Superoptimal Preconditioned Conjugate Gradient Iteration for Image Deblurring

We study the superoptimal Frobenius operators in the two-level circulant algebra. We consider two specific viewpoints: ( 1) the regularizing properties in imaging and ( 2) the computational effort in connection with the preconditioned conjugate gradient method. Some numerical experiments illustrating the effectiveness of the proposed technique are given and discussed

A modified T. Chan’s preconditioner for Toeplitz systems

AbstractWe present a modified T. Chan’s preconditioner for solving Toeplitz linear systems by the preconditioned conjugate gradient (PCG) method in this paper. Especially, we give some results when the matrices are Hermitian positive definite Toeplitz matrices. The operation and convergence of the PCG method are discussed. Numerical examples presented illustrate the effectiveness of the preconditioner obtained

The shifted classical circulant and skew circulant splitting iterative methods for Toeplitz matrices

It is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting (denoted by CSCS), i.e., T=C-S with C a circulant matrix and S a skew circulant matrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss-Seidel (GS) iterative methods if the CSCS is convergent and that there is always a constant $\alpha$ such that the shifted CSCS iteration converges much faster than the Gauss-Seidel iteration, no matter whether the CSCS itself is convergent or not.National Natural Science Foundation of China No. 11371075The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the research innovation program of Hunan province for postgraduate students under Grant No. CX2015B374, the Portuguese Funds through FCT–Fundac˜ao para a Ciˆencia, within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Block diagonal and schur complement preconditioners for block-toeplitz systems with small size blocks

In this paper we consider the solution of Hermitian positive definite block-Toeplitz systems with small size blocks. We propose and study block diagonal and Schur complement preconditioners for such block-Toeplitz matrices. We show that for some block-Toeplitz matrices, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers where this fixed number depends only on the size of the block. Hence, conjugate gradient type methods, when applied to solving these preconditioned block-Toeplitz systems with small size blocks, converge very fast. Recursive computation of such block diagonal and Schur complement preconditioners is considered by using the nice matrix representation of the inverse of a block-Toeplitz matrix. Applications to block-Toeplitz systems arising from least squares filtering problems and queueing networks are presented. Numerical examples are given to demonstrate the effectiveness of the proposed method. © 2007 Society for Industrial and Applied Mathematics.published_or_final_versio

A preconditioned MINRES method for nonsymmetric Toeplitz matrices

Circulant preconditioning for symmetric Toeplitz linear systems is well established; theoretical guarantees of fast convergence for the conjugate gradient method are descriptive of the convergence seen in computations. This has led to robust and highly efficient solvers based on use of the fast Fourier transform exactly as originally envisaged in [G. Strang, Stud. Appl. Math., 74 (1986), pp. 171--176]. For nonsymmetric systems, the lack of generally descriptive convergence theory for most iterative methods of Krylov type has provided a barrier to such a comprehensive guarantee, though several methods have been proposed and some analysis of performance with the normal equations is available. In this paper, by the simple device of reordering, we rigorously establish a circulant preconditioned short recurrence Krylov subspace iterative method of minimum residual type for nonsymmetric (and possibly highly nonnormal) Toeplitz systems. Convergence estimates similar to those in the symmetric case are established

Matrix algebras in optimal preconditioning

AbstractThe theory and the practice of optimal preconditioning in solving a linear system by iterative processes is founded on some theoretical facts understandable in terms of a class V of spaces of matrices including diagonal algebras and group matrix algebras. The V-structure lets us extend some known crucial results of preconditioning theory and obtain some useful information on the computability and on the efficiency of new preconditioners. Three preconditioners not yet considered in literature, belonging to three corresponding algebras of V, are analyzed in detail. Some experimental results are included

Optimal rank matrix algebras preconditioners

When a linear system Ax = y is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner P and move to the preconditioned system P-1 Ax = P(-1)y. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. The construction of optimal rank preconditioners is strongly related to the possibility of splitting A as A = P R E. where E is a small perturbation and R is of low rank (Tyrtyshnikov, 1996) [1]. In the present work we extend the black-dot algorithm for the computation of such splitting for P circulant (see Oseledets and Tyrtyshnikov, 2006 [2]), to the case where P is in A, for several known low-complexity matrix algebras A. The algorithm so obtained is particularly efficient when A is Toeplitz plus Hankel like. We finally discuss in detail the existence and the properties of the decomposition A = P+R+E when A is Toeplitz, also extending to the phi-circulant and Hartley-type cases some results previously known for P circulant

On the best least squares fit to a matrix and its applications

The best least squares fit L_A to a matrix A in a space L can be useful to improve the rate of convergence of the conjugate gradient method in solving systems Ax=b as well as to define low complexity quasi-Newton algorithms in unconstrained minimization. This is shown in the present paper with new important applications and ideas. Moreover, some theoretical results on the representation and on the computation of L_A are investigated

Stationary splitting iterative methods for the matrix equation AX B = C

Stationary splitting iterative methods for solving AXB = Care considered in this paper. The main tool to derive our new method is the induced splitting of a given nonsingular matrix A = M −N by a matrix H such that (I −H) invertible. Convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants. In particular, for certain surface fitting applications, our method is much more efficient than the progressive iterative approximation (PIA), a conventional iterative method often used in computer-aided geometric design (CAGD).The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, and the Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013