Standard algebraic multigrid
methods assume explicit knowledge of so-called
algebraically-smooth or near-kernel components,
which loosely speaking are errors that give
relatively small residuals. Typically, these
methods automatically generate a sequence of
coarse problems under the assumption that the
near-kernel is locally constant. The difficulty
in applying algebraic multigrid to lattice QCD
is that the near-kernel components can be far
from constant, often exhibiting little or no
apparent smoothness. In fact, the local character
of these components appears to be random, depending
on the randomness of the so-called "gauge"
group. Hence, no apriori knowledge of the local
character of the near-kernel is readily available. |

This talk develops an adaptive
algebraic multigrid (AMG) preconditioner suitable
for the linear systems arising in lattice QCD.The
method is a recently developed extension of
smoothed aggregation, aSA, in which good convergence
properties are achieved in situations where
explicit knowledge of the near-kernel components
may not be available. This extension is accomplished
using the method itself to determine near-kernel
components automatically, by applying it carefully
to the homogeneous matrix equation, Ax=0. The
coarsening process is modified to use and improve
the computed components. Preliminary results
with model 2D QCD problems suggest that this
approach may yield optimal multigrid-like performance
that is uniform in matrix dimension and gauge-group
randomness. |