Optimization and coarse-grid selection for algebraic multigrid

Multigrid methods are often the most efficient approaches for solving the very large linear systems that arise from discretized PDEs and other problems. Algebraic multigrid (AMG) methods are used when the discretization lacks the structure needed to enable more efficient geometric multigrid techniques. AMG methods rely in part on heuristic graph algorithms to achieve their performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these heuristics. The main focus of this thesis is to develop e↵ective algebraic multigrid methods. A key step in all AMG approaches is the choice of the coarse/fine partitioning, aiming to balance the convergence of the iteration with its cost. In past work (MacLachlan and Saad, A greedy strategy for coarse-grid selection, SISC 2007), a constrained combinatorial optimization problem was used to define the “best” coarse grid within the setting of two-level reduction-based AMG and was shown to be NP-complete. In the first part of the thesis, a new coarsening algorithm based on simulated annealing has been developed to solve this problem. The new coarsening algorithm gives better results than the greedy algorithm developed previously. The goal of the second part of the thesis is to improve the classical AMGr method. Convergence factor bounds do not hold when AMGr algorithms are applied to matrices that are not diagonally dominant. In this part of our research, we present modifications to the classical AMGr algorithm that improve its performance on such matrices. For non-diagonally dominant matrices, we find that strength of connection plays a vital role in the performance of AMGr. To generalize the diagonal approximations of AFF used in classical AMGr, we use a sparse approximate inverse (SPAI) method, with nonzero pattern determined by strong connections, to define the AMGr-style interpolation operator, coupled with rescaling based on relaxed vectors. We present numerical results demonstrating the robustness of this approach for non-diagonally dominant systems. In the third part of this research, we have developed an improved deterministic coarsening algorithm that generalizes an existing technique known as Lloyd’s algorithm. The improved algorithm provides better control of the number of clusters than classical approaches and attempts to provide more “compact” groupings.

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