<div dir="ltr">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.<div><div><div><br></div><div>Jeff<br><div class="gmail_extra"><br><div class="gmail_quote">On Fri, Jul 31, 2015 at 4:45 PM, Hoemmen, Mark <span dir="ltr"><<a href="mailto:mhoemme@sandia.gov" target="_blank">mhoemme@sandia.gov</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">



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<div>The point I'm trying to make is that we could use tools like Precimonious to improve accuracy for minimal extra cost.</div>
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<div>On 7/31/15, 5:38 PM, "Jeff Hammond" <<a href="mailto:jeff.science@gmail.com" target="_blank">jeff.science@gmail.com</a>> wrote:</div>
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<div dir="ltr">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).
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<div>It should be possible to implement DDP or some form of compensated summation more efficiently.<br>
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<div>Jeff<br>
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<div class="gmail_quote">On Fri, Jul 31, 2015 at 4:18 PM, Brian Van Straalen <span dir="ltr">
<<a href="mailto:bvstraalen@lbl.gov" target="_blank">bvstraalen@lbl.gov</a>></span> wrote:<br>
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<div>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.</div>
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<div>HPGMG uses max norms, so we should be ok for that part.</div>
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<div>Brian</div>
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<div>On Jul 31, 2015, at 3:27 PM, Hoemmen, Mark <<a href="mailto:mhoemme@sandia.gov" target="_blank">mhoemme@sandia.gov</a>> wrote:</div>
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<div><br style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px">
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 7/31/15, 3:45 PM, "Jed Brown" <</span><a href="mailto:jed@jedbrown.org" style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px" target="_blank">jed@jedbrown.org</a><span style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline!important">>
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Brian Van Straalen <<a href="mailto:bvstraalen@lbl.gov" target="_blank">bvstraalen@lbl.gov</a>> writes:<br>
<blockquote type="cite">The concern is not trivial.  I¹ve spent some time re-reading<br>
Precimonious paper (<a href="http://eecs.berkeley.edu/~rubio/includes/sc13.pdf" target="_blank">eecs.berkeley.edu/~rubio/includes/sc13.pdf</a><br>
<<a href="http://eecs.berkeley.edu/~rubio/includes/sc13.pdf" target="_blank">http://eecs.berkeley.edu/~rubio/includes/sc13.pdf</a>>) and I realize<br>
that it would not be hard to make a faster version of FMG using mixed<br>
precision.  <br>
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Just a quick comment now.  I think there's not as much fat to trim as<br>
you think.  In general, the precision needs to be as accurate as the<br>
discretization.  Most flops occur on fine grids where the discretization<br>
is more accurate than single precision.  I challenge you to speed up<br>
HPGMG by more than, say, 15%, while maintaining order of accuracy on<br>
fine grids.<br>
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<blockquote type="cite">There have been papers over the last few years using 4-byte AMG as a<br>
preconditioner<span> </span><br>
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So much fat already.  Then you have a Krylov method and full-accuracy<br>
residuals, but HPGMG solves in the cost of a few residual evaluations.<br>
Also, these low-accuracy preconditioners are usually used for problems<br>
that are only modestly ill-conditioned.  Try it with an operator with<br>
condition number 10^{12} like you see in solid mechanics or geodynamics<br>
and it doesn't look so hot any more.<br>
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<span style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline!important">It
 could be fun to use such a tool to find out the best places to put</span><br style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px">
<span style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline!important">128-bit
 floating-point arithmetic.  That could help with some really hard</span><br style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px">
<span style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline!important">problems,
 or at least avoid some reproducibility issues.</span><br style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px">
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<span style="font-family:Helvetica;font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline!important">mfh</span></div>
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<div><font face="Courier New">Brian Van Straalen         Lawrence Berkeley Lab</font></div>
<div><font face="Courier New"><a href="mailto:BVStraalen@lbl.gov" target="_blank">BVStraalen@lbl.gov</a>         Computational Research</font></div>
<div><font face="Courier New">(510) 486-4976             Division (<a href="http://crd.lbl.gov" target="_blank">crd.lbl.gov</a>)</font></div>
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_______________________________________________<br>
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-- <br>
<div>Jeff Hammond<br>
<a href="mailto:jeff.science@gmail.com" target="_blank">jeff.science@gmail.com</a><br>
<a href="http://jeffhammond.github.io/" target="_blank">http://jeffhammond.github.io/</a></div>
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</blockquote></div><br><br clear="all"><div><br></div>-- <br><div class="gmail_signature">Jeff Hammond<br><a href="mailto:jeff.science@gmail.com" target="_blank">jeff.science@gmail.com</a><br><a href="http://jeffhammond.github.io/" target="_blank">http://jeffhammond.github.io/</a></div>
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