refactor interpolation

[leo-collins]

Simplifies interpolation code and introduces new features. So far:

• Implemented assembly of cross-mesh interpolation operators, both forward and adjoint.
• We can now matfree adjoint interpolate cross-mesh and VomOntoVom. The renumbering malarkey is completely removed.
• Removed interp_data dictionary in favour of the InterpolateOptions dataclass. We get type hinting, better IDE support, single source of truth for these options.
• Interpolator becomes an internal object. I removed the __new__ method and dispatch instead with a get_interpolator function.
• Removed frozen_interpolator logic

[connorjward]

@leo-collins this looks like it includes changes from the other ongoing interpolation PRs. This will be much easier to review if you point this PR at that branch instead of main. Then we are only seeing the new changes.

[pbrubeck]

• Interpolator becomes an internal object. I removed the __new__ method and dispatch instead with a _get_interpolator function.

This is fine, but we might want to preserve a user-facing interface for reusable interpolators.

[pbrubeck]

For the 0-form interpolate we should also allow Constant here, it also supports assign.

[pbrubeck]

The commit history includes commits from other PRs. Would it be much of a problem to drop them?

[pbrubeck]

Does VomOntoVomInterpolator support MixedFunctionSpace, or are we never expecting to build one for this case? There should be a ValueError in the constructor if we go for the latter.

[pbrubeck]

"zero form" is too close to 0-form, which is not necesarily equal to 0.

Suggested change

	The symbolic interpolation expression, or a zero form. Zero forms
	The symbolic interpolation expression, or a ZeroBaseForm. ZeroBaseForms

[pbrubeck]

For the 1-form adjoint on a mixed space, we might be adding different sub-interpolates onto the same tensor. The zeroing should be handled outside the for loop of line 633

[leo-collins]

tensor is already the indexed sub-tensor in this case. Can you write a test for this?

[pbrubeck]

I'll do that

[connorjward]

I haven't reviewed this in great detail but this seems really good!

[connorjward]

Suggested change

	@dataclass
	@dataclass(kw_only=True)

I think we should disallow positional arguments for this.

[connorjward]

This seems bizarre. A user could pass a matfree option that gets ignored.

[leo-collins]

We can't assemble the cross-mesh interpolation matrix if matfree=True. By doing this the user can do assemble(interpolate(TrialFunction(V), U)) and get the operator without having to pass an additional matfree=False.

[connorjward]

In which case I think we should change (and document) the interface to

def interpolate(expr, ..., *, matfree=None):
  if matfree is None:
    if some_condition:
      matfree = False
    else:
      matfree = True

because currently the user could run

assemble(interpolate(TrialFunction(V), U, matfree=True))

and the option would be silently ignored.

[leo-collins]

The same could be said about the allow_missing_dofs and default_missing_val parameters if the user is doing same-mesh interpolation. I think it's fine if the docstring says something like "this parameter only applies if interpolating between a VertexOnlyMesh and its input_ordering".

[connorjward]

Well the right answer there is to disallow those options for same-mesh interpolation. I don't want to force you to make big changes here though so I'd be happy provided that this behaviour is made clear in the docstring and you open an issue about it.

[connorjward]

This is a great idea. Big fan.

[connorjward]

.

[dham]

subject to checking that adjoint interpolation on mixed spaces is really correct, this is great. thanks.