Multigrid Methods for Block Toeplitz Matrices with Small Size Blocks

Huckle, Thomas; Staudacher, Jochen

doi:10.1007/s10543-006-0047-2

Cited by 14 publications

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“…Like the standard projection b(x, y) = (1 + cos(x))(1 + cos(y)) an interpolation has to have the zeros (0, π), (π, 0) and (π, π) ( [7,8,1]). A simple function with these properties can be derived by consideringb(x, y) := f (x, y + π)f (x + π, y)f (x + π, y + π) ( [13,9,10]). To obtain the block symbol forb we use the block symbol for the three factors that are only slight modifications of the given function f 5 :…”

Section: The General Solution Is Given Bymentioning

confidence: 99%

“…If this cannot be achieved then at least we can try to adjust the projection and the smoother in such a way that the TGS-symbol is of small rank and norm. The tool we are using in the following is a compact Fourier analysis developed for Multigrid methods for Toeplitz matrices ( [7,8,4,1,20,13,14,9,10]). In contrast to the classical Fourier analysis, here we consider the constant coefficient case with Dirichlet boundary conditions resulting in the 1D case in a Toeplitz matrix The analysis is then based on the symbol or generating function of T n :…”

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confidence: 99%

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Compact Fourier Analysis for Designing Multigrid Methods

Huckle¹

2008

SIAM J. Sci. Comput.

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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.

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Section: The General Solution Is Given Bymentioning

confidence: 99%

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confidence: 99%

Compact Fourier Analysis for Designing Multigrid Methods

Huckle¹

2008

SIAM J. Sci. Comput.

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“…In the last 20 years multigrid methods have gained a remarkable reputation as fast solvers for structured matrices associated to shift invariant operators, where the size n is large and the system shows a conditioning growing polynomially with n (see [10,24,11,5,15,16,29,2,27,14] and references therein). Under suitable mild assumptions, the considered techniques are optimal showing linear or almost linear (O(n log n) arithmetic operations as the celebrated fast Fourier transform (FFT)) complexity for reaching the solution within a preassigned accuracy and a convergence rate independent of the size n of the involved system.…”

Section: Introductionmentioning

confidence: 99%

“…The latter is not compulsory so that, by choosing a different size reduction from n to n/g and g > 2, we can overcome the pathology induced by the mirror points. A different approach for dealing with such pathologies was proposed in [5] and further analyzed in [16], by exploiting a projection strategy based on matrix-valued symbols.…”

Section: Introductionmentioning

confidence: 99%

Multigrid methods for Toeplitz linear systems with different size reduction

2011

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Starting from the spectral analysis of g-circulant matrices, we study the convergence of a multigrid method for circulant and Toeplitz matrices with various size reductions. We assume that the size n of the coefficient matrix is divisible by g ≥ 2 such that at the lower level the system is reduced to one of size n/g, by employing g-circulant based projectors. We perform a rigorous two-grid convergence analysis in the circulant case and we extend experimentally the results to the Toeplitz setting, by employing structure preserving projectors. The optimality of the two-grid method and of the multigrid method is proved, when the number θ ∈ N of recursive calls is such that 1 < θ < g. The previous analysis is used also to overcome some pathological cases, in which the generating function has zeros located at "mirror points" and the standard two-grid method with g = 2 is not optimal. The numerical experiments show the correctness and applicability of the proposed ideas, both for circulant and Toeplitz matrices.

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“…circulant matrices, and trigonometric functions. It was analyzed by S. Serra Capizzano, R. Chan, T. Huckle, and coauthors in [10,11,5,14,18,15,20,21]. In this paper we complete this formal approach in order to represent a full two-grid step in terms of the block symbol, called also block generating (matrix) function.…”

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Compact Fourier Analysis for Multigrid Methods based on Block Symbols

Huckle¹,

Kravvaritis²

2012

SIAM J. Matrix Anal. & Appl.

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Abstract. The notion of Compact Fourier Analysis (CFA) is discussed. The CFA allows description of multigrid (MG) in a nutshell and offers a clear overview on all MG components. The principal idea of CFA is to model the MG mechanisms by means of scalar generating functions and matrix functions (block symbols). The formalism of the CFA approach is presented by describing the symbols of the fine and coarse grid problems, the prolongation and restriction, the smoother, and the coarse grid correction, resp. smoothing corrections. CFA uses matrix functions and their features (e.g. product, inverse, adjoint, norm, spectral radius, eigenvectors, eigenvalues), and scalar functions and their roots. This leads to an elementary description and allows for an easy analysis of MG algorithms. A first application is to utilize the CFA for deriving MG as a direct solver, i.e. an MG cycle that will converge in just one iteration step. Necessary and sufficient conditions that have to be fulfilled by the MG components are given for obtaining MG functioning as a direct solver. Furthermore, new general and practical smoothers and transfer operators that lead to efficient MG methods are introduced. In addition, we study sparse approximations of the Galerkin coarse grid operator yielding efficient and practicable MG algorithms (approximately direct solvers). Numerical experiments demonstrate the theoretical results.Key words. multigrid, Fourier analysis, generating function, block symbol, Toeplitz matrices AMS subject classifications. 65N55, 65F10, 65F15, 65N12, 15A121. Introduction. A crucial point for the efficiency of a multigrid (MG) method is the appropriate choice of its components, which allows for an efficient interplay between smoother and coarse grid correction. In many cases, this coordination can be made by use of Local Fourier Analysis (LFA), which is an important quantitative tool for the development of powerful MG methods [3,24,23,13,4]. This approach has been generalized for structured matrices by employing generating functions expressing, e.g. the symbol of the smoother, of the projection or of the standard discrete Laplace operator in terms of trigonometric polynomials. It is based on the connection between Toeplitz, resp. circulant matrices, and trigonometric functions. It was analyzed by S. Serra Capizzano, R. Chan, T. Huckle, and coauthors in [10,11,5,14,18,15,20,21]. In this paper we complete this formal approach in order to represent a full two-grid step in terms of the block symbol, called also block generating (matrix) function. Furthermore, we show how to use the block symbol formalism for a multigrid Fourier analysis.We consider the notion of Compact Fourier Analysis (CFA), which can be seen as a reformulation and generalization of LFA based on matrix functions. Instead of discrete operators on a grid we consider analytic matrix functions (block symbols) of small order that capture the behavior of the full matrices [14]. This allows the use of matrix features, such as product and eigendecomposition, for descr...

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Multigrid Methods for Block Toeplitz Matrices with Small Size Blocks

Cited by 14 publications

References 18 publications

Compact Fourier Analysis for Designing Multigrid Methods

Compact Fourier Analysis for Designing Multigrid Methods

Multigrid methods for Toeplitz linear systems with different size reduction

Compact Fourier Analysis for Multigrid Methods based on Block Symbols

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