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.     

Convergence conditions for a restarted GMRES method augmented with eigenspaces

We consider the GMRES(m,k) method for the solution of linear systems Ax=b, i.e. the restarted GMRES with restart m where to the standard Krylov subspace of dimension m the other subspace of dimension k is added, resulting in an augmented Krylov subspace. This additional subspace approximates usually an A‐invariant subspace. The eigenspaces associated with the eigenvalues closest to zero are commonly used, as those are thought to hinder convergence the most. The behaviour of residual bounds is described for various situations which can arise during the GMRES(m,k) process. The obtained estimates for the norm of the residual vector suggest sufficient conditions for convergence of GMRES(m,k) and illustrate that these augmentation techniques can remove stagnation of GMRES(m) in many cases. All estimates are independent of the choice of an initial approximation. Conclusions and remarks assessing numerically the quality of proposed bounds conclude the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations

Linear partial differential algebraic equations (PDAEs) of the form Au _t(t, x) + Bu _xx(t, x) + Cu(t, x) = f(t, x) are studied where at least one of the matrices A, B R ^n×n is singular. For these systems we introduce a uniform differential time index and a differential space index. We show that in contrast to problems with regular matrices A and B the initial conditions and/or boundary conditions for problems with singular matrices A and B have to fulfill certain consistency conditions. Furthermore, two numerical methods for solving PDAEs are considered. In two theorems it is shown that there is a strong dependence of the order of convergence on these indexes. We present examples for the calculation of the order of convergence and give results of numerical calculations for several aspects encountered in the numerical solution of PDAEs.     

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.     

Singular continuous spectrum under rank one perturbations and localization for random hamiltonians

We consider a selfadjoint operator, A, and a selfadjoint rank-one projection, P, onto a vector, φ, which is cyclic for A. In terms of the spectral measure dμ^A_φ, we give necessary and sufficient conditions for A + λ P to have empty singular continuous spectrum or to have only point spectrum for a.e. λ. We apply these results to questions of localization in the one- and multi-dimensional Anderson models.     

Generalization of convergence conditions for a restarted GMRES

We consider the GMRES(s), i.e. the restarted GMRES with restart s for the solution of linear systems Ax = b with complex coefficient matrices. It is well known that the GMRES(s) applied on a real system is convergent if the symmetric part of the matrix A is positive definite. This paper introduces sufficient conditions implying the convergence of a restarted GMRES for a more general class of non‐Hermitian matrices. For real systems these conditions generalize the known result initiated as above. The discussion after the main theorem concentrates on the question of how to find an integer j such that the GMRES(s) converges for all s ≥ j. Additional properties of GMRES obtained by derivation of the main theorem are presented in the last section. Copyright © 2000 John Wiley & Sons, Ltd.     

Preconditioned Krylov subspace method for the solution of least-squares problems

We consider preconditioned Krylov subspace iteration methods, e.g., CG, LSQR and GMRES, for the solution of large sparse least-squares problems min ∥Ax – b ∥₂, with A ∈ R ^m×n, based on the Krylov subspaces K_k (BA, Br) and K_k (AB, r), respectively, where B ∈ R ^n×m is the preconditioning matrix. More concretely, we propose and implement a class of incomplete QR factorization preconditioners based on the Givens rotations and analyze in detail the efficiency and robustness of the correspondingly preconditioned Krylov subspace iteration methods. A number of numerical experiments are used to further examine their numerical behaviour. It is shown that for both overdetermined and underdetermined least-squares problems, the preconditioned GMRES methods are the best for large, sparse and ill-conditioned matrices in terms of both CPU time and iteration step. Also, comparisons with the diagonal scaling and the RIF preconditioners are given to show the superiority of the newly-proposed GMRES-type methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lanczos,Householder transformations,and implicit deflation for fast and reliable dominant singular subspace computation

Many applications, such as subspace‐based models in information retrieval and signal processing, require the computation of singular subspaces associated with the k dominant, or largest, singular values of an m×n data matrix A, where k?min(m,n). Frequently, A is sparse or structured, which usually means matrix–vector multiplications involving A and its transpose can be done with much less than ??(mn) flops, and A and its transpose can be stored with much less than ??(mn) storage locations. Many Lanczos‐based algorithms have been proposed through the years because the underlying Lanczos method only accesses A and its transpose through matrix–vector multiplications. We implement a new algorithm, called KSVD, in the Matlab environment for computing approximations to the singular subspaces associated with the k dominant singular values of a real or complex matrix A. KSVD is based upon the Lanczos tridiagonalization method, the WY representation for storing products of Householder transformations, implicit deflation, and the QR factorization. Our Matlab simulations suggest it is a fast and reliable strategy for handling troublesome singular‐value spectra. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

