unconditional convergence theory. In particular, we derive a sharp upper bound about its asymptotic convergence rate and deduct the optimal value of its iteration parameter. The diagonal and Toeplitz splitting iteration method naturally leads to a diagonal and circulant splitting preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1, especially when the discretization step‐size h is small. Numerical results exhibit that the diagonal and circulant splitting preconditioner can significantly improve the convergence properties of GMRES and BiCGSTAB, and these preconditioned Krylov subspace iteration methods outperform the conjugate gradient method preconditioned by the approximate inverse circulant‐plus‐diagonal preconditioner proposed recently by Ng and Pan (M.K. Ng and J.‐Y. Pan, SIAM J. Sci. Comput. 2010;32:1442‐1464). Moreover, unlike this preconditioned conjugate gradient method, the preconditioned GMRES and BiCGSTAB methods show h‐independent convergence behavior even for the spatial fractional diffusion equations of discontinuous or big‐jump coefficients.     

Preconditioners for nonsymmetric block toeplitz-like-plus-diagonal linear systems

Summary. We consider the system of linear equations Lu=f, where L is a nonsymmetric block Toeplitz-like-plus-diagonal matrix, which arises from the Sinc-Galerkin discretization of differential equations. Our main contribution is to construct effective preconditioners for this structured coefficient matrix and to derive tight bounds for eigenvalues of the preconditioned matrices. Moreover, we use numerical examples to show that the new preconditioners, when applied to the preconditioned GMRES method, are efficient for solving the system of linear equations. Mathematics Subject Classification (2000):65F10, 65F15, 65T10Research subsidized by The Special Funds for Major State Basic Research Projects G1999032803Research supported in part by RGC Grant Nos. 7132/00P and 7130/02P, and HKU CRCG Grant Nos 10203501, 10203907 and 10203408     

Spectral behaviour of GMRES applied to singular systems

The purpose of this paper is to develop a spectral analysis of the Hessenberg matrix obtained by the GMRES algorithm used for solving a linear system with a singular matrix. We prove that the singularity of the Hessenberg matrix depends on the nature of A and some other criteria such as the zero eigenvalue multiplicity and the projection of the initial residual on particular subspaces. We also show some new results about the distinct kinds of breakdown which may occur in the algorithm when the system is singular.      

BCCB preconditioners for systems of BVM‐based numerical integrators

Boundary value methods (BVMs) for 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 block‐circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is A_k1,k2‐stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block‐circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.     

Some observations on the l2 convergence of the additive Schwarz preconditioned GMRES method

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.     

Preconditioned GMRES methods with incomplete Givens orthogonalization method for large sparse least-squares problems

We propose to precondition the GMRES method by using the incomplete Givens orthogonalization (IGO) method for the solution of large sparse linear least-squares problems. Theoretical analysis shows that the preconditioner satisfies the sufficient condition that can guarantee that the preconditioned GMRES method will never break down and always give the least-squares solution of the original problem. Numerical experiments further confirm that the new preconditioner is efficient. We also find that the IGO preconditioned BA-GMRES method is superior to the corresponding CGLS method for ill-conditioned and singular least-squares problems.     

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.     

Strang-type preconditioners for systems of LMF-based ODE codes

We consider the solution of ordinary differential equations(ODEs) using boundary value methods. These methods require thesolution of one or more unsymmetric, large and sparse linearsystems. The GMRES method with the Strang-type block-circulantpreconditioner is proposed for solving these linear systems.We show that if an A_k1,k2 -stable boundary value method is usedfor an m-by-m system of ODEs, then our preconditioners are invertibleand all the eigenvalues of the preconditioned systems are 1except for at most 2m(k₁ + k₂) outliers. It follows that whenthe GMRES method is applied to solving the preconditioned systems,the method will converge in at most 2m(k₁ + k₂) + 1 iterations.Numerical results are given to illustrate the effectivenessof our methods. Received 8 October 1999. Accepted 30 May 2000.     

A new adaptive GMRES algorithm for achieving high accuracy

GMRES(k) is widely used for solving non-symmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(k) procedure outperforms standard GMRES(k), several other GMRES-like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified Gram–Schmidt process in GMRES, can fail causing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(k) with the orthogonalization performed by Householder transformations succeeds whenever GMRES(k) with the orthogonalization performed by the modified Gram–Schmidt process fails, and the extra cost of computing Householder transformations is justified for these applications. © 1998 John Wiley & Sons, Ltd.     

Block-circulant preconditioners for systems arising from discretization of the three-dimensional convection–diffusion equation

We consider the system of equations arising from finite difference discretization of a three-dimensional convection–diffusion model problem. This system is typically nonsymmetric. The GMRES method with the Strang block-circulant preconditioner is proposed for solving this linear system. We show that our preconditioners are invertible and study the spectra of the preconditioned matrices. Numerical results are reported to illustrate the effectiveness of our methods.     

