4. Algebraic Multigrid

Ruge, John W.; Stüben, Klaus

doi:10.1137/1.9781611971057.ch4

Cited by 950 publications

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“…In the first stage, the coarse grids and their connectivities are setup using an agglomeration or coarsening algorithm 28,29,30,31,32 . In the second stage, a multigrid cycling procedure is used with a smoother to yield the solution at the finest grid.…”

Section: Multigrid Methodsmentioning

confidence: 99%

“…The AMG extended the main idea of geometric multigrid to a purely algebraic setting, yielding robustness and algorithmic simplicity. AMG can be used to build highly efficient and robust linear solver 14,15,16,17,18 by combining iterative solvers efficient in damping high frequency errors with multi-level grids that transform low frequency errors of the fine grids to highfrequency errors on coarser grids. This dual approach has been quite successful in tackling relatively large CFD simulations, but reaches its limits with large scale simulations for which the use of parallel processing becomes essential.…”

Section: Introductionmentioning

confidence: 99%

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A high scalability parallel algebraic multigrid solver

Saad¹,

Darwish²

2009

Computational Fluid Dynamics 2006

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Abstract. This paper deals with the implementation and performance analysis of a parallel Algebraic Multigrid Solver (pAMG) for a finite volume unstructured CFD code. The parallelization of the solver is based on the domain decomposition approach using the single program multiple data paradigm. The Message passing interface Library (MPI) is used for communication of data. An ILU(0) iterative solver is used for smoothing the errors arising within each partition at the different grid levels, and a multi-level synchronization across the computational domain partitions is enforced in order to improve the performance of the parallelized Multigrid solver. Two synchronization strategies are evaluated: in the first the synchronization is applied across the multigrid levels during the restriction step in addition to the base level, while in the second the synchronization is enforced during the restriction and prolongation steps. To increase robustness gathering of coefficients across partitions for the coarsest level is investigated. Tests on a number of grids from 100,00 to 800,000 elements for diffusion and advection problems have been conducted on up to 20 processors.

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Section: Multigrid Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A high scalability parallel algebraic multigrid solver

Saad¹,

Darwish²

2009

Computational Fluid Dynamics 2006

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“…Thanks to the Ruge-Stüben theory [15], the TGM and the V-cycle convergence analysis can be split into two independent conditions, one on the smoother and the other on p i , for i = 0, . .…”

Section: Remarkmentioning

confidence: 99%

“…An MGM for circulant matrices was introduced in [13]. Furthermore in [8,14], generalizing the techniques used in [7], and using the Ruge and Stüben theory [15] and the Perron-Frobenius theorem, a complete proof of the optimality of the V -cycle for multilevel circulant and matrices was proposed. This analysis leads to a stronger condition with respect to the previous two-grid analysis.…”

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confidence: 99%

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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“…Ghosal and Vanȇk [14] proposed using a smoothed aggregate restriction/prolongation operator inspired by the algebraic multigrid concept introduced by Ruge of strongly coupled nodes [29]. Although similar to the technique that we present here in the sense that a data-driven aggregation operator is proposed, an important difference between our work and [14] is that their aggregation technique will not preserve a lattice structure at coarse levels.…”

Section: Introductionmentioning

confidence: 97%

A Lattice-Preserving Multigrid Method for Solving the Inhomogeneous Poisson Equations Used in Image Analysis

Grady

2008

Lecture Notes in Computer Science

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Abstract. The inhomogeneous Poisson (Laplace) equation with internal Dirichlet boundary conditions has recently appeared in several applications ranging from image segmentation [1-3] to image colorization [4], digital photo matting [5,6] and image filtering [7,8]. In addition, the problem we address may also be considered as the generalized eigenvector problem associated with Normalized Cuts [9], the linearized anisotropic diffusion problem [10,11,8] solved with a backward Euler method, visual surface reconstruction with discontinuities [12,13] or optical flow [14]. Although these approaches have demonstrated quality results, the computational burden of finding a solution requires an efficient solver. Design of an efficient multigrid solver is difficult for these problems due to unpredictable inhomogeneity in the equation coefficients and internal Dirichlet boundary conditions with unpredictable location and value. Previous approaches to multigrid solvers have typically employed either a data-driven operator (with fast convergence) or the maintenance of a lattice structure at coarse levels (with low memory overhead). In addition to memory efficiency, a lattice structure at coarse levels is also essential to taking advantage of the power of a GPU implementation [15,16,5,3]. In this work, we present a multigrid method that maintains the low memory overhead (and GPU suitability) associated with a regular lattice while benefiting from the fast convergence of a data-driven coarse operator.

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4. Algebraic Multigrid

Cited by 950 publications

References 0 publications

A high scalability parallel algebraic multigrid solver

A high scalability parallel algebraic multigrid solver

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

A Lattice-Preserving Multigrid Method for Solving the Inhomogeneous Poisson Equations Used in Image Analysis

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