Robust Parallel Smoothing for Multigrid Via Sparse Approximate Inverses

Bröker, Oliver; Grote, Marcus J.; Mayer, Carsten; Reusken, Arnold

doi:10.1137/s1064827500380623

Cited by 53 publications

(55 citation statements)

References 27 publications

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“…Also, incomplete factorization preconditioners have been shown to provide good smoothers for multigrid (see, e.g., [290]). Recent studies [69,70,270] have shown that sparse approximate inverses can also provide effective and robust smoothers for both geometric and algebraic multigrid, especially for tough problems (see also [19,29,30,234] for early work in this direction). Conversely, the ever-increasing size of the linear systems arising in countless applications has exposed the limitations of nonscalable methods (such as incomplete factorization and sparse approximate inverse preconditioners), and it is now generally agreed that the multilevel paradigm must somehow be incorporated into these techniques in order for these to remain competitive.…”

Section: Algebraic Multilevel Variantsmentioning

confidence: 99%

Preconditioning Techniques for Large Linear Systems: A Survey

Benzi¹

2002

Journal of Computational Physics

1,004

659

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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper. c 2002 Elsevier Science (USA)

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Section: Algebraic Multilevel Variantsmentioning

confidence: 99%

Preconditioning Techniques for Large Linear Systems: A Survey

Benzi¹

2002

Journal of Computational Physics

1,004

659

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“…Sparse approximate inverses have been proposed as parallel smoothers [14]. However, their setup time and memory cost are too high.…”

Section: Definition Of Matrix-free Smoothersmentioning

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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“…They also analyzed the smoothing factor for constant coefficient PDEs on a two-dimensional regular grid. Some additional theoretical results are given in [23], including for a diagonal approximate inverse smoother, which may be preferable over damped Jacobi. Experimental results in the algebraic multigrid context are given in [22].…”

Section: Polynomial Smoothingmentioning

confidence: 99%

10. A Survey of Parallelization Techniques for Multigrid Solvers

Chow¹,

Falgout²,

Hu³

et al. 2006

Parallel Processing for Scientific Computing

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This paper surveys the techniques that are necessary for constructing computationally efficient parallel multigrid solvers. Both geometric and algebraic methods are considered. We first cover the sources of parallelism, including traditional spatial partitioning and more novel additive multilevel methods. We then cover the parallelism issues that must be addressed: parallel smoothing and coarsening, operator complexity, and parallelization of the coarsest grid solve.

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Robust Parallel Smoothing for Multigrid Via Sparse Approximate Inverses

Cited by 53 publications

References 27 publications

Preconditioning Techniques for Large Linear Systems: A Survey

Preconditioning Techniques for Large Linear Systems: A Survey

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

10. A Survey of Parallelization Techniques for Multigrid Solvers

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