DQGMRES: a Direct Quasi-minimal Residual Algorithm Based on Incomplete Orthogonalization

We describe a Krylov subspace technique, based on incomplete orthogonalization of the Krylov vectors, which can be considered as a truncated version of GMRES. Unlike GMRES(m), the restarted version of GMRES, the new method does not require restarting. Like GMRES, it does not break down. Numerical experiments show that DQGMRES(k) often performs as well as the restarted GMRES using a subspace of dimension m=2k. In addition, the algorithm is flexible to variable preconditioning, i.e., it can accommodate variations in the preconditioner at every step. In particular, this feature allows the use of any iterative solver as a right-preconditioner for DQGMRES(k). This inner-outer iterative combination often results in a robust approach for solving indefinite non-Hermitian linear systems.     

Possible numbers of ones in the squares of 0–1 matrices with a given rank

For a symmetric 0–1 matrix A, we give the number of ones in A ² when rank(A) = 1, 2, and give the maximal number of ones in A ² when rank(A) = k (3 ≤ k ≤ n). The sufficient and necessary condition under which the maximal number is achieved is also obtained. For generic 0–1 matrices, we only study the cases of rank 1 and rank 2.     

π-Complementation in the Unitisation and Multiplication Algebras of a Semiprime Algebra

π-complemented algebras are defined as those (not necessarily associative or unital) algebras such that each annihilator ideal is complemented by other annihilator ideal. For a given semiprime algebra A, we discuss the π-complementation of the unitisation algebra A ¹ of A. Moreover, if in addition the multiplication algebra ?(A) of A is also semiprime, we study the π-complementation in the algebras ?(A) and ?^?(A) (the multiplication ideal of A). In associative setting, we prove that A is π-complemented if and only if ?^?(A) is π-complemented, and that A ¹ π-complemented if and only if ?(A) is π-complemented.     

Coloring uniform hypergraphs with few colors

Let m(r, k) denote the minimum number of edges in an r‐uniform hypergraph that is not k‐colorable. We give a new lower bound on m(r, k) for fixed k and large r. Namely, we prove that if k ≥ 2ⁿ, then m(r, k) ≥ ?(k)k^r(r/ln r)^n/(n+1). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A _g associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σ _k (A _g ), 1 ≤ k ≤ n} of the eigenvalues of A _g with respect to g; we call σ _k (A _g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A _g is semi-positive definite and σ _k (A _g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ₂(A _g ) is a non-negative constant and (M ⁿ , g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.     

Large sets of 3-designs from psl(2, q), with block sizes 4 and 5

We determine the distribution of 3?(q + 1,k,λ) designs, with k ? {4,5}, among the orbits of k-element subsets under the action of PSL(2,q), for q ? 3 (mod 4), on the projective line. As a consequence, we give necessary and sufficient conditions for the existence of a uniformly-PSL(2,q) large set of 3?(q + 1,k,λ) designs, with k ? {4,5} and q ≡ 3 (mod 4). © 1995 John Wiley & Sons, Inc.     

A note on preconditioned GMRES for solving singular linear systems

For solving a singular linear system Ax=b by GMRES, it is shown in the literature that if A is range-symmetric, then GMRES converges safely to a solution. In this paper we consider preconditioned GMRES for solving a singular linear system, we construct preconditioners by so-called proper splittings, which can ensure that the coefficient matrix of the preconditioned system is range-symmetric.     

Geometric computation of the numerical radius of a matrix

We use methods of geometric computing combined with hermitean matrix eigenvalue/eigenvector evaluations to find the numerical radius w(A) of a real or complex square matrix A simply, quickly, and accurately. The numerical radius w(A) is defined as the maximal distance of points in the field of values F(A) = { x^* A x | ||x||₂ = 1 }F(A) = \{ x^* A x \mid \|x\|_2 = 1 \} from zero in ℂ. Its value is an indicator of the transient behavior of the discrete dynamical system f _k + 1 = Af _k . We describe and test a MATLAB code for solving this optimization problem that can have multiple and even infinitely many solutions with maximal distance.     

Jacobians among abelian threefolds: a geometric approach

Let (A, θ) be a principally polarized abelian threefold over a perfect field k, not isomorphic to a product over [`(k)]{\bar k} . There exists a canonical extension k′/k, of degree ≤ 2, such that (A, θ) becomes isomorphic to a Jacobian over k′. The aim of this note is to give a geometric construction of this extension.