Strength-of-Connection In Algebraic Multigrid
Research on strength-of-connection in algebraic multigrid
has been done in conjunction with
Dr. Raymond Tuminaro,
of Sandia National Labs. In general, strength-of-connection is not
a well understood concept. Also, classic strength-of-connection measures
are limited to M-matrices or near M-matrices. This is problematic.
For instance, even simple rotated anisotropies in 2D can
easily generate non M-matrices. Our goal is to develop a more
general strength-of-connection measure and to provide a framework for
interpreting and understanding existing strength-of-connection measures.
Our research has produced a promising and
novel strength-of-connection measure that is available through
PyAMG. The new strength-of-connection
measure only assumes a definite matrix and novelly combines local
knowledge about relaxation with knowledge about the local behavior of interpolation.
By developing the new measure, we also developed a novel ODE-based framework
for analyzing the new strength-of-connection measure and existing strength-of-connection measures.
The publication,
A New Perspective on Strength Measures in Algebraic Multigrid,
and the presentation,
A General Strength-of-Connection Concept in AMG,
describe some of the contributions of this work.
Another problem with the classic strength measure is sensitivity to the
drop tolerance used. In the below plot, work per digit of accuracy for stand-alone smoothed
aggregation solvers is plotted against the drop tolerance used by each solver's
strength measure. The test problem is rotated anisotropic
diffusion, where the angle of rotation is pi/8. The methods compared
are
- Smoothed aggregation using the classic strength measure (Classic SA)
- Smoothed aggregation using the classic strength measure, but with an enhanced prolongation smoothing method (Enhanced Prol)
- Smoothed aggregation using our new strength measure, and an enhanced prolongation smoothing method (Evolution Str)
Last updated September, 2009