We will present an adaptive algebraic multigrid method for
the solution of positive definite linear systems arising from the
discretizations of elliptic partial differential equations.  The
proposed method uses compatible relaxation to adaptively construct the
set of coarse variables.  The nonzero supports of the basis spanning
the coarse space is determined by approximation of the so-called
two-level ``ideal'' interpolation operator.  Then, an energy
minimizing basis is formed using an approach aimed to minimize the
trace of the coarse-level operator.  The presented approach maintains
multigrid-like optimality, without the need for parameter tuning, for
problems where current algorithms exhibit degraded performance.
Numerical experiments are presented that demonstrate the efficacy of
the approach.  In addition, it is demostrated that this method can be
extended in a straightforward manner to handle more complicated
situations, for example higher-order scalar partial differential
equaitons or systems of paritial differential equations.  Some other
adaptive methods will also be briefly considered.