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

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

Preconditioners for nondefinite Hermitian Toeplitz systems

This paper is concerned with the construction of circulant preconditioners for Toeplitz systems arising from a piecewise continuous generating function with sign changes. If the generating function is given, we prove that for any Σ > 0, only Ο(log N) eigenvalues of our preconditioned Toeplitz systems of size N x N are not contained in [-1- Σ, -1+Σ]U [1-Σ, 1+Σ]. The result can be modified for trigonometric preconditioners. We also suggest circulant preconditioners for the case that the generating function is not explicitly known and show that only Ο(log N) absolute values of the eigenvalues of the preconditioned Toeplitz systems are not contained in a positive interval on the real axis. Using the above results, we conclude that the preconditioned minimal residual method requires only Ο(N log² N) arithmetical operations to achive a solution of prescribed precision if the spectral condition numbers of the Toeplitz systems increase at most polynomial in N. We present various numerical tests

BTTB preconditioners for BTTB least squares problems

AbstractIn this paper, we consider solving the least squares problem minx‖b-Tx‖2 by using preconditioned conjugate gradient (PCG) methods, where T is a large rectangular matrix which consists of several square block-Toeplitz–Toeplitz-block (BTTB) matrices and b is a column vector. We propose a BTTB preconditioner to speed up the PCG method and prove that the BTTB preconditioner is a good preconditioner. We then discuss the construction of the BTTB preconditioner. Numerical examples, including image restoration problems, are given to illustrate the efficiency of our BTTB preconditioner. Numerical results show that our BTTB preconditioner is more efficient than the well-known Level-1 and Level-2 circulant preconditioners

Spectral analysis for preconditioning of multi-dimensional Riesz fractional diffusion equations

In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which will generate symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2,3/2) and thus the preconditioned conjugate gradient method converges linearly. The proposed method can be extended to multi-level Toeplitz matrices generated by functions with zeros of fractional order. Our theoretical results fill in a vacancy in the literature. Numerical examples are presented to demonstrate our new theoretical results in the literature and show the convergence performance of the proposed preconditioner that is better than other existing preconditioners

Circulant preconditioners from B-splines and their applications.

by Tat-Ming Tso.Thesis (M.Phil.)--Chinese University of Hong Kong, 1997.Includes bibliographical references (p. 43-45).Chapter Chapter 1 --- INTRODUCTION --- p.1Chapter §1.1 --- Introduction --- p.1Chapter §1.2 --- Preconditioned Conjugate Gradient Method --- p.3Chapter §1.3 --- Outline of Thesis --- p.3Chapter Chapter 2 --- CIRCULANT AND NON-CIRCULANT PRECONDITIONERS --- p.5Chapter §2.1 --- Circulant Matrix --- p.5Chapter §2.2 --- Circulant Preconditioners --- p.6Chapter §2.3 --- Circulant Preconditioners from Kernel Function --- p.8Chapter §2.4 --- Non-circulant Band-Toeplitz Preconditioners --- p.9Chapter Chapter 3 --- B-SPLINES --- p.11Chapter §3.1 --- Introduction --- p.11Chapter §3.2 --- New Version of B-splines --- p.15Chapter Chapter 4 --- CIRCULANT PRECONDITIONERS CONSTRUCTED FROM B-SPLINES --- p.24Chapter Chapter 5 --- NUMERICAL RESULTS AND CONCLUDING REMARKS --- p.28Chapter Chapter 6 --- APPLICATIONS TO SIGNAL PROCESSING --- p.37Chapter §6.1 --- Introduction --- p.37Chapter §6.2 --- Preconditioned regularized least squares --- p.39Chapter §6.3 --- Numerical Example --- p.40REFERENCES --- p.4

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

Fast algorithms for integral equations.

by Wing-Fai Ng.Thesis (M.Phil.)--Chinese University of Hong Kong, 1996.Includes bibliographical references (leaves 7-8).Abstract --- p.1-2Introduction --- p.3-6References --- p.7-8Paper I --- p.9-32Paper II --- p.33-6

Fast algorithm for ill-conditioned toeplitz and toeplitz-like systems.

by Hai-Wei Sun.Thesis (Ph.D.)--Chinese University of Hong Kong, 1996.Includes bibliographical references.Abstracts --- p.1Summary --- p.3Introduction --- p.3Summary of the papers A-C --- p.5Paper A --- p.19Paper B --- p.34Paper C --- p.6

Preconditioners for Krylov subspace methods: An overview

When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind