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

Hoemmen, Mark mhoemme at sandia.gov
Fri Jul 31 22:27:26 UTC 2015



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



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