Independent Quality Measures for Symmetric Algebraic Multigrid Components

A new algebraic multigrid (AMG) method is developed to replace a fast, parallel direct solver used for the coarse-grid problem in a massively parrallel (P ≥ 10) implementation of a multilevel method, resulting in a dramatic improvement in overall efficiency. In addition to being sparse and symmetric positive definite (SPD), these coarse-grid problems are characterized by having few degrees of freedom per processor, n/P = O(1). For the target processor counts, the coarse problem is large and a challenge to solve efficiently. The AMG method developed aims to produce the most efficient AMG hierarchy possible, without regard to setup costs, because the target applications require the approximate solution of a single, unchanging system hundreds of thousands of times within a single computation. The thrust of the approach rests on proposed measures of quality for each AMG component: coarsening, interpolation, and smoothing. Heuristic procedures are developed for constructing near-optimum components in turn, targeting approximations of the proposed theoretical quality measures. Crucially, the measures do not reference those components yet to be constructed. For example, the proposed measure of coarsening quality is independent of both the interpolation and the smoother. Moreover, these measures are grounded in theory; in particular a two-grid convergence bound in terms of them is proven. Numerical results comparing optimized AMG with a fast parallel direct solver intended for coarse problems show efficiency gains up to nearly two orders of magnitude. While coarse-grid problems motivated this research, the theory presented applies generally and provides a framework for deriving AMG strategies for general SPD systems.