Block {ω}-circulant preconditioners¶for the systems of differential equations

Bertaccini, Daniele; Ng, Michael K.

doi:10.1007/s100920300004

Cited by 27 publications

(13 citation statements)

References 26 publications

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“…The approach discussed here has to be contrasted with other approaches for the solution of systems arising from the discretization of general ODEs. One approach to solving systems with matrices of the form A ⊗ I m − ∆tB ⊗ L, obtained by using BVMs, is to use circulant approximations for the matrices A and B to precondition a Krylov subspace iteration (see for example [1,6,2,12]). This essentially comes down to splitting the matrices A and B, whereas we split the matrix L. The approaches are of course not mutually exclusive, but for the subsystems that have to be solved at each grid point in our multigrid smoothers, direct methods are hard to beat.…”

Section: Remarksmentioning

confidence: 99%

Multigrid Methods for Implicit Runge--Kutta and Boundary Value Method Discretizations of Parabolic PDEs

lent¹,

Vandewalle²

2005

SIAM J. Sci. Comput.

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Sophisticated high order time discretization methods, such as implicit Runge-Kutta and boundary value methods, are often disregarded when solving time dependent partial differential equations, despite several appealing properties. This is mainly because it is considered hard to develop efficient methods for the more complex linear systems involved. We show here that for implicit Runge-Kutta and boundary value method discretizations of time dependent parabolic problems, multigrid methods for the elliptic case can be extended in a straightforward way. The key to this approach is the use of a smoother that updates several unknowns at a spatial grid point simultaneously. Combination of the multigrid principle with both time stepping and waveform relaxation techniques are described, together with a convergence analysis. Numerical results are presented for the isotropic heat equation and a general diffusion equation with variable coefficients.Abstract. Sophisticated high order time discretization methods, such as implicit Runge-Kutta and boundary value methods, are often disregarded when solving time dependent partial differential equations, despite several appealing properties. This is mainly because it is considered hard to develop efficient methods for the more complex linear systems involved. We show here that for implicit Runge-Kutta and boundary value method discretizations of time dependent parabolic problems, multigrid methods for the elliptic case can be extended in a straightforward way. The key to this approach is the use of a smoother that updates several unknowns at a spatial grid point simultaneously. Combination of the multigrid principle with both time stepping and waveform relaxation techniques are described, together with a convergence analysis. Numerical results are presented for the isotropic heat equation and a general diffusion equation with variable coefficients.

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Section: Remarksmentioning

confidence: 99%

Multigrid Methods for Implicit Runge--Kutta and Boundary Value Method Discretizations of Parabolic PDEs

lent¹,

Vandewalle²

2005

SIAM J. Sci. Comput.

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“…Toeplitz-type linear systems arise from numerical approximation of differential equations (see, e.g., [7], [23]). Moreover, in restoration of blurred images, it is often dealt with Toeplitz matrices (see, e.g., [10], [11], [20], [24]).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, there have been several studies about simultaneously diagonalizable real matrices. In particular, in investigating the preconditioning of Toeplitz matrices (see, e.g., [21]), several matrix algebras are considered, among which the class of all circulant matrices (see, e.g., [6], [15], [16], [17], [26], [32], [42], [45]), the family of all ω-circulant matrices (see, e.g., [7], [29]), the set of all τ-matrices (see, e.g., [8]) and the family of all matrices diagonalized by the Hartley transform (see, e.g., [3], [9], [12]).…”

Section: Introductionmentioning

confidence: 99%

Gamma-matrices: a new class of simultaneously diagonalizable matrices

Boccuto¹,

Gerace²,

Giorgetti³

et al. 2021

Preprint

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In order to precondition Toeplitz systems, we present a new class of simultaneously diagonalizable real matrices, the γ-matrices, which include both symmetric circulant matrices and a subclass of the set of all reverse circulant matrices. We define some algorithms for fast computation of the product between a γ-matrix and a real vector and between two γ-matrices. Moreover, we illustrate a technique of approximating a real symmetric Toeplitz matrix by a γ-matrix, and we show that the eigenvalues of the preconditioned matrix are clustered around zero with the exception of at most a finite number of terms.

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“…The well-known circulant, block circulant-type matrices and operator norms have set up the strong basis with the work in [6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Equalities and Inequalities for Norms of Block Imaginary Circulant Operator Matrices

Jiang¹,

Hong

2015

Abstract and Applied Analysis

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Circulant, block circulant-type matrices and operator norms have become effective tools in solving networked systems. In this paper, the block imaginary circulant operator matrices are discussed. By utilizing the special structure of such matrices, several norm equalities and inequalities are presented. The normτin consideration is the weakly unitarily invariant norm, which satisfiesτA=τ(UAV). The usual operator norm and Schattenp-norm are included. Furthermore, some special cases and examples are given.

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Block {ω}-circulant preconditioners¶for the systems of differential equations

Cited by 27 publications

References 26 publications

Multigrid Methods for Implicit Runge--Kutta and Boundary Value Method Discretizations of Parabolic PDEs

Multigrid Methods for Implicit Runge--Kutta and Boundary Value Method Discretizations of Parabolic PDEs

Gamma-matrices: a new class of simultaneously diagonalizable matrices

Equalities and Inequalities for Norms of Block Imaginary Circulant Operator Matrices

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