Questions on the roundtrip error measurements

[mreineck]

I have seen your preprint on arxiv and would like to get a better understanding for the methodology you are using for measuring the transform roundtrip error. I hope this is the correct place to ask!

• Did you use https://github.com/NVlabs/cuHPX/blob/main/tests/reproduce_cuhpx_sht_round_trip_error.py for measuring the roundtrip error or was this done using some other script?
• If you used this file, do I understand correctly that in the input array only 100 values are set to a nonzero value?
• Also, was an lmax of 64 used for all different choices of nside in the paper? This appears to be hardcoded in the script.

If so, then I fear that the resulting error values are not really representative of the actual round trip error. For a rigorous measurement, you will have to

• choose an lmax that scales with the requested Nside value (e.g. the lmax=3*Nside-1 that is stated in sect IV.A of the paper
• fill the entire coefficient array with random numbers, not just a subset.
• (to avoid potential systematics: use random numbers in the [-1;1] range instead of [0;1])

Using this approach, the results will be admittedly much worse than the ones from the preprint, but everyone working on a daily basis with Healpix grids knows that the round trip errors shown there are way too good to be true (unfortunately).

Concerning the weak performance of pixel weights in your analysis: this is expected, at least for the situation described in the preprint. They only give good results if the function is band limited to lmax<=1.5*Nside. For functions with higher band limit they should not be used.