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**1 - 2**of**2**### COARSENING INVARIANCE AND BUCKET-SORTED INDEPENDENT SETS FOR ALGEBRAIC MULTIGRID ∗

"... Abstract. Independent set-based coarse-grid selection algorithms for algebraic multigrid are defined by their policies for weight initialization, independent set selection, and weight update. In this paper, we develop theory demonstrating that algorithms employing the same policies produce identical ..."

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Abstract. Independent set-based coarse-grid selection algorithms for algebraic multigrid are defined by their policies for weight initialization, independent set selection, and weight update. In this paper, we develop theory demonstrating that algorithms employing the same policies produce identical coarse grids, regardless of the implementation. The coarse-grid invariance motivates a new coarse-grid selection algorithm, called Bucket-Sorted Independent Sets (BSIS), that is more efficient than an existing algorithm (CLJP-c) using the same policies. Experimental results highlighting the efficiency of two versions of the new algorithm are presented, followed by a discussion of BSIS in a parallel setting. Key words. Algebraic multigrid, parallel, coarse-grid selection. AMS subject classifications. 65Y05, 65Y20, 65F10. 1. Introduction. The algebraic multigrid (AMG) method [5, 24] is an efficient numerical algorithm to iteratively approximate the solution to linear systems of the form Ax = b. Often, these algebraic systems arise from the discretization of partial differential equations on structured and unstructured meshes. In many cases, the computational complexity of AMG isO(n), wherenis the number of unknowns in the linear system. The linear cost property of

### Efficient Setup . . .

, 2007

"... Solving partial differential equations (PDEs) using analytical techniques is intractable for all but the simplest problems. Many computational approaches to approximate solutions to PDEs yield large systems of linear equations. Algorithms known as linear solvers then compute an approximate solution ..."

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Solving partial differential equations (PDEs) using analytical techniques is intractable for all but the simplest problems. Many computational approaches to approximate solutions to PDEs yield large systems of linear equations. Algorithms known as linear solvers then compute an approximate solution to the linear system. Multigrid methods are one class of linear solver and find an approximate solution to a linear system through two complementary processes: relaxation and coarse-grid correction. Relaxation cheaply annihilates portions of error from the approximate solution, while coarse-grid correction constructs a lower dimensional problem to remove error remaining after relaxation. In algebraic multigrid (AMG), the lower dimensional space is constructed by coarse-grid selection algorithms. In this thesis, an introduction and study of independent set-based parallel coarse-grid selection algorithms is presented in detail, following a review of algebraic multigrid. The behavior of the Cleary-Luby-Jones-Plassmann (CLJP) algorithm is analyzed and modifications to the initialization phase of CLJP are recommended, resulting in the CLJP in Color (CLJP-c) algorithm, which achieves large performance gains over CLJP for problems on uniform grids. CLJP-c is then extended to the Parallel Modified Independent Set (PMIS) coarse-grid selection algorithm producing