Parallel Smoothed Aggregation Multigrid : Aggregation Strategies on Massively Parallel Machines

Abstract

lgebraic multigrid methods offer the hope that multigrid convergence can be achieved (for at least some important applications) without a great deal of effort from engineers and scientists wishing to solve linear systems. In this paper we consider parallelization of the smoothed aggregation multigrid method. Smoothed aggregation is one of the most promising algebraic multigrid methods. Therefore, developing parallel variants with both good convergence and efficiency properties is of great importance. However, parallelization is nontrivial due to the somewhat sequential aggregation (or grid coarsening) phase. In this paper, we discuss three different parallel aggregation" algorithms and illustrate the advantages and disadvantages of each variant in terms of parallelism and convergence. Numerical results will be shown on the Intel Teraflop computer for some large problems coming from nontrivial codes: quasi-static electric potential simulation and a fluid flow calculation.

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

“…In a standard algebraic multigrid method the most common way to create the aggregates is to resort to a greedy algorithm, where an initial node is chosen along with all of its nearest neighbors. The net effect of this type of procedure is to generate aggregates which are 'ball-like' with an approximate diameter of three graph vertices, see [50]. This means that on a 3D regular Cartesian grid, each aggregate contains 3×3×3 = 27 vertices.…”

10. A Survey of Parallelization Techniques for Multigrid Solvers

“…For the RS coarsening, it generally violates condition (C1) by generating strong F -F connections without common coarse neighbors and leads to poor convergence [51]. While in practice this approach might lead to fairly good results for coarsening by aggregation [79], it can produce many aggregates near processor boundaries that are either smaller or larger than an ideal aggregate and so lead to larger complexities or have a negative effect on convergence. Another disadvantage of this approach is that it cannot have fewer coarse points or aggregates than processors, which can lead to a large coarsest grid.…”

“…Finally, if there are any remaining points, new local aggregates are formed. This process yields significantly better aggregates and does not limit the coarseness of grids to the number of processors, see [79]. Another aggregation scheme suggested in [79] is based on a parallel maximally independent set algorithm, since the goal is to find an initial set of aggregates with as many points as possible with the restriction that no root point can be adjacent to an existing aggregate.…”

“…This process yields significantly better aggregates and does not limit the coarseness of grids to the number of processors, see [79]. Another aggregation scheme suggested in [79] is based on a parallel maximally independent set algorithm, since the goal is to find an initial set of aggregates with as many points as possible with the restriction that no root point can be adjacent to an existing aggregate. Maximizing the number of aggregates is equivalent to finding the largest number of root points such that the distance between any two root points is at least three.…”

Analysis of two-level domain decomposition preconditioners based on aggregation

“…In general, when the ratio between the number of nodes and the number of aggregates is large enough, like for the two-level methods here presented, the decoupled aggregation offers good partitioning. If a large number of aggregates are required (like, for instance, in multilevel methods), decoupled aggregation may result in a somewhat irregular decomposition, and in this case it is usually worth to re-equilibrate the partitioning among the subdomains to minimize the dependency of the resulting algorithm on the subdomain decomposition; see [21]. …”