  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.
