[HPGMG Forum] Acceptable rounding errors

Jed Brown jed at jedbrown.org
Fri Jul 31 21:45:50 UTC 2015

Brian Van Straalen <bvstraalen at lbl.gov> writes:
> The concern is not trivial.  I’ve spent some time re-reading
> Precimonious paper (eecs.berkeley.edu/~rubio/includes/sc13.pdf
> <http://eecs.berkeley.edu/~rubio/includes/sc13.pdf>) and I realize
> that it would not be hard to make a faster version of FMG using mixed
> precision.  

Just a quick comment now.  I think there's not as much fat to trim as
you think.  In general, the precision needs to be as accurate as the
discretization.  Most flops occur on fine grids where the discretization
is more accurate than single precision.  I challenge you to speed up
HPGMG by more than, say, 15%, while maintaining order of accuracy on
fine grids.

> There have been papers over the last few years using 4-byte AMG as a
> preconditioner 

So much fat already.  Then you have a Krylov method and full-accuracy
residuals, but HPGMG solves in the cost of a few residual evaluations.
Also, these low-accuracy preconditioners are usually used for problems
that are only modestly ill-conditioned.  Try it with an operator with
condition number 10^{12} like you see in solid mechanics or geodynamics
and it doesn't look so hot any more.
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