Fast computation of two-level circulant preconditioners

Mee, Cornelis van der; Rodriguez, Giuseppe; Seatzu, Sebastiano

doi:10.1007/s11075-005-9011-5

Cited by 5 publications

(2 citation statements)

References 141 publications

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“…The code in the functions strang, optimal, and superopt (see Table 3) was developed by one of the authors during the research which led to [22], and the details of the algorithms are described in that paper. To compute the superoptimal preconditioner, it is possible to use either the method introduced in [3], or the one from [20].…”

Section: Linear Systems and Preconditionersmentioning

confidence: 99%

smt: a Matlab toolbox for structured matrices

Redivo–Zaglia

Rodriguez

2012

Numer Algor

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The full exploitation of the structure of large scale algebraic problems is often crucial for their numerical solution. Matlab is a computational environment which supports sparse matrices, besides full ones, and allows one to add new types of variables (classes) and define the action of arithmetic operators and functions on them. The smt toolbox for Matlab introduces two new classes for circulant and Toeplitz matrices, and implements optimized storage and fast computational routines for them, transparently to the user. The toolbox, available in Netlib, is intended to be easily extensible, and provides a collection of test matrices and a function to compute three circulant preconditioners, to speed up iterative methods for linear systems. Moreover, it incorporates a simple device to add to the toolbox new routines for solving Toeplitz linear systems

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Section: Linear Systems and Preconditionersmentioning

confidence: 99%

smt: a Matlab toolbox for structured matrices

Redivo–Zaglia

Rodriguez

2012

Numer Algor

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“…We will use a preconditioner to increase the rate of convergence of the iterative method. BTTB matrices are commonly preconditioned by block circulant with circulant block (BCCB) matrices; see [1,3,12,19,20] for discussions, illustrations, and further references. The use of BCCB preconditioners for BTTB matrices is attractive both due to the spectral properties of the preconditioned matrix and because of the possibility to evaluate a matrix-vector product with a preconditioned matrix of order n 1 n 2 in only O(n 1 n 2 log 2 (n 1 n 2 )) flops with the aid of the FFT.…”

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confidence: 99%

Circulant preconditioners for discrete ill-posed Toeplitz systems

2016

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Dedicated to Ken Hayami on the occasion of his 60th birthday.Abstract. Circulant preconditioners are commonly used to accelerate the rate of convergence of iterative methods when solving linear systems of equations with a Toeplitz matrix. Block extensions that can be applied when the system has a block Toeplitz matrix with Toeplitz blocks also have been developed. This paper is concerned with preconditioning of linear systems of equations with a symmetric block Toeplitz matrix with symmetric Toeplitz blocks that stem from the discretization of a linear ill-posed problem. The right-hand side of the linear systems represents available data and is assumed to be contaminated by error. These kinds of linear systems arise, e.g., in image deblurring problems. It is important that the preconditioner does not affect the invariant subspace associated with the smallest eigenvalues of the block Toeplitz matrix to avoid severe propagation of the error in the right-hand side. A perturbation result indicates how the dimension of the subspace associated with the smallest eigenvalues should be chosen and allows the determination of a suitable preconditioner when an estimate of the error in the right-hand side is available. This estimate also is used to decide how many iterations to carry out by a minimum residual iterative method. Applications to image restoration are presented.

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