Add NdBSpline: n-dim tensor product b-spline object

[ev-br]

Reference issue

cross-ref #18010, #7903

What does this implement/fix?

Add a new class NdBSpline, to represent an N-dimensional tensor product b-spline. This class only knows how to evaluate a tensor product given coefficients and knot vectors. This way it generalizes BSpline for 1D data to N-D, and parallels NdPPoly (which represents N-D tensor product polynomials in the power basis).

This class is intended to serve as a basis for a further class/routine to construct an N-D interpolator given multidim data, possibly internally in RegularGridInterpolator. The latter currently uses a recursive application of make_interp_spline, which is very slow.
However constructing an interpolator from data is separate and is not a part of this PR.

One other thing we get for free from this class is evaluation of gradients, cf #7903 --- which gets messy for the recursive approach.

Additional information

This class uses the localized nature of b-splines, and goes to some lengths to accomodate a general ND case.
A straightforward implementation is available in https://github.com/ev-br/ndbsplne_playground/blob/main/NdBSpline_playground.ipynb , see NdBSpline and NdBSpline0 classes there.

This might be of interest to @Kai-Striega @j-bowhay and @AtsushiSakai who worked on similar things recently.
I'll also take liberty to ping @charris who discussed these things before.

[Kai-Striega]

Looks really cool. Well done!

I don't have the time for a full review tonight, but I'll try to get one done this weekend.

One thing: I don't have much of a background in this; do you have any resources to get an understanding of the maths going on here?

[ev-br]

One thing: I don't have much of a background in this; do you have any resources to get an understanding of the maths going on here?

Do you mean the general math behind tensor products or the tricks this PR does with jumping through the hoops of a dealing with arrays of runtime-defined dimensionality? For the general background, Chap 7 of Ref [1] of BSpline docstring is nice (https://www.uio.no/studier/emner/matnat/ifi/nedlagte-emner/INF-MAT5340/v05/undervisningsmateriale/hele.pdf) these lecture notes did help me a lot when I first looked at this; For what' s done here I just added some blurb to the top of https://github.com/ev-br/ndbsplne_playground/blob/main/NdBSpline_playground.ipynb, let me know if it's what you're looking for and what's missing.

[ev-br]

No idea what mypy wants from me.
4549d9e does not seem to fix it.

[Kai-Striega]

Do you mean the general math behind tensor products or the tricks this PR does with jumping through the hoops of a dealing with arrays of runtime-defined dimensionality?

I meant the general maths. I haven't done any real work with splines before and would like to learn more. I'll take a look at the link and get back to you. It does seem like a good introduction.

[Kai-Striega]

I still haven't had the time to review the maths literature. But the code looks good to me. I would prefer it if someone familiar with the maths could give this a review before we merge it.

[Kai-Striega]

@ev-br any progress on getting mypy happy? I think this is a mypy issue, not an issue with your code. If you need to, you can append # type: ignore to line 6 to make the tests pass. We should probably make sure it's documented somewhere.

[ev-br]

any progress on getting mypy happy? I think this is a mypy issue, not an issue with your code. If you need to, you can append # type: ignore to line 6 to make the tests pass. We should probably make sure it's documented somewhere.

I rebased on main and the lint CI seems to pass now. Documenting this would be great actually.

EDIT: Spoke too soon. Mypy still shows the same error.

EDIT2: adding # type: ignore to import _bspl seems to do the trick indeed. Thank you @Kai-Striega !

[Kai-Striega]

@ev-br is there anything else that you are wanting to add to this PR, or are we only waiting on a reviewer for the maths?

[AtsushiSakai]

Sorry for my late review. My comments are only related to docs, so can be ignored to merge.

[ev-br]

Thank you @AtsushiSakai , the last commit addresses your suggestions.

@Kai-Striega :

is there anything else that you are wanting to add to this PR, or are we only waiting on a reviewer for the maths?

I think this PR is done from my side. Constructing an interpolator from data + retrofitting it into RegularGridInterpolator can I think be done separately. If there is a preference the contrary, this PR can of course wait for that other part to be implemented.

[Kai-Striega]

I'm happy with the PR as is. Let's leave it for a couple of days and if there's no more feedback I'll merge it. Thanks for your work @ev-br!

[j-bowhay]

Some minor points, the rest LGTM

[ev-br]

Thanks Jake, merged in your suggestions.

[Kai-Striega]

Merged :) Thanks for your work @ev-br, @AtsushiSakai and @j-bowhay!