Constrained spline fit, exact discrete adjoint and QoI estimates for unstructured nearby problems

[bwhewe-13]

The per-cell polynomial fit on unstructured meshes fits each side of a material interface independently, and the two sides disagree by about the size of the scheme error. That is enough to throw off the estimate near interfaces (1.55-2.07x the true error on the triangle MMS test), and on multi-material cores the k estimate can come out with the wrong sign.

_spline_fit.py fits one quintic tensor B-spline per connected material region, all regions and groups in one sparse least-squares solve, with weighted rows for flux and current continuity on interface faces and the Robin condition on boundary faces. Two configs:

• smooth: also the diffusion equation (and its gradient) at the face points. Good when interfaces run straight to the boundary.
• corner: used when interfaces meet at a corner (>30 deg) or enclose a small region (<8 knot spans across, e.g. a rod). The face PDE rows make the fit ill-posed there, so the PDE is applied softly at the cell centres instead, and r^alpha Phi(theta) corner functions are added, cut off radially before the shortest interface out of the corner ends.

Knots are 2.5 shortest diffusion lengths apart, clipped to 2-3 cells.

API: nearby_fixed_source / nearby_k_eigenvalue take bc= and use the spline fit when it is given (fit="local" keeps the old one, and nothing changes for calls without bc). nearby_k_eigenvalue(adjoint_solver=...) adds k_error_adjoint and the per-cell k_error_cells; nearby=False skips the nearby power iteration. Fixed-source fits use the corner config since only its interior rows can carry the source. NearbyKResult gets its new fields at the end with None defaults.

Second part: fv_operator.unstructured_operator builds the unstructured solver's operator as a sparse matrix, including the deferred non-orthogonal correction, so a converged solve satisfies it to the solver tolerance. Two things use it:

• nearby_k_eigenvalue(method="direct") factorizes the operator once for the nearby power iteration and gets the adjoint for the k estimate from a bordered system instead of a second k solve. No fixed-source or adjoint solver needed. On 7k-15k cell meshes the whole estimate now takes about half a base solve (it was 1-2 base solves with SOR).
• nearby_qoi estimates the error in ratios of reaction-rate integrals (region powers, absorber rates) from a generalized adjoint, several at once off one factorization, and returns the per-cell contributions.

The transpose has to be of the actual discrete operator. Running a solver on make_adjoint_materials transposes scatter and fission but not the least-squares gradient correction, which isn't symmetric, so on triangles and hexes adjoint-weighted estimates are off by 1-2% at every mesh size. On orthogonal quads the correction is zero and both agree.

Also fixes _xs_arrays reading a one-group nusigf as a fission matrix, which broke one-group nearby_k_eigenvalue.

Checks: the fit is identical (to the bit) to the research version on IAEA, a curved rodded core, a slab and a disk. MMS: quads 1.000, triangles 1.04-1.07. The assembled operator matches the research version exactly and its assembly-power estimates agree to 1e-12% on IAEA triangles. test_fv_operator.py checks the matrix against fixed-source and k solves on regular and jittered meshes, that the one-shot QoI estimate equals a central difference of the perturbed problem, and that it tracks the nearby estimate. Full suite passes, docs build with -W.

master is merged in, so this includes the round-off sweep stop from #1.

[codecov-commenter]

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Codecov Report

❌ Patch coverage is 90.78014% with 78 lines in your changes missing coverage. Please review.
✅ Project coverage is 81.49%. Comparing base (af2ed02) to head (302771d).

Files with missing lines	Patch %	Lines
src/ndiffusion/_spline_fit.py	89.87%	57 Missing ⚠️
src/ndiffusion/nearby.py	90.15%	13 Missing ⚠️
src/ndiffusion/fv_operator.py	95.33%	7 Missing ⚠️
src/ndiffusion/__init__.py	0.00%	1 Missing ⚠️
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Additional details and impacted files

@@            Coverage Diff             @@
##           master       #2      +/-   ##
==========================================
+ Coverage   74.87%   81.49%   +6.62%     
==========================================
  Files          10       12       +2     
  Lines        1190     2027     +837     
==========================================
+ Hits          891     1652     +761     
- Misses        299      375      +76     

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