Circulant block-factorization preconditioning of anisotropic elliptic problems

Lirkov, Ivan; Margenov, Svetozar; Zikatanov, Ludmil T.

doi:10.1007/bf02684392

Cited by 7 publications

(4 citation statements)

References 16 publications

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“…Our approach relies on the embedding of the model problem into a periodic problem. Similar ideas have also been explored in works on circulant preconditioners for elliptic problems [4,5] and also for preconditioning the indefinite Helmholtz equation [6]. We introduce a class of operators called LFA-compatible operators here and prove that for such operators the LFA gives the exact multigrid convergence factors.…”

Section: Introductionmentioning

confidence: 89%

On the Validity of the Local Fourier Analysis

Rodrigo

Gaspar

Zikatanov³

2019

JCM

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Local Fourier analysis (LFA) is a useful tool in predicting the convergence factors of geometric multigrid methods (GMG). As is well known, on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such methods. In this work, using the Fourier method, we extend these results by proving that such analysis yields the exact convergence factors for a wider class of problems.

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Section: Introductionmentioning

confidence: 89%

On the Validity of the Local Fourier Analysis

Rodrigo

Gaspar

Zikatanov³

2019

JCM

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“…Remark If the fine‐grid operator is block‐circulant with circulant blocks (BCCB), each block of can be diagonalized using Fourier modes. Thus, in the BCCB case, SAMA coincides with rigorous Fourier analysis. Remark While SAMA is most naturally considered for the BTCB case, we focus primarily on the special case of BTTB, where the Toeplitz blocks are diagonalized exactly by the sine Fourier basis. Remark The use of circulant or block‐circulant‐structured preconditioners for non‐circulant problems has a long history , but is primarily focused on the case of problems that retain elliptic character; here, we consider the space‐time discretization of parabolic problems, for which simple (multilevel) circulant preconditioners do not yield good performance.…”

Section: Semi‐algebraic Mode Analysismentioning

confidence: 99%

“…The use of circulant or block-circulant-structured preconditioners for non-circulant problems has a long history [51,52], but is primarily focused on the case of problems that retain elliptic character; here, we consider the space-time discretization of parabolic problems, for which simple (multilevel) circulant preconditioners do not yield good performance.…”

Section: Remarkmentioning

confidence: 99%

A generalized predictive analysis tool for multigrid methods

Friedhoff

MacLachlan

2015

Numerical Linear Algebra App

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Summary Multigrid and related multilevel methods are the approaches of choice for solving linear systems that result from discretization of a wide class of PDEs. A large gap, however, exists between the theoretical analysis of these algorithms and their actual performance. This paper focuses on the extension of the well‐known local mode (often local Fourier) analysis approach to a wider class of problems. The semi‐algebraic mode analysis (SAMA) proposed here couples standard local Fourier analysis approaches with algebraic computation to enable analysis of a wider class of problems, including those with strong advective character. The predictive nature of SAMA is demonstrated by applying it to the parabolic diffusion equation in one and two space dimensions, elliptic diffusion in layered media, as well as a two‐dimensional convection‐diffusion problem. These examples show that accounting for boundary conditions and heterogeneity enables accurate predictions of the short‐term and asymptotic convergence behavior for multigrid and related multilevel methods. Copyright © 2015 John Wiley & Sons, Ltd.

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“…The relative condition number of the CBF preconditioner for the model (Laplace) 3D problem is analyzed using the technique from [5] and the following estimate is derived:…”

Section: Circulant Block Factorizationmentioning

confidence: 99%

Parallel Performance of a 3D Elliptic Solver

Lirkov

Margenov

Paprzycki

2001

Lecture Notes in Computer Science

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Abstract.It was recently shown that block-circulant preconditioners applied to a conjugate gradient method used to solve structured sparse linear systems arising from 2D or 3D elliptic problems have good numerical properties and a potential for high parallel efficiency. The asymptotic estimate for their convergence rate is as for the incomplete factorization methods but the efficiency of the parallel algorithms based on circulant preconditioners are asymptotically optimal. In this paper parallel performance of a circulant block-factorization based preconditioner applied to a 3D model problem is investigated. The aim of this presentation is to analyze the performance and to report on the experimental results obtained on shared and distributed memory parallel architectures. A portable parallel code is developed based on Message Passing Interface (MPI) and OpenMP (Open Multi Processing) standards. The performed numerical tests on a wide range of parallel computer systems clearly demonstrate the high level of parallel efficiency of the developed parallel code.

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Circulant block-factorization preconditioning of anisotropic elliptic problems

Cited by 7 publications

References 16 publications

On the Validity of the Local Fourier Analysis

On the Validity of the Local Fourier Analysis

A generalized predictive analysis tool for multigrid methods

Parallel Performance of a 3D Elliptic Solver

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