Distributed Fast Fourier Transforms

[ClaudiaComito]

Resolves #1097 and can be used for #1248

Due Diligence

• General:
  • base branch must be main for new features, latest release branch (e.g. release/1.3.x) for bug fixes
  • title of the PR is suitable to appear in the Release Notes
• Implementation:
  • unit tests: all split configurations tested
  • unit tests: multiple dtypes tested
  • documentation updated where needed

Description

Introducing fft module with all relevant operations.

Main points:

• Transforms along the split dimension are handled by redistributing to a different axis first, transforming the original split dimension locally, and then redistributing back.
• Multi-axis transforms on real input (rfftn, ihfft etc.) will be performed as the corresponding generic FFT operation if transform of the split axis is required. This is due to the intrinsic recursion built into these operations (transpose, resplit, transform, resplit, transpose, transform), where the complex output of the first real transform cannot act as input to the second real transform.
• Multi-axis transforms on real input still return the correct shape, i.e. for the last transformed axis only up to the Nyquist frequency.

Still TODO:

• expand documentation esp. of real ffts
• implement fftfreq, fftshift etc.
• fix test failure (on 7 processes) EDIT: solved by setting correct output dtype of empty local tensor)

Issue/s resolved: #1097

Changes proposed:

• implemented FFT functions
• implemented tests

Type of change

• New feature (non-breaking change which adds functionality)

Memory requirements

Performance

Does this change modify the behaviour of other functions? If so, which?

no

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Codecov Report

Attention: 23 lines in your changes are missing coverage. Please review.

Comparison is base (3302a44) 91.83% compared to head (e4409cb) 91.82%.

Files	Patch %	Lines
heat/fft/fft.py	91.15%	23 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1218      +/-   ##
==========================================
- Coverage   91.83%   91.82%   -0.02%     
==========================================
  Files          77       79       +2     
  Lines       11193    11457     +264     
==========================================
+ Hits        10279    10520     +241     
- Misses        914      937      +23     

Flag	Coverage Δ
unit	91.82% <91.48%> (-0.02%)	⬇️

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[ClaudiaComito]

From my point of view this looks fine. Thanks for the work 👍 I only have added some minor questions/comment that we can address in the PR talk; after that I guess we can merge.

In addition, we could already add a benchmark for FFT if this makes sense.

@mrfh92 thanks a lot for the review. I think I've addressed all of your comments. I will add the benchmarks next week.

[mrfh92]

As far as I can see, everything is fine now.
Thanks for your work! 👍

[mrfh92]

It looks like that we have the same problem with the old PyTorch versions as in #1262 (where no changes to the code have been done) ... so failure on many old PyTorch-versions seems to be a more general problem

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