Right preconditioned MINRES for singular systems†

We consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right‐hand‐side vector and an arbitrary initial vector even if the linear system is singular and inconsistent. For the special case when the system is consistent, we prove that the iterates converge to a min‐norm solution with respect to the preconditioner if the initial vector is in the range space of the right preconditioned coefficient matrix. Furthermore, we propose a right preconditioned MINRES using symmetric successive over‐relaxation (SSOR) with Eisenstat's trick. Some numerical experiments on semidefinite systems in electromagnetic analysis and so forth indicate that the method is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the iterations.     

Preconditioners for non‐Hermitian Toeplitz systems

In this paper, we construct new ω‐circulant preconditioners for non‐Hermitian Toeplitz systems, where we allow the generating function of the sequence of Toeplitz matrices to have zeros on the unit circle. We prove that the eigenvalues of the preconditioned normal equation are clustered at 1 and that for (N, N)‐Toeplitz matrices with spectral condition number 𝒪(N^α) the corresponding PCG method requires at most 𝒪(N log² N) arithmetical operations. If the generating function of the Toeplitz sequence is a rational function then we show that our preconditioned original equation has only a fixed number of eigenvalues which are not equal to 1 such that preconditioned GMRES needs only a constant number of iteration steps independent of the dimension of the problem. Numerical tests are presented with PCG applied to the normal equation, GMRES, CGS and BICGSTAB. In particular, we apply our preconditioners to compute the stationary probability distribution vector of Markovian queuing models with batch arrival. Copyright © 2001 John Wiley & Sons, Ltd.     

Modified hierarchy basis for solving singular boundary value problems

In this paper, we develop an efficient preconditioning method on the basis of the modified hierarchy basis for solving the singular boundary value problem by the Galerkin method. After applying the preconditioning method, we show that the condition number of the linear system arising from the Galerkin method is uniformly bounded. In particular, the condition number of the preconditioned system will be bounded by 2 for the case q(x)=0 (see Eq. (1) in the paper). Numerical results are presented to confirm our theoretical results.     

A geometric view of Krylov subspace methods on singular systems

We give a geometric framework for analysing iterative methods on singular linear systems A x = b and apply them to Krylov subspace methods. The idea is to decompose the method into the ?(A) component and its orthogonal complement ?(A)^?, where ?(A) is the range of A. We apply the framework to GMRES, GMRES(k) and GCR(k), and derive conditions for convergence without breakdown for inconsistent and consistent singular systems. The approach also gives a geometric interpretation and different proofs of the conditions obtained by Brown and Walker for GMRES. We also give examples arising in the finite difference discretization of two‐point boundary value problems of an ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Regularized DPSS Preconditioners for Singular Saddle Point Problems

Recently, Cao proposed a regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for the non-Hermitian nonsingular saddle point problem. In this paper, we consider applying RDPSS preconditioner to solve the singular saddle point problem. Moreover, we propose a two-parameter accelerated variant of the RDPSS (ARDPSS) preconditioner to further improve its efficiency. Theoretical analysis proves that the RDPSS and ARDPSS methods are semi-convergent unconditionally. Some spectral properties of the corresponding preconditioned matrices are analyzed. Numerical experiments indicate that better performance can be achieved when applying the ARDPSS preconditioner to accelerate the GMRES method for solving the singular saddle point problem.     

A simpler GMRES

The generalized minimal residual (GMRES) method is widely used for solving very large, nonsymmetric linear systems, particularly those that arise through discretization of continuous mathematical models in science and engineering. By shifting the Arnoldi process to begin with Ar₀ instead of r₀, we obtain simpler Gram–Schmidt and Householder implementations of the GMRES method that do not require upper Hessenberg factorization. The Gram–Schmidt implementation also maintains the residual vector at each iteration, which allows cheaper restarts of GMRES(m) and may otherwise be useful.     

Preconditioning iterative algorithm for the electromagnetic scattering from a large cavity

A preconditioning iterative algorithm is proposed for solving electromagnetic scattering from an open cavity embedded in an infinite ground plane. In this iterative algorithm, a physical model with a vertically layered medium is employed as a preconditioner of the model of general media. A fast algorithm developed in (SIAM J. Sci. Comput. 2005; 27 :553–574) is applied for solving the model of layered media and classical Krylov subspace methods, restarted GMRES, COCG, and BiCGstab are employed for solving the preconditioned system. Our numerical experiments on cavity models with large numbers of mesh points and large wave numbers show that the algorithm is efficient and the number of iterations is independent of the number of mesh points and dependent upon the wave number. Copyright © 2008 John Wiley & Sons, Ltd.     

Semiconvergence of block SOR method for singular linear systems with p-cyclic matrices

In this paper, we discuss semiconvergence of the block SOR method for solving singular linear systems with p-cyclic matrices. Some sufficient conditions for the semiconvergence of the block SOR method for solving a general p-cyclic singular system are proved.