  Algebraic multigrid (AMG) methods are powerful solvers with linear or
near-linear computational complexity for certain classes of linear systems,
Ax=b. Broadening the scope of problems that AMG can effectively solve requires
the development of improved interpolation operators. Such development is often
based on AMG convergence theory. However, convergence theory in AMG tends to
have a disconnect with AMG in practice due to the practical constraints of (i)
maintaining matrix sparsity in transfer and coarse-grid operators, and (ii)
retaining linear complexity in the setup and solve phase. This paper presents a
review of fundamental results in AMG convergence theory, followed by a
discussion on how these results can be used to motivate interpolation operators
in practice. A general weighted energy minimization functional is then proposed
to form interpolation operators, and a novel `diagonal' preconditioner for
Sylvester- or Lyapunov-type equations developed simultaneously. Although
results based on the weighted energy minimization typically underperform
compared to a fully constrained energy minimization, numerical results provide
new insight into the role of energy minimization and constraint vectors in AMG
interpolation.
