Fix heat recovery mismatch

[andrinr]

Area

Results / reporting

Solver (if applicable)

all heat solvers

What happened?

As stated in the paper:

The cause is a gradient inconsistency in the adjoint-based solvers: the forward pass reports [..] (nodal sum), while the adjoint differentiates ∫ (𝑇 − 𝑇 ∗)2 dΩ (an area-weighted integral)
and applies a scalar correction factor 𝑛nodes |Ω| to bridge the two. This correction is only
approximate, so the returned gradient is not the exact derivative of the reported objective.
First-order methods tolerate this mismatch because they use only gradient direction; L-BFGS
accumulates gradient differences to estimate curvature, so an inconsistent gradient corrupts
the Hessian approximation and causes convergence to a suboptimal point.

Expected behavior

All solver should converge to the same solution when fixed.

Environment

No response