Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems

Vaněk, Petr; Mandel, Jan; Brezina, Marian

doi:10.1007/bf02238511

Cited by 604 publications

(651 citation statements)

References 13 publications

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“…In the AMGe variant information of the finite element mesh is taken into account in the coarsening procedure [12]. In smoothed aggregation (SA) multigrid [51,52] nodes of a certain level are grouped into contiguous subsets, called aggregates, that form the nodes of the next coarser level. Both, classical and aggregation based approaches, are implemented in open-source high-performance libraries [19,35,43].…”

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confidence: 99%

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Arbenz

Lenthe

Mennel

et al. 2007

Numerical Meth Engineering

132

106

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Abstract. The recent advances in microarchitectural bone imaging are disclosing the possibility to assess both the apparent density and the trabecular microstructure of intact bones in a single measurement. Coupling these imaging possibilities with microstructural finite element (µFE) analysis offers a powerful tool to improve bone stiffness and strength assessment for individual fracture risk prediction.Many elements are needed to accurately represent the intricate microarchitectural structure of bone; hence, the resulting µFE models possess a very large number of degrees of freedom. In order to be solved quickly and reliably on state-of-the-art parallel computers, the µFE analyses require advanced solution techniques. In this paper, we investigate the solution of the resulting systems of linear equations by the conjugate gradient algorithm, preconditioned by aggregation-based multigrid methods. We introduce a variant of the preconditioner that does not need assembling the system matrix but uses element-by-element techniques to build the multilevel hierarchy. The preconditioner exploits the voxel approach that is common in bone structure analysis, it has modest memory requirements, while being at the same time robust and scalable.Using the proposed methods, we have solved in less than 10 minutes a model of trabecular bone composed of 247'734'272 elements, leading to a matrix with 1'178'736'360 rows, using only 1024 CRAY XT3 processors. The ability to solve, for the first time, large biomedical problems with over 1 billion degrees of freedom on a routine basis will help us improve our understanding of the influence of densitometric, morphological and loading factors in the etiology of osteoporotic fractures such as commonly experienced at the hip, the spine and the wrist.

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mentioning

confidence: 99%

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Arbenz

Lenthe

Mennel

et al. 2007

Numerical Meth Engineering

132

106

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“…For PDEs with heterogeneous coefficients that are discretized on unstructured meshes, algebraic multigrid (AMG) approaches [40,44,49] offer similar scalability, although at a higher cost per iteration (see, for example, [34] for a comparison of structured and unstructured multigrid approaches). While the fundamental complementarity of the multigrid approach doesn't change within AMG, the way in which the coarse-grid problems are defined does.…”

Section: Algebraic Multigrid Methodsmentioning

confidence: 99%

“…Several high quality, well parallelisable public domain direct solvers exist [32,38,42,10,6]. The FETI and AMG methods are also robust but are often much less expensive than direct solution methods and have been discussed in [23] and [49]. As a comparison to DPCG, we focus on the best AMG adaptation, smoothed aggregation (SA), as it has been demonstrated to be a successful parallel preconditioner for a number of structural mechanics applications [2,4,14].…”

Section: Introductionmentioning

confidence: 99%

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

et al. 2011

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Many applications in computational science and engineering concern composite materials, which are characterized by large discontinuities in the material properties. Such applications require fine-scale finite-element meshes, which lead to large linear systems that are challenging to solve with current direct and iterative solutions algorithms. In this paper, we consider the simulation of asphalt concrete, which is a mixture of components with large differences in material stiffness. The discontinuities in material stiffness give rise to many small eigenvalues that negatively affect the convergence of iterative solution algorithms such as the preconditioned conjugate gradient (PCG) method. This paper considers the deflated preconditioned conjugate gradient (DPCG) method in which the rigid body modes of sets of elements with homogeneous material properties are used as deflation vectors. As preconditioner we consider several variants of the algebraic multigrid smoothed aggregation method. We evaluate the performance of the DPCG method on a parallel computer using up to 64 processors. Our test problems are derived from real asphalt core samples, obtained using CT scans. We show that the DPCG method is an efficient and robust technique for solving these challenging linear systems. T. B. Jönsthövel (B)·

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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%

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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Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems

Cited by 604 publications

References 13 publications

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

A high scalability parallel algebraic multigrid solver

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