Prototype approximate parallel FE diagnostics

[apoorvalal]

Summary

• add spectral graph diagnostics to the Correia CSV approximate-parallel benchmark
• commit full benchmark CSV/Markdown outputs with lambda2, Fiedler conductance, one-shot residuals, and solver recommendations
• add the approximate FE residualization memo under docs/approx-fe with rendered HTML

Verification

• uv run python -m py_compile benchmarks/approx_parallel_csv.py
• uv run python benchmarks/approx_parallel_csv.py --output benchmarks/results/approx_parallel_correia.csv --markdown-output benchmarks/results/approx_parallel_correia.md
• quarto render docs/approx-fe/paper.qmd

[s3alfisc]

Hi @schroedk - my very brief summary of @apoorvalal's note:

• There are three core ideas: (i) component-wise parallel solves when the FE graph truly decomposes, (ii) a one-shot Schwarz / “approximate within” solve, and (iii) graph diagnostics for solver routing.
• The one-shot approximation is basically: build the same Schwarz preconditioner, apply it once, stop, and report the remaining FE normal-equation residual as a diagnostic. For ordinary linear regressions this may not buy much if preconditioner setup dominates and Krylov already converges in ~20 iterations, though I am not sure.
• It could be more useful for GLM/IWLS settings: early IWLS iterations could use loose/one-shot residualization or a stale preconditioner, then tighten later, similar in spirit to ppmlhdfe/PyFixest tolerance schedules.
• The diagnostics are probably the most useful part for us. Apoorva proposes components, component shares, the second eigenvalue (lambda2) of the normalized Laplacian, and possibly Fiedler/conductance cuts to detect weak graph connectivity. Here the core question is how efficient all these measures can be computed?
• He then suggests a router that chooses among MAP, and Schwarz-preconditioned