[HPGMG Forum] [EXTERNAL] Re: Acceptable rounding errors

Hoemmen, Mark mhoemme at sandia.gov
Fri Jul 31 23:45:45 UTC 2015


The point I'm trying to make is that we could use tools like Precimonious to improve accuracy for minimal extra cost.

mfh

On 7/31/15, 5:38 PM, "Jeff Hammond" <jeff.science at gmail.com<mailto:jeff.science at gmail.com>> wrote:

If by 128b floats, you mean IEEE754 quad precision implemented in SW, then the associated dot product will run ~50x slower on conventional hardware (that is, hardware that does not support QP).

It should be possible to implement DDP or some form of compensated summation more efficiently.

Jeff

On Fri, Jul 31, 2015 at 4:18 PM, Brian Van Straalen <bvstraalen at lbl.gov<mailto:bvstraalen at lbl.gov>> wrote:

I would think that we could probably implement a reproducible dot product in the krylov code since it only happens on the coarse grid which should be small enough.

HPGMG uses max norms, so we should be ok for that part.

Brian


On Jul 31, 2015, at 3:27 PM, Hoemmen, Mark <mhoemme at sandia.gov<mailto:mhoemme at sandia.gov>> wrote:



On 7/31/15, 3:45 PM, "Jed Brown" <jed at jedbrown.org<mailto:jed at jedbrown.org>> wrote:

Brian Van Straalen <bvstraalen at lbl.gov<mailto:bvstraalen at lbl.gov>> writes:
The concern is not trivial.  I¹ve spent some time re-reading
Precimonious paper (eecs.berkeley.edu/~rubio/includes/sc13.pdf<http://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.

It could be fun to use such a tool to find out the best places to put
128-bit floating-point arithmetic.  That could help with some really hard
problems, or at least avoid some reproducibility issues.

mfh

Brian Van Straalen         Lawrence Berkeley Lab
BVStraalen at lbl.gov<mailto:BVStraalen at lbl.gov>         Computational Research
(510) 486-4976             Division (crd.lbl.gov<http://crd.lbl.gov>)





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Jeff Hammond
jeff.science at gmail.com<mailto:jeff.science at gmail.com>
http://jeffhammond.github.io/
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