Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems

Serra, Stefano

doi:10.1090/s0025-5718-97-00833-8

Cited by 62 publications

(30 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The combination of equations (17) and (19), in light of the powerful Lemma 6.1, allows one to state that {P k(n) (A n (f ))} and {A n (f )} are both distributed as f and that their difference is clustered at zero. Therefore the main result holds for the continuous functions.…”

Section: Main Theoremsmentioning

confidence: 99%

Korovkin tests, approximation, and ergodic theory

Serra‐Capizzano

2000

Math. Comp.

View full text Add to dashboard Cite

Abstract. We consider sequences of s · k(n) × t · k(n) matrices {An(f )} with a block structure spectrally distributed as an L 1 p-variate s × t matrix-valued function f , and, for any n, we suppose that An(·) is a linear and positive operator. For every fixed n we approximate the matrix An(f ) in a suitable linear space Mn of s · k(n) × t · k(n) matrices by minimizing the Frobenius norm of An(f ) − Xn when Xn ranges over Mn. The minimizerXn is denoted by P k(n) (An(f )). We show that only a simple Korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts:1. the sequence {P k(n) (An(f ))} is distributed as f , 2. the sequence {An(f )−P k(n) (An(f ))} is distributed as the constant function 0 (i.e. is spectrally clustered at zero). The first result is an ergodic one which can be used for solving numerical approximation theory problems. The second has a natural interpretation in the theory of the preconditioning associated to cg-like algorithms.

show abstract

Section: Main Theoremsmentioning

confidence: 99%

Korovkin tests, approximation, and ergodic theory

Serra‐Capizzano

2000

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…Let us collect some useful properties of such matrices ( [22,18,19,12])(in the following R j denote matrices of small rank, independent of n):…”

mentioning

confidence: 99%

Compact Fourier Analysis for Designing Multigrid Methods

Huckle¹

2008

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. The convergence of Multigrid methods can be analyzed based on a Fourier analysis of the method or by proving certain inequalities that have to be fulfilled by the smoother and by the coarse grid correction separately. Here, we analyze the Multigrid method for the constant coefficient Poisson equation with a compact Fourier analysis using the formalism of multilevel Toeplitz matrices and their generating functions or symbols. The Fourier analysis is applied for determining the smoothing factor and the overall error of the combined smoothing and coarse grid correction error reduction of a Twogrid step by representing the Twogrid step explicitly by a symbol. If the effects of the smoothing correction and the coarse grid correction are orthogonal to each other, then in a Twogrid step the error is removed in one step and the Twogrid method can be considered as a direct solver. If the coarse linear system is identical to the original matrix, then the same projection and smoother again make the Twogrid step a direct solver. Hence, the multigrid cycle has to be applied only once with one smoothing step on each level and therefore the whole Multigrid method can be considered as a direct solver. In this paper we want to identify Multigrid as a direct solver in 1D with coarse system derived by linear and constant interpolation, and in 2D for a modified projection related to [21, paragraph A.2.3]. By studying not only smoothing and coarse grid correction separately but the symbol of the full Twogrid step as well, we are furthermore able to derive better convergence estimates and information on how to find efficient combinations of projection and smoother. As smoothers we consider also approximate inverse smoothers, colored smoothers that take into account the different character of grid points, and rank reducing smoothers that coincide with the given matrix on a large number of entries. Numerical examples show the effectiveness of the new approach.

show abstract

“…Moreover, if the convergence is also uniform, that is M ǫ does not depend on ǫ, the number of iterations initially decreases as the order N increases and then it stabilizes to a constant value. Therefore the associated PCG method is comparable with the one devised in [32].…”

Section: Tr(p Un (A)) = Tr(a)mentioning

confidence: 84%

Fast and numerically stable algorithms for discrete Hartley transforms and applications to preconditioning

Aricò

Serra‐Capizzano

Tasche

2005

Communications in Information and Systems

View full text Add to dashboard Cite

Abstract. The discrete Hartley transforms (DHT) of types I -IV and the related matrix algebras are discussed. We prove that any of these DHTs of length N = 2 t can be factorized by means of a divide-and-conquer strategy into a product of sparse, orthogonal matrices where in this context sparse means at most two nonzero entries per row and column. The sparsity joint with orthogonality of the matrix factors is the key for proving that these new algorithms have low arithmetic costs equal to 5 2 N log 2 (N )+ O(N ) arithmetic operations and an excellent normwise numerical stability. Further, we consider the optimal Frobenius approximation of a given symmetric Toeplitz matrix generated by an integrable symbol in a Hartley matrix algebra. We give explicit formulas for computing these optimal approximations and discuss the related preconditioned conjugate gradient (PCG) iterations. By using the matrix approximation theory, we prove the strong clustering at unity of the preconditioned matrix sequences under the sole assumption of continuity and positivity of the generating function. The multilevel case is also briefly treated. Some numerical experiments concerning DHT preconditioning are included.

show abstract

Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems

Cited by 62 publications

References 29 publications

Korovkin tests, approximation, and ergodic theory

Korovkin tests, approximation, and ergodic theory

Compact Fourier Analysis for Designing Multigrid Methods

Fast and numerically stable algorithms for discrete Hartley transforms and applications to preconditioning

Contact Info

Product

Resources

About