Circulant preconditioners for solving differential equations with multidelays

AbstractWe consider the solution of differential equations with multidelays by using boundary value methods (BVMs). These methods require the solution of some nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed to solve these linear systems. If an Ak1,k2-stable BVM is used, we show that our preconditioner is invertible and the spectrum of the preconditioned matrix is clustered. It follows that when the GMRES method is applied to solving the preconditioned systems, the method would converge fast. Numerical results are given to show the effectiveness of our methods

Recursive-Based PCG Methods for Toeplitz Systems with Nonnegative Generating Functions

In this paper, we consider the solutions of symmetric positive definite, but ill-conditioned, Toeplitz systems An x = b. Here we propose to solve the system by the recursive-based preconditioned conjugate gradient method. The idea is to use the inverse of Am (the principal submatrix of An with the Gohberg--Semencul formula as a preconditioner for An. The inverse of Am can be generated recursively by using the formula until m is small enough. The construction of the preconditioners requires only the entries of An and does not require the explicit knowledge of the generating function f of An. We show that if f is a nonnegative, bounded, and piecewise continuous even function with a finite number of zeros of even order, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers. Hence the conjugate gradient method, when applied to solving the preconditioned system, converges very quickly. Numerical results are included to illustrate the effectiveness of our approach.published_or_final_versio

Preconditioners for ill-conditioned Toeplitz matrices

This paper is concerned with the solution of systems of linear equations AN&#967

Symmetrization Techniques in Image Deblurring

This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary conditions. More precisely, we first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. The results of many numerical experiments are reported to show the effectiveness of the new preconditioning techniques, including when considering the deblurring of sparse images

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

smt: a Matlab structured matrices toolbox

We introduce the smt toolbox for Matlab. It implements optimized storage and fast arithmetics for circulant and Toeplitz matrices, and is intended to be transparent to the user and easily extensible. It also provides a set of test matrices, computation of circulant preconditioners, and two fast algorithms for Toeplitz linear systems.Comment: 19 pages, 1 figure, 1 typo corrected in the abstrac

Strang-type Preconditioners for Solving Linear Systems from Neutral Delay Di®erential Equations

We study the solution of neutral delay di®erential equations (NDDEs) by using boundary value methods (BVMs). The BVMs require the solution of nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed to solve these linear systems. We show that if an Ak1;k2-stable BVM is used for solving an m-by-m system of NDDEs, then our preconditioner is invertible and the spectrum of the preconditioned system is clustered. It follows that when the GMRES method is applied to the preconditioned systems, the method could converge fast. Numerical results are given to show that our method is e®ective