Parallel multigrid smoothing: polynomial versus Gauss–Seidel

Adams, Mark F.; Brezina, Marian; Hu, Jonathan J.; Tuminaro, Raymond S.

doi:10.1016/s0021-9991(03)00194-3

Cited by 171 publications

(193 citation statements)

References 12 publications

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“…It is convenient (and efficient) to adapt Chebyshev polynomial preconditioners to become multigrid smoothers [2]. Sparse approximate inverses have been proposed as parallel smoothers [14].…”

Section: Definition Of Matrix-free Smoothersmentioning

confidence: 99%

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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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Section: Definition Of Matrix-free Smoothersmentioning

confidence: 99%

“…The upper end of this interval is obtained by a few (here 10) steps of the Lanczos algorithm together with a security margin [2]. The lower end is simply set to a fixed fraction of the upper bound.…”

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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“…For example, if one applies this approach to Gauß-Seidel, Q k are lower triangular matrices (we call this particular smoother hybrid Gauß-Seidel; it has also been referred to as Processor Block Gauß-Seidel [5]). While this approach is easy to implement, it has the disadvantage of being more similar to a block Jacobi method, albeit worse, since the block systems are not solved exactly.…”

Section: Hybrid Gauß-seidel With Relaxation Weightsmentioning

confidence: 99%

“…Thus, it is often sufficient to take the smallest eigenvalue as a fraction of the largest eigenvalue. Experience has shown that this fraction can be chosen to be the coarsening rate (the ratio of the number of coarse grid unknowns to fine grid unknowns), meaning more aggressive coarsening requires the smoother to address a larger range of high frequencies [5].…”

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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“…The generalized form offers even greater robustness, but relies on an expensive preprocessing step. One important issue is the choice of parallel smoother; this was studied in depth in [3], comparing parallel hybrid Gauss-Seidel orderings with polynomial (Chebyshev) smoothers and concluding that polynomial smoothers offer many advantages.…”

Section: Parallel Amgmentioning

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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Parallel multigrid smoothing: polynomial versus Gauss–Seidel

Cited by 171 publications

References 12 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

10. A Survey of Parallelization Techniques for Multigrid Solvers

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

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