[HPGMG Forum] Acceptable rounding errors

Jed Brown jed at jedbrown.org
Mon Aug 3 18:21:19 UTC 2015

Benson Muite <benson.muite at ut.ee> writes:
> Fixing algorithm seems not such a good idea. A level of accuracy for
> final result is a better specification, then encourage people to explain
> or post the implementations they have used. This doesn't mean CS people
> need to work on the algorithmic/math part, just that they will likely
> have more choices when choosing what to use to optimize and obtain
> results with. 

I'd agree with this if we were hosting an algorithm competition or
trying to determine the best architecture for solving a particular
science/engineering problem.  But this is a benchmark that is supposed
to be representative of a broad range of applications while being
simpler than any of them.

> It would also mean that they may be better able to advise people
> running applications using multigrid what algorithms work well on
> their systems. Finally, this allows comparison of FE and FV
> implementations.

If I gave you a smooth elliptic problem specification defined in a box,
along with a target accuracy, you'd probably consider full spectral or
other very high-order methods.  But such methods are more fragile in the
sense that they don't generalize to less smooth problems or those with
complicated geometry, for example.  Moreover, the computational
structure is significantly different from the low-order methods that are
used for those more general problems encountered in production.

We could perhaps choose problem data (like coefficients and forcing)
such that third or fourth order methods deliver the best results, but
even that usually depends on the target accuracy (thus problem size and
machine size).  I.e., as target accuracy changes, you *should* change
the discretization and algorithms.

Even if we specify fine-grid discretization, if the problem is small
enough, one could use a higher order method to solve the problem to
target accuracy on a much coarser grid, then simply interpolate to the
target fine grid and use the answer.

We don't want tricks like this in a benchmark for computers because it
makes the benchmark less representative and it makes it more complicated
for people to understand and optimize.  With HPGMG, we've tried to make
the algorithm as lean as possible in that there shouldn't be easy
shortcuts that would require arbitrary legislation to exclude.  But I
don't think it makes sense to eliminate rules about the basic
algorithmic structure.
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