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Multigrid methods (MG) are known to be optimal solvers for large sparse linear systems arising from the finite element, finite difference or finite volume discretization of a partial differential equation (PDE). Algebraic multigrid methods (AMG) extend this approach to wide a class of problems, e.g. anisotropic operators or unstructured grids. Howe...
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... we expect that the setup times for the CGC and CGC-E schemes are very similar. In Table 1 we give the setup times for this prob- lem. We see that both CGC variants are faster than the Falgout algorithm. ...
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... the CGC and CGC-E algorithm, this means that more edges are constructed in the coarse grid selec- tion graph, while the Falgout algorithm needs to employ the boundary treatment on twice as many points. For the operator complexities, c.f. Table 10, we see an almost equal increase for the Falgout and CGC-E algorithm. In contrast, the CGC algorithm shows a larger deterioration than in the grid-aligned case, which can be explained from the large amount of strongly coupled coarse grid points on different processor subdomains. ...
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... contrast, the CGC algorithm shows a larger deterioration than in the grid-aligned case, which can be explained from the large amount of strongly coupled coarse grid points on different processor subdomains. From the solution times given in Table 11 and the convergence factors given in Table 12 we see that the classical CGC algorithm again does not provide an effi- cient multigrid cycle without agglomeration. The other algorithms only exhibit some deterioration as we go from the sequential to the parallel setting, but the parallel con- vergence factors are independent of the number of pro- cessors. ...
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... contrast, the CGC algorithm shows a larger deterioration than in the grid-aligned case, which can be explained from the large amount of strongly coupled coarse grid points on different processor subdomains. From the solution times given in Table 11 and the convergence factors given in Table 12 we see that the classical CGC algorithm again does not provide an effi- cient multigrid cycle without agglomeration. The other algorithms only exhibit some deterioration as we go from the sequential to the parallel setting, but the parallel con- vergence factors are independent of the number of pro- cessors. ...
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... Table 13 we show the setup times for this example. We see that while both CGC variants exhibit compara- ble times up to 64 processors, this is no longer true for 512 processors any more. ...
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... contrast, the CGC-E algorithm, which combines an efficient coarse grid matching and fast coarsening, achieves the fastest setup. From Table 14 we see that the operator complexity increases for all coarsening methods as the number of processors grows. Hence, we cannot expect solution-time scalability. ...
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... the solution time mainly depends on the com- putational work done per multigrid cycle which in turn is related to the operator complexity. This can be seen from the solution times given in Table 15. As we increase the number of processors from 64 to 512, we see a large jump in the solution times due to hardware and inter- connect bandwidth limitations. ...
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Citations
... In addition, it should be possible to apply the operator in parallel in order to ensure scalability. The tested coarsening techniques include the classical Ruge Stueben technique (RGST) (Ruge and Stüben, 1987) using a third pass for correcting inconsistencies on the interprocessor boundaries, the Cleary Luby Jones Plassman scheme (CLJP) (Cleary et al., 1998), the Falgout scheme (FALG) (Henson and Meier Yang, 2002), the hybrid modified independent set technique (HMIS) and the extended coarse grid classification algorithm (ECGC) of Griebel et al. (2006b). The RGST method shows a good sequential performance (Griebel et al., 2006a, Section 4) and is often used as a basis for enhanced coarsening techniques. ...
... It has been shown that the construction of coarsenings for highly partitioned test cases is difficult to achieve. The ECGC algorithm addresses this problem showing robust coarsening rates that are independent of the number of partitions (Griebel et al., 2006b). Table 1 gives results for the combination of several operator coarsening methods. ...
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... For example, one can apply multigrid techniques to the whole system (1)- (2) to solve problems in areas of computational fluid dynamics [22], [30], [40], [50], [52], [24], [25] constrained optimization [43], [44], [45], [46], mixed finite elements [1], [23] and elsewhere. For parallel multigrid see e.g [26], [27], [28]. ...
... In caseλ 1 andλ N are the two conjugate complex roots of (19), it follows that (26) must also hold for ρ(H(τ 1 , τ 2 , ω 2 , a)) to be minimized. So, (27) holds if either (19) has real or conjugate complex roots. However, if (27) holds, then (13) becomes ...
... So, (27) holds if either (19) has real or conjugate complex roots. However, if (27) holds, then (13) becomes ...
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... The drawback of the CGC algorithm is that it can not handle arbitrary small coarsenings, i.e. the algorithm fails, if the number of nodes is relatively small compared to the number of processors. [Griebel et al., 2006b] mentions that this leads to smallest coarsenings that scale with the number of cpus n proc , so that the coarsest node size N behaves like N = O(n proc ). ...
... The extended coarse grid classification (ECGC) algorithm remedies this problem. It is proposed in [Griebel et al., 2006b]. The ECGC method extends the original CGC method, by replacing local coarsenings that can be interpolated by outside grid points using empty grids in the global selection step. ...
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