Modern scientists must solve increasingly large linear systems, with hundreds of millions, or even billions, of unknowns. This has necessitated the use of massively parallel computers, which has in turn led to the investigation of scalable linear solvers, such as multigrid. For unstructured grid problems, an attractive choice is algebraic multigrid (AMG). We have implemented BoomerAMG, an AMG code for massively parallel computers with distributed memory.
Much of the classical AMG algorithm can be parallelized in a straightforward fashion; for example, the entire "solve" phase can be implemented using parallel matvec or matvec-like operations. However, the classical coarse-grid selection algorithm is highly sequential. BoomerAMG provides the choice of several parallel coarsening strategies: the "CLJP" method, based on a parallel algorithm by Luby, Jones and Plassmann for finding maximal independent sets; the "RS" approach, a parallel variant of the classical algorithm in which the sequential algorithm is performed on each processor and several options are available to treat the points near the processor boundaries; and finally, the "Falgout" algorithm, a combination of the two approaches above.
BoomerAMG also offers two choices for the parallel smoother: the first is weighted Jacobi while the alternative is a "chaotic" Gauss-Seidel relaxation. This latter method uses Gauss-Seidel locally, within each processor, but updates the values on the processor boundary only after each sweep over the processor domain.
We describe the code and present test results and timings for a variety of test problems. We compare the performance of the coarsening strategies for various settings of the input parameters. We describe some alterations to the basic algorithm that improve convergence factors (or prevent divergence) for certain test problems. We examine the question of how to define the "strong connections" and choose the "strength threshold" for the coarsening algorithms. Results are given showing the influence of the strength threshold on convergence and computation time.
Finally, we indicate the current direction our research is taking and describe our plans for the future of BoomerAMG.
*This work was performed under the auspices of the U.S. Department of
Energy by Lawrence Livermore National Laboratory under contract number