Adaptive Algebraic Multigrid

Brezina, Marian; Falgout, Robert D.; Manteuffel, S. MacLachlanT.; McCormick, Steve; Ruge, John W.

doi:10.1137/040614402

Cited by 126 publications

(151 citation statements)

References 19 publications

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“…Weak connections are not important in interpolation, so they are collapsed directly to the diagonal. Strong connections to fine-grid points are accounted for through an indirect interpolation, based on an assumption on the errors that are slow to be reduced by relaxation (as in [28]) or, possibly, further knowledge of such errors [9]. Making choices consistent with classical AMG gives the interpolation formula [10] for i ∈ F ,…”

Section: Combination With Amg Coarseningmentioning

confidence: 99%

“…The classical AMG algorithm [28] does this by collapsing certain off-diagonal connections based on the assumption that the near-null space of A is accurately represented by the constant vector. If the nature of errors that are slow to be reduced by relaxation is not known, the adaptive algebraic multigrid method [9] may be used to expose prototypes of such errors to be used in the definition of interpolation. Energy-minimization principles may also be used to define the multigrid interpolation operators [35,36].…”

Section: Algebraic Multigridmentioning

confidence: 99%

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A Greedy Strategy for Coarse-Grid Selection

MacLachlan¹,

Saad²

2007

SIAM J. Sci. Comput.

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Efficient solution of the very large linear systems that arise in numerical modelling of real-world applications is often only possible through the use of multilevel techniques. While highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considered, much recent interest has been in the development of purely algebraic approaches that may be applied in many situations, without problem-specific tuning. Here, we consider an algebraic approach to finding the fine/coarse partitions needed in multilevel approaches. The algorithm is motivated by recent theoretical analysis of the performance of two common multilevel algorithms, multilevel block factorization and algebraic multigrid. While no guarantee on the rate of coarsening is given, the splitting is shown to always yield an effective preconditioner in the two-level sense. Numerical performance of two-level and multilevel variants of this approach is demonstrated in combination with both algebraic multigrid and multilevel block factorizations, and the advantages of each of these two algorithmic settings are explored.

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Section: Combination With Amg Coarseningmentioning

confidence: 99%

Section: Algebraic Multigridmentioning

confidence: 99%

A Greedy Strategy for Coarse-Grid Selection

MacLachlan¹,

Saad²

2007

SIAM J. Sci. Comput.

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“…The same problem is known for the classical Ruge-Stüben AMG, but a fix is for instance mentioned by Brezina et al in [2]. The idea of their argumentation will be repeated in the following subsection.…”

Section: Modified Objective Functionmentioning

confidence: 95%

Filtering algebraic multigrid and adaptive strategies

Nägel

Falgout

Wittum

2007

Comput. Visual Sci.

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Abstract. Solving linear systems arising from systems of partial differential equations, multigrid and multilevel methods have proven optimal complexity and efficiency properties. Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner. This paper proposes a variant of Wagner's method with substantially improved robustness properties. The method is used in an adaptive, selfcorrecting framework and tested numerically.

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“…The AMG methods, originated in [5], together with the smoothed aggregation AMG (or SA AMG) [20], have become a powerful tool for solving problems of linear algebraic equations that typically arise from discretization of elliptic PDEs. In recent years substantial progress has been made to extend the applicability of AMG to more general sparse linear systems by developing methods that use appropriate adaptive strategies (cf., [2,3,7,8,14], etc.) that are aimed at capturing the near-null components of the error (sometimes referred to as algebraically smooth components) that the current solver cannot efficiently handle so that they are then used to improve the solver by modifying its hierarchy of coarse spaces.…”

Section: Introductionmentioning

confidence: 99%

Adaptive AMG with coarsening based on compatible weighted matching

D'Ambra

Vassilevski

2013

Comput. Visual Sci.

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We introduce a new composite adaptive Algebraic Multigrid (composite αAMG) method to solve systems of linear equations without a-priori knowledge or assumption on characteristics of near-null components of the AMG preconditioned problem referred to as algebraic smoothness. Our version of αAMG is a composite solver built through a bootstrap strategy aimed to obtain a desired convergence rate. The coarsening process employed to build each new solver component relies on a pairwise aggregation scheme based on maximum weighted matching in a graph and on principles of compatible relaxation. The latter replaces the commonly used characterization of strength of connection in both the coarse space selection and in the interpolation scheme. The goal is to design a method leading to scalable AMG for a wide class of problems that go beyond the standard elliptic Partial Differential Equations (PDEs). In the present work, we introduce the method and demonstrate its potential when applied to symmetric positive definite linear systems arising from finite element discretization of highly anisotropic elliptic PDEs on structured and unstructured meshes.

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Adaptive Algebraic Multigrid

Cited by 126 publications

References 19 publications

A Greedy Strategy for Coarse-Grid Selection

A Greedy Strategy for Coarse-Grid Selection

Filtering algebraic multigrid and adaptive strategies

Adaptive AMG with coarsening based on compatible weighted matching

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