Per discussion with Perry: this is not really about "performance", in the sense of timing, and I've updated the description to say so. The issue really is that he believes the ngmix code is not robust and fails on some simple PSF shapes, and he wants to characterize that in more detail. In that sense, this is not analogous to DM-4368.

I am now finding the EmPsfApprox to be reasonably fast and reliable. See the notes under DM-6124.

I did run a series of "robustness" test, which seem to indicate that the algorithm no longer fails at all with the PhoSim Psf simulations we have been testing with. I ran 15000 Psfs without failure over 15000 exposures with different Psfs. The Pixel variation from the original input PSF found by the model is consistent with what we found for the ShapeletPsfApprox results found in DM-6124

SUMMARY using PsfApprox nGauss1 with 15000 exposures
PsfApprox runtime: 0.0015 sec. +- 0.000049 sec.
CModel runtime: 0.0113 sec. +- 0.008044 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00016254
PSF std pixel dff stdev: 0.00231341

Here is set of runs to show that the EmPsfApprox algorithm is stable with 3 different seeing values of the
PhoSim simulated Psfs:

1. With 200 exposure with seeing 0.5
SUMMARY using PsfApprox nGauss1 with 200 exposures
PsfApprox runtime: 0.0014 sec. +- 0.000100 sec.
CModel runtime: 0.0084 sec. +- 0.006040 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00016376
PSF std pixel dff stdev: 0.00292722
SUMMARY using PsfApprox nGauss2 with 200 exposures
PsfApprox runtime: 0.0140 sec. +- 0.000661 sec.
CModel runtime: 0.0113 sec. +- 0.002151 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00015757
PSF std pixel dff stdev: 0.00301078
SUMMARY using PsfApprox nGauss3 with 200 exposures
PsfApprox runtime: 0.2525 sec. +- 0.029623 sec.
CModel runtime: 0.0159 sec. +- 0.003145 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00015643
PSF std pixel dff stdev: 0.00302485

2. With 200 exposure with seeing 0.7
SUMMARY using PsfApprox nGauss1 with 200 exposures
PsfApprox runtime: 0.0015 sec. +- 0.000051 sec.
CModel runtime: 0.0178 sec. +- 0.017543 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00016231
PSF std pixel dff stdev: 0.00231461
SUMMARY using PsfApprox nGauss2 with 200 exposures
PsfApprox runtime: 0.0193 sec. +- 0.000415 sec.
CModel runtime: 0.0120 sec. +- 0.002745 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00015718
PSF std pixel dff stdev: 0.00238253
SUMMARY using PsfApprox nGauss3 with 200 exposures
PsfApprox runtime: 0.2405 sec. +- 0.027247 sec.
CModel runtime: 0.0172 sec. +- 0.003851 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00015615
PSF std pixel dff stdev: 0.00239068

3. With 200 exposure with seeing 0.9
SUMMARY using PsfApprox nGauss1 with 200 exposures
PsfApprox runtime: 0.0015 sec. +- 0.000056 sec.
CModel runtime: 0.0225 sec. +- 0.022049 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00015982
PSF std pixel dff stdev: 0.00193976
SUMMARY using PsfApprox nGauss2 with 200 exposures
PsfApprox runtime: 0.0270 sec. +- 0.000659 sec.
CModel runtime: 0.0136 sec. +- 0.004662 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00015543
PSF std pixel dff stdev: 0.00199526
SUMMARY using PsfApprox nGauss3 with 200 exposures
PsfApprox runtime: 0.2525 sec. +- 0.073085 sec.
CModel runtime: 0.0191 sec. +- 0.005762 sec
Falure count for PsfApprox: 0
PSF avg pixel diff: -0.00015451
PSF std pixel dff stdev: 0.00200097

The EmPsfApprox now seems to be robust over both a large number of Psf samples, all three fits (nGauss1, 2, and 3) and over the set of seeing values we have in our PhoSim simulation (raw seeing 0.5, 0.7, and 0.9 arcsec)

Looks great. I'm especially encouraged to see that the runtime is now consistently much less than the CModel runtime, which what I'd hoped.