JuliaApproximation / JuliaApproximation/MultivariateOrthogonalPolynomials.jl
How to make rotationally invariant solves fast?
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Description
Rotationally invariant operators are typically represented by ModalInterlace which combines matrices acting only on Fourier modes into a diagonal-block-banded matrix (that is, block banded matrix with diagonal blocks, i.e. subblockbandwidths are (0,0)).
At the moment we aren't taking advantage of this structure. There are two approaches:
- Do it at the level of factorizing a
BandedBlockBandedMatrixby checking whether the subblockbandwidths are(0,0). The easiest way would be to copy to a banded matrix, do a QR, and then copy the data back. - Introduce a
DiagonalBlockBandedMatrixthough this doesn't seem to have any obvious benefits over (1). - Do it directly on a
ModalInterlace. This has the benefit of being able to callqron each of the banded operators separately (and potentially taking advantage of@threads). Though probably (1) could also be parallelised.
@TSGut @ioannisPApapadopoulos any thoughts? Do you think this is important right now? It won't help with the variable coefficients problem but could be important for fractional DEs and time-stepping.
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First steps
- Read the whole issue, then the project's contributing guide.
- Comment on the issue to say you are picking it up — it saves two people doing the same work.
- Fork the repository and make your change on a branch.
- Open a pull request that references the issue number.
Research direction
Start by reading ModalInterlace, BandedBlockBandedMatrix factorization, and the qr paths mentioned in the issue. Compare the three proposed approaches, including whether copying to a banded matrix or parallelizing separate banded operators is viable. Done means selecting and implementing a fast path for rotationally invariant solves, with evidence that it improves the relevant fractional differential equation or time-stepping workloads.
Written by the indexing model from the issue text.
Assessment
- Tech stack
- julia
- Domain
- performance
- Issue type
- Feature
- Difficulty
- 5/5
- Estimated time
- Over a week
- Activity status
- Stale
- Clarity
- Needs clarification
- Newbie friendliness
- 25/100