Multigrid methods for toeplitz matrices

Fiorentino, Giuseppe; Serra, Stefano

doi:10.1007/bf02575816

Cited by 100 publications

(136 citation statements)

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“…MGMs for multilevel positive-definite Toeplitz matrices have been developed in the last 20 years by considering the linear system (2), independently of the continuous problem [5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we show that the techniques used in [5][6][7]13] represent a linear algebra generalization of the LFA [2][3][4], in the case of the Galerkin approach. Indeed, the matrix algebra approach leads to a condition analogous to (5) that involves the order of the zeros of the generating functions of the grid transfer operators instead of the orders of those operators.…”

mentioning

confidence: 99%

“…Therefore, by using the Toeplitz approach, other cases can be considered including the case when the ill-conditioning arises in high frequencies: we recall that the latter characterizes certain discretized integral problem, in particular those employed in the restoration of blurred signals and images. We will show that for problem (1), the LFA performed in [2][3][4] and the Toeplitz approach [5,7] (which was introduced independently) are essentially equivalent. As MGMs for Toeplitz matrices usually adopt the Galerkin approach, in the literature we find the limitation that the restriction must be proportional to the transpose of the prolongation, but we will show that in practical implementations this condition is not necessary as already observed for PDEs, e.g.…”

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confidence: 99%

“…An MGM for Toeplitz matrices was originally defined in [5] exploiting a powerful eigenvalue interlacing property connecting a band symmetric Toeplitz matrix with its counterpart in the algebra (the class of matrices diagonalized by discrete sine transforms of type I). This first proposal was extended to the multilevel case in [6,7,9].…”

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confidence: 99%

“…First, in Section 2 we present the results provided in [3]. In Section 3 we describe the MGM defined in [5][6][7] for multilevel Toeplitz matrices using the zeros of the generating functions and we compare the two-grid analysis with the results in [3]. In Section 4 we generalize the TGM described in Section 3 to the case where the restriction is not necessarily proportional to the transpose of the prolongation.…”

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confidence: 99%

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An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices

Donatelli¹

2010

Numerical Linear Algebra App

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SUMMARYLocal Fourier analysis (LFA) is a classical tool for proving convergence theorems for multigrid methods (MGMs). In particular, we are interested in optimal convergence, i.e. convergence rates that are independent of the problem size. For elliptic partial differential equations (PDEs), a well-known optimality result requires that the sum of the orders of the grid transfer operators is not lower than the order of the PDE approximated. Analogously, when dealing with MGMs for Toeplitz matrices, a well-known optimality condition concerns the position and the order of the zeros of the symbols of the grid transfer operators. In this work we show that in the case of elliptic PDEs with constant coefficients, the two different approaches lead to an equivalent condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA, which allows to deal not only with differential problems but also for instance with integral problems. The equivalence of the two approaches gives the possibility of using grid transfer operators with different orders also for MGMs for Toeplitz matrices. We give also a class of grid transfer operators related to the B-spline's refinement equation and study their geometric properties. Numerical experiments confirm the correctness of the proposed analysis.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices

Donatelli¹

2010

Numerical Linear Algebra App

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Algebraic description and automatic generation of multigrid methods in SPIRAL

Bolten

Franchetti

Lengauer

et al. 2017

Concurrency and Computation

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SPIRAL is an autotuning, program generation and code synthesis system that offers a fully automatic generation of highly optimized target codes, customized for the specific execution platform at hand. Initially, SPIRAL was targeted at problem domains in digital signal processing, later also at basic linear algebra. We open SPIRAL up to a new, practically relevant and challenging domain: multigrid solvers. SPIRAL is driven by algebraic transformation rules. We specify a set of such rules for a simple multigrid solver with a Richardson smoother for a discretized square 2D Poisson equation with Dirichlet boundary conditions. We present the target code that SPIRAL generates in static single-assignment form and discuss its performance. While this example required no changes of or extensions to the SPIRAL system, more complex multigrid solvers may require small adaptations

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Preconditioners for Krylov subspace methods: An overview

Pearson

Pestana

2020

GAMM-Mitteilungen

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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.

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Multigrid methods for toeplitz matrices

Cited by 100 publications

References 5 publications

An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices

An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices

Algebraic description and automatic generation of multigrid methods in SPIRAL

Preconditioners for Krylov subspace methods: An overview

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