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

Jeff Hammond jeff.science at gmail.com
Fri Jul 31 23:57:42 UTC 2015


I am merely suggesting that QP is potentially the wrong way to maximize the
ratio of accuracy to cost.  You specifically said 128b FP, not "some form
of post-64b precision", which is what provoked my comment.

Jeff

On Fri, Jul 31, 2015 at 4:45 PM, Hoemmen, Mark <mhoemme at sandia.gov> wrote:

> 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> 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>
> 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> wrote:
>>
>>
>>
>> On 7/31/15, 3:45 PM, "Jed Brown" <jed at jedbrown.org> wrote:
>>
>> Brian Van Straalen <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>) 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         Computational Research
>> (510) 486-4976             Division (crd.lbl.gov)
>>
>>
>>
>>
>>
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>>
>
>
> --
> Jeff Hammond
> jeff.science at gmail.com
> http://jeffhammond.github.io/
>
>


-- 
Jeff Hammond
jeff.science at gmail.com
http://jeffhammond.github.io/
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