Toeplitz Preconditioners for Toeplitz Systems with Nonnegative Generating Functions

The preconditioned conjugate gradient method is employed tosolve Toeplitz systems T_nx = b where the generating functionsof the n-by-n Toeplitz matrices T_n are continuous nonnegativeperiodic functions defined in [–,]. The preconditionedC_n are band Toeplitz matrices with band-widths independent ofn. We prove that the spectra of C_n^-1T_n are uniformly boundedby constants independent of n. In particular, we show that thesolutions of T_nx = b can be obtained in O(nlogn) operations.     

Locally Optimal Block Preconditioned Conjugate Gradient Method for Hierarchical Matrices

We present a method of almost linear complexity to approximate some (inner) eigenvalues of symmetric self-adjoint integral or differential operators. Using ℋ-arithmetic the discretisation of the operator leads to a large hierarchical (ℋ-) matrix M. We assume that M is symmetric, positive definite. Then we compute the smallest eigenvalues by the locally optimal block preconditioned conjugate gradient method (LOBPCG), which has been extensively investigated by Knyazev and Neymeyr. Hierarchical matrices were introduced by W. Hackbusch in 1998. They are data-sparse and require only O(nlog₂ n) storage. There is an approximative inverse, besides other matrix operations, within the set of ℋ-matrices, which can be computed in linear-polylogarithmic complexity. We will use the approximative inverse as preconditioner in the LOBPCG method. Further we combine the LOBPCG method with the folded spectrum method to compute inner eigenvalues of M. This is equivalent to the application of LOBPCG to the matrix M_μ = (M − μI)² . The matrix M_μ is symmetric, positive definite, too. Numerical experiments illustrate the behavior of the suggested approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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 convergence results and preconditioning strategies for the conjugate gradient method

Instead of the standard estimate in terms of the spectral condition number we develop a new CG iteration number estimate depending on the quantity B = 1/ntr M/(det M)^1/n, where M is an n × n preconditioned matrix. A new family of iterative methods for solving symmetric positive definite systems based on B-reducing strategies is described. Numerical results are presented for the new algorithms and compared with several well-known preconditioned CG methods.     

Fastiterative methods for least squares estimations

Least squares estimations have been used extensively in many applications, e.g. system identification and signal prediction. When the stochastic process is stationary, the least squares estimators can be found by solving a Toeplitz or near-Toeplitz matrix system depending on the knowledge of the data statistics. In this paper, we employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Our proposed circulant preconditioners are derived from the spectral property of the given stationary process. In the case where the spectral density functions() of the process is known, we prove that ifs() is a positive continuous function, then the spectrum of the preconditioned system will be clustered around 1 and the method converges superlinearly. However, if the statistics of the process is unknown, then we prove that with probability 1, the spectrum of the preconditioned system is still clustered around 1 provided that large data samples are taken. For finite impulse response (FIR) system identification problems, our numerical results show that annth order least squares estimator can usually be obtained inO(n logn) operations whenO(n) data samples are used. Finally, we remark that our algorithm can be modified to suit the applications of recursive least squares computations with the proper use of sliding window method arising in signal processing applications.Research supported in part by HKRGC grant no. 221600070, ONR contract no. N00014-90-J-1695 and DOE grant no. DE-FG03-87ER25037.     

Semi-convergence analysis of preconditioned deteriorated PSS iteration method for singular saddle point problems

In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.     

An error bound for the modified successive overrelaxation method

In this paper we consider the modified successive overrelaxation(MSOR)methodto appropriate the solution of the linear system D^-1/2 Ax =D^-1/2b, where A is a symmetric, positive definite and consistentlyordered matrix and D is a diagonal matrix with the diagonalidentical to that of A. The main purpose of this paper is to obtain some theoreticalresults, namely a bound for the norm of _n = v –v_n in termsof the norms _n –v_n-1, _n+1 –v_n and their inner product,where v =D^-1/2 x and v_n is the nth iteration vector, obtainedusing the (MSOR)method.     

A splitting preconditioner for saddle point problems

For large sparse systems of linear equations iterative techniques are attractive. In this paper, we study a splitting method for an important class of symmetric and indefinite system. Theoretical analyses show that this method converges to the unique solution of the system of linear equations for all t>0 (t is the parameter). Moreover, all the eigenvalues of the iteration matrix are real and nonnegative and the spectral radius of the iteration matrix is decreasing with respect to the parameter t. Besides, a preconditioning strategy based on the splitting of the symmetric and indefinite coefficient matrices is proposed. The eigensolution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomials for the preconditioned matrix is obtained. Numerical experiments of a model Stokes problem and a least‐squares problem with linear constraints presented to illustrate the effectiveness of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedurefor computing analogues of Young's seminormal representationsof the symmetric groups. The method is to generalize the Jucys-Murphyelements in the group algebras of the symmetric groups to arbitraryWeyl groups and Iwahori-Hecke algebras. The combinatorics ofthese elements allows one to compute irreducible representationsexplicitly and often very easily. In this paper we do thesecomputations for Weyl groups and Iwahori-Hecke algebras of typesA_n, B_n, D_n, G₂. Although these computations are in reach fortypes F₄, E₆ and E₇, we shall postpone this to another work.1991 Mathematics Subject Classification: primary 20F55, 20C15;secondary 20C30, 20G05.     

Symmetric Groups as Products of Abelian Subgroups

A proof is given that the full symmetric group over any infiniteset is the product of finitely many Abelian subgroups. In fact,289 subgroups suffice. Sharp bounds are also obtained on theminimal number k, such that the finite symmetric group S_n isthe product of k Abelian subgroups. Using this, S_n is provedto be the product of 72n^1/2(log n)^3/2 cyclic subgroups. 2000Mathematics Subject Classification 20B30, 20D40.     

Greville��s method for preconditioning least squares problems

In this paper, we propose a preconditioning algorithm for least squares problems $\displaystyle{\min_{x\in{{\mathbb{R}}}^n}}\|b-Ax\|_2$ , where A can be matrices with any shape or rank. The preconditioner is constructed to be a sparse approximation to the Moore?CPenrose inverse of the coefficient matrix A. For this preconditioner, we provide theoretical analysis to show that under our assumption, the least squares problem preconditioned by this preconditioner is equivalent to the original problem, and the GMRES method can determine a solution to the preconditioned problem before breakdown happens. In the end of this paper, we also give some numerical examples showing the performance of the method.     

Preconditioning by inverting the Laplacian: an analysis of the eigenvalues

Wolfgang Hackbusch ^{We study the eigenvalues of the operator generated by usingthe inverse of the Laplacian as a preconditioner for self-adjointsecond-order elliptic partial differential equations with smoothcoefficients. It is well-known that the spectral condition numberof the preconditioned operator can be bounded by , where k is the uniformly positive coefficientof the second-order elliptic equation. The purpose of this paperis to study the spectrum of the preconditioned operator. Wewill show that there is a strong relation between the spectrumof this operator and the range of the coefficient function.In the continuous case, we prove, both for mappings definedon Sobolev spaces and in terms of generalized functions, thatthe spectrum of the preconditioned operator contains the rangeof the coefficient function k. In the discrete case, we indicateby numerical examples that the entire discrete spectrum is approximatelygiven by values of k.}     

关于具优势对称部分的不定线性代数方程组的分裂极小残量算法

１．引 言 考虑大型稀疏线性代数方程组 为利用系数矩阵的稀疏结构以尽可能减少存储空间和计算开销，Ｋｒｙｌｏｖ子空间迭代算法［１，１６，２３］及其预处理变型［６，８，１３，１８，１９］通常是求解（１）的有效而实用的方法．当系数矩阵对称正定时，共轭梯度法（ＣＧ（     

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.     

Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations

<正>Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.     

A comparison of abstract versions of deflation,balancing and additive coarse grid correction preconditioners

In this paper we consider various preconditioners for the conjugate gradient (CG) method to solve large linear systems of equations with symmetric positive definite system matrix. We continue the comparison between abstract versions of the deflation, balancing and additive coarse grid correction preconditioning techniques started in (SIAM J. Numer. Anal. 2004; 42 :1631–1647; SIAM J. Sci. Comput. 2006; 27 :1742–1759). There the deflation method is compared with the abstract additive coarse grid correction preconditioner and the abstract balancing preconditioner. Here, we close the triangle between these three methods. First of all, we show that a theoretical comparison of the condition numbers of the abstract additive coarse grid correction and the condition number of the system preconditioned by the abstract balancing preconditioner is not possible. We present a counter example, for which the condition number of the abstract additive coarse grid correction preconditioned system is below the condition number of the system preconditioned with the abstract balancing preconditioner. However, if the CG method is preconditioned by the abstract balancing preconditioner and is started with a special starting vector, the asymptotic convergence behavior of the CG method can be described by the so‐called effective condition number with respect to the starting vector. We prove that this effective condition number of the system preconditioned by the abstract balancing preconditioner is less than or equal to the condition number of the system preconditioned by the abstract additive coarse grid correction method. We also provide a short proof of the relationship between the effective condition number and the convergence of CG. Moreover, we compare the A‐norm of the errors of the iterates given by the different preconditioners and establish the orthogonal invariants of all three types of preconditioners. Copyright © 2008 John Wiley & Sons, Ltd.     

Enumerating Transitive Finite Permutation Groups

Denote by f(n) the number of subgroups of the symmetric groupSym(n) of degree n, and by f_trans(n) the number of its transitivesubgroups. It was conjectured by Pyber [9] that almost all subgroupsof Sym(n) are not transitive, that is, f_trans(n)/f(n) tendsto 0 when n tends to infinity. It is still an open questionwhether or not this conjecture is true. The difficulty comesfrom the fact that, from many points of view, transitivity isnot a really strong restriction on permutation groups, and thereare too many transitive groups [9, Sections 3 and 4]. In thispaper we solve the problem in the particular case of permutationgroups of prime power degree, proving the following result.1991 Mathematics Subject Classification 20B05, 20D60.     

Vector Majorization Via Hessenberg Matrices

In this paper it is proved that, for real n-vectors x and y,x is majorized by y if and only if x = PHQy for some permutationmatrices P, Q, and for some doubly stochastic matrix H whichis a direct sum of doubly stochastic Hessenberg matrices. Thisresult reveals that any n-vector which is majorized by a vectory can be expressed as a convex combination of at most (n² –n + 2)/2 permutations of y.