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

Brian Van Straalen bvstraalen at lbl.gov
Sat Aug 1 00:04:25 UTC 2015


Once you reach the bottom solver for full multigrid you should not be in the network at all anymore.  You would be computing a dot product over a few hundred doubles in L1 memory.  I’m just guessing here but I don’t think the cost of a reproducible dot product would be that bad to implement in software as a compensated sum or distillation.   There was a talk at ISC this year showing some results.  From DRAM compensated summation is about the same, and gets more expensive in faster memory levels.  The impact was not 50x ever though.



> On Jul 31, 2015, at 4:38 PM, Jeff Hammond <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 <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/ <http://jeffhammond.github.io/>
Brian Van Straalen         Lawrence Berkeley Lab
BVStraalen at lbl.gov         Computational Research
(510) 486-4976             Division (crd.lbl.gov)




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