Add smoothed linear interpolation

[SouthEndMusic]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

I first described this method here. I'm not sure whether I should put all of that in the DataInterpolations docs.

[SouthEndMusic]

[SouthEndMusic]

A question related to SparseConnectivityTracer: The DataInterpolations extension there handles each interpolation type individually, so new interpolation types do not work out of the box with SparseConnectivityTracer. Specifically, SparseConnectivityTracer needs to know whether the first and second derivative of an interpolation type are identically zero. Would it be ok with you @sathvikbhagavan @ChrisRackauckas if this information lives in DataInterpolations instead of in the SparseConnectivityTracer extension for DataInterpolations?

[adrhill]

The alternative would be to move the extension from SCT to this repo.

[sathvikbhagavan]

Just left a small comment. Otherwise LGTM!

[sathvikbhagavan]

This is done by mistake? It should be for SmoothedLinearInterpolation?

[sathvikbhagavan]

The alternative would be to move the extension from SCT to this repo.

I am fine moving the extension here if that helps.

[ChrisRackauckas]

That's probably the right way to do it. It's easier to maintain it here.

[adrhill]

How should we handle this? I haven't "swapped" extensions before, so I'm not sure what to watch out for.
SCT has an upcoming v0.7.0 release (refactoring the internal type system, but we'll conservatively tag it as breaking), maybe that helps.

[ChrisRackauckas]

Let's use that. The old one supports up to v0.6 final, delete from the v0.7, and then have this one only start with v0.7 support. That makes it pretty clean

[adrhill]

Tagged SCT v1.0.0 without the extension.

[ChrisRackauckas]

Continuing that thread here: #444

[sathvikbhagavan]

LGTM! Just needs a run with the formatter.

[SouthEndMusic]

The formatter (I use JuliaFormatter 2.1.2 locally) doesn't converge on the SCT extension

[SouthEndMusic]

@adrhill do you know why some of the SCT tests for the new interpolation type are failing?

[sathvikbhagavan]

The formatter (I use JuliaFormatter 2.1.2 locally) doesn't converge on the SCT extension

Using more maxiters in the format call should work

[adrhill]

@adrhill do you know why some of the SCT tests for the new interpolation type are failing?

The tests only fail on local Hessian sparsity patterns using SCT's Dual numbers.
Up until now, the test suite was able to make the assumption that for the given inputs t, u etc., interpolations have either zero or non-zero second-order derivatives everywhere. That's how the reference Hessian is determined:

DataInterpolations.jl/test/sparseconnectivitytracer_tests.jl

Line 134 in c181596

Href = t.is_der2_zero ? zeros(N_IN, N_IN) : ones(N_IN, N_IN)

SmoothedLinearInterpolation breaks that pattern by having 2nd-order derivatives that are non-zero around datapoints but zero otherwise. So you are going to have to update the tests, or at least Href.

[adrhill]

The formatter (I use JuliaFormatter 2.1.2 locally) doesn't converge on the SCT extension

I'm guessing the extension still had SCT's Runic formatting applied. 🤷‍♂️