Give each pull back an inverse

[pbrubeck]

Description

A pull back maps a function on the reference cell to the physical cell. UFL had no map in the other direction, so a form compiler that has to evaluate a physical expression on the reference cell — to dual-evaluate an Interpolate, for instance — carried its own copy of the Piola maps. TSFC has one.

This adds apply_inverse to AbstractPullback and implements it for every pull back that has an inverse:

pull back	apply	apply_inverse
IdentityPullback	g	g
ContravariantPiola	(1/detJ) J g	detJ K g
CovariantPiola	Kᵀ g	Jᵀ g
L2Piola	g / detJ	g detJ
DoubleContravariantPiola	(1/detJ)² J g Jᵀ	detJ² K g Kᵀ
DoubleCovariantPiola	Kᵀ g K	Jᵀ g J
CovariantContravariantPiola	(1/detJ) Kᵀ g Jᵀ	detJ Jᵀ g Kᵀ
MixedPullback	per sub-element	per sub-element
SymmetricPullback	per unique block	per unique block

PhysicalPullback and CustomPullback are their own inverse. UndefinedPullback inherits the NonStandardPullbackException from AbstractPullback, as it does for apply.

The test asserts the property that matters rather than the formulas: pushing a reference function forward and mapping it back returns the original, once remove_component_tensors and cancel_jacobian_products have cancelled the Jacobian products the round trip leaves behind. That closes exactly for eight of the nine, mixed and symmetric elements included. The L2Piola round trip leaves detJ / detJ standing, because cancelling a scalar factor is not something cancel_jacobian_products does, so that one is asserted directly instead.

Stacking

Based on pbrubeck/interpolate-holes (FEniCS#497), so that the branch pairs with pbrubeck/form-interp-tsfc in Firedrake. The pull back work itself is independent of FEniCS#497, and the same commit is proposed upstream against main as FEniCS#511.

Nothing calls apply_inverse yet. #69 uses it to lower interpolation in compute_form_data, which lets TSFC delete its copy of the maps.

🤖 Generated with Claude Code

https://claude.ai/code/session_0155wQMTMSGiF9a6TbyTVK5P

[pbrubeck]

Superseded by #71, which uses the same branch as FEniCS#511 instead of a duplicate of it.

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Written by Claude (Claude Code), posted from @pbrubeck's account.