Which circulant preconditioner is better?

Strela, Vasily; Тыртышников, Е. Е.

doi:10.1090/s0025-5718-96-00682-5

Cited by 40 publications

(20 citation statements)

References 15 publications

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“…In the case where there is strong convergence (strong or proper clustering in the terminology used in [65]) and the function f is strictly positive, we have a superlinear convergence of the related PCG methods having {P Un (A n (f ))} n as preconditioner, but we may have a sublinear behaviour when the weak convergence (weak or general clustering [65]) case occurs [60,24].…”

Section: Definition 32mentioning

confidence: 99%

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Serra¹

1999

Numerische Mathematik

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Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system A n x = b [59,9,20,5,34,62,6,10,28,45,44,46,49]. Frobenius-optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the Frobenius distance from A n . The convergence features of these PCG have been naturally studied by means of the Weierstrass-Jackson Theorem [17,36,49], owing to the profound relationship between the spectral features of the matrices A n , generated by the Fourier coefficients of a continuous function f , and the analytical properties of the symbol f itself. In this paper, we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f . On the other hand, by modifying the Korovkin Theorem to study the Frobenius-optimal preconditioning problem, we provide a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms. Finally, the multilevel case is sketched and discussed by showing that a Korovkin-type Theory also holds in a multivariate sense. Subject Classification (1991): 65F10, 47B25, 41A36, 15A18 Mathematics

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Section: Definition 32mentioning

confidence: 99%

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Serra¹

1999

Numerische Mathematik

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“…In fact, it is known that if the spectrum of a Hermitian matrix A exhibits a proper cluster at 1 as the dimension n grows, that is if all its eigenvalues lie in a fixed neighbourhood of 1 except a finite number of outliers independent on n, then the conjugate gradient method applied to the linear system Ax ¼ b converges superlinearly [1,19] (see also [22]). …”

Section: Numerical Resultsmentioning

confidence: 99%

Structured matrix algorithms for inverse scattering on the line

Mee

Seatzu

Theis

2007

Calcolo

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In this paper we present an algorithm for the construction of the superoptimal circulant preconditioner for a two-level Toeplitz linear system. The algorithm is fast, in the sense that it operates in FFT time. Numerical results are given to assess its performance when applied to the solution of two-level Toeplitz systems by the conjugate gradient method, compared with the Strang and optimal circulant preconditioners.

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“…From Table 6.4 we see that the superoptimal preconditioner is not as effective as the optimal or Strang preconditioners. This may be related to the lack of satisfaction of our theory, but it is also known in the symmetric case that when A n is ill conditioned the superoptimal preconditioner may preserve small eigenvalues that hamper convergence [15,44]. The condition numbers of the preconditioned matrices are given in Table 6.5.…”

Section: Extension To Block Matricesmentioning

confidence: 99%

A Preconditioned MINRES Method for Nonsymmetric Toeplitz Matrices

Pestana¹,

Wathen²

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. 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 Gil Strang's 'Proposal for Toeplitz Matrix Calculations' (Studies in Applied Mathematics, 74, pp. 171-176, 1986.). 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.

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Which circulant preconditioner is better?

Cited by 40 publications

References 15 publications

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Structured matrix algorithms for inverse scattering on the line

A Preconditioned MINRES Method for Nonsymmetric Toeplitz Matrices

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