[HPGMG Forum] HPGMG 1.0: A Benchmark for Ranking High Performance Computing Systems
Jed Brown
jed at jedbrown.org
Fri Jun 13 19:45:57 UTC 2014
Mark Adams <mfadams at lbl.gov> writes:
> We have discussed ways to get more coverage out of HPGMG but have a long
> way to go before we take any extensions seriously. More interesting
> equations is one dimension that we have considered but I see our biggest
> weakness currently as particle codes (MC, etc.); this might be the first
> extension.
I think irregularity induced by unpredictable nonlinearities are also a
useful dimension. I wrote about that some here:
https://hpgmg.org/fe/#linear-or-nonlinear
>> I agree with many of your comments in your pre-print, but it would
>> seem that limiting one multigrid (or for example conjugate gradient)
>> may be a choice that is not optimal on all architectures and hence
>> reduce the motivation for manufacturers to produce computers that
>> solve large problems quickly rather than simply have good dense
>> matrix operations or good performance for one multigrid algorithm?
I challenge you to write a performance model for a hypothetical machine
that can solve this problem faster using a method substantially
different from FMG.
One of our objectives has been to choose a problem that is fundamentally
globally coupled and to use a method representative of the best known
algorithms for solving globally coupled problems. A lot of real
problems will be more "difficult" to solve, but that usually manifests
itself as needing more iterations and stronger smoothers. A machine
that can change scale rapidly (as in FMG) can also change scale slowly
(as in stronger smoothers).
There are definitely algorithms that handle global coupling as an O(N
log N) rather than O(N) manner. FFT, treecodes, and H-matrices are
examples. These generally have higher bisection bandwidth demands.
O(N) algorithms such as (F)MG, FMM, and H^2 matrices have somewhat
smaller bisection bandwidth demands, but still require low-latency
long-range messaging. It is unknown which users of O(N log N)
algorithms will be able to switch to O(N) algorithms in the future, but
there is no question that we need future machines to be able to execute
O(N) algorithms fast.
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