Parallel Energy-Minimization Prolongation for Algebraic Multigrid

Abstract

Algebraic multigrid (AMG) is one of the most widely used solution techniques for linear systems of equations arising from discretized partial differential equations. The popularity of AMG stems from its potential to solve linear systems in almost linear time, that is with an O(n) complexity, where n is the problem size. This capability is crucial at the present, where the increasing availability of massive HPC platforms pushes for the solution of very large problems. The key for a rapidly converging AMG method is a good interplay between the smoother and the coarse-grid correction, which in turn requires the use of an effective prolongation. From a theoretical viewpoint, the prolongation must accurately represent near kernel components and, at the same time, be bounded in the energy norm. For challenging problems, however, ensuring both these requirements is not easy and is exactly the goal of this work. We propose a constrained minimization procedure aimed at reducing prolongation energy while preserving the near kernel components in the span of interpolation. The proposed algorithm is based on previous energy minimization approaches utilizing a preconditioned restricted conjugate gradients method, but has new features and a specific focus on parallel performance and implementation. It is shown that the resulting solver, when used for large real-world problems from various application fields, exhibits excellent convergence rates and scalability and outperforms at least some more traditional AMG approaches.