Multigrid methods for indefinite Toeplitz matrices

Fiorentino, Giuseppe; Serra, Stefano

doi:10.1007/bf02576002

Cited by 4 publications

(6 citation statements)

References 34 publications

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“…Unfortunately, as remarked in the last section of [13] the lower level matrix A k = (p k n ) T T n (f )p k n is not Toeplitz unless the degree of p does not exceed 1. Actually the matrix A k can be written as T k (f ) + R k n wheref is the same function introduced in Proposition 2.2 and where R k n is a matrix of constant rank independent of n and only depending linearly on the degree of p. The problem is computationally not negligible since at each level the rank of the correction grows so that in [15] we proposed modified cutting matrices T k n [t] in order to preserve the exact Toeplitzness at each level. More precisely, for any t ≥ 0, T k n [t] is of dimension n × (k − 2t) and coincides with the submatrix of T k n obtained by deleting its first and last t columns.…”

Section: Ifmentioning

confidence: 98%

“…The matrix T n (f ) has external dimension n 1 × n 1 , with (d − 1)-level Toeplitz blocks of dimension (n 2 · · · n d )×(n 2 · · · n d ) so that the i-th pair (s i , t i ) denotes a block at the i-th level of Toeplitzness. The description is naturally recursive so that when we arrive at expressing the deepest level, then we find the Fourier coefficients {a k } k formally determined in (15).…”

Section: Multilevel Structured Matricesmentioning

confidence: 99%

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Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences

Serra‐Capizzano

2002

Numerische Mathematik

130

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The solution of large Toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers [13,14,22]. The technique was modified in [6,36] and a rigorous proof of convergence of the TGM (two-grid method) was given in the special case where the generating function has only a zero at x0 = 0 of order at most two. Here, by extending the latter approach, we perform a complete analysis of convergence of the TGM under the sole assumption that / is nonnegative and with a zero at x0 = 0 of finite order. An extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic PDEs with nonconstant coefficients and of any order is then discussed

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Section: Ifmentioning

confidence: 98%

Section: Multilevel Structured Matricesmentioning

confidence: 99%

Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences

Serra‐Capizzano

2002

Numerische Mathematik

130

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“…We are using multigrid V‐cycle method to solve linear system arising from the discretization of HOC difference scheme. To show the performance and to match the results of HOC scheme, we use the full weighting projection operator on uniform grids as proposed in the previous researches 44; 57–59 for Toeplitz matrices. Let Au = b be the linear system with u , b ∈ R n ; the smoother is

u^{j + 1} = S u^{j} + b_{1} = (u^{(j)}, b_{1}),

where S = I − M −1 A is the iteration matrix,

b_{1} = M^{- 1} b \in R^{n} .

Consider that P is the projection operator.…”

Section: Multigrid Methodsmentioning

confidence: 99%

A higher‐order unconditionally stable scheme for the solution of fractional diffusion equation

Ghaffar

Ullah

Badshah

et al. 2020

Math Methods in App Sciences

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In this paper, a higher‐order compact finite difference scheme with multigrid algorithm is applied for solving one‐dimensional time fractional diffusion equation. The second‐order derivative with respect to space is approximated by higher‐order compact difference scheme. Then, Grünwald–Letnikov approximation is used for the Riemann–Liouville time derivative to get an implicit scheme. The scheme is based on a heptadiagonal matrix with eighth‐order accurate local truncation error. Fourier analysis is used to analyze the stability of higher‐order compact finite difference scheme. Matrix analysis is used to show that the scheme is convergent with the accuracy of eighth‐order in space. Numerical experiments confirm our theoretical analysis and demonstrate the performance and accuracy of our proposed scheme.

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Section: Introductionmentioning

confidence: 99%

“…The Toeplitz structure allows for a straightforward formulation of multigrid in the language of linear algebra that is suitable for implementation using SPIRAL. Multigrid methods for Toeplitz matrices have been studied by Fiorentino and Serra 14,15 and by Serra-Capizzano. 16 Multigrid methods consist of a smoother and a multilevel representation of the solution at the finest level and of the error on the coarser levels.…”

Section: Introductionmentioning

confidence: 99%

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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Multigrid methods for indefinite Toeplitz matrices

Cited by 4 publications

References 34 publications

Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences

Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences

A higher‐order unconditionally stable scheme for the solution of fractional diffusion equation

Algebraic description and automatic generation of multigrid methods in SPIRAL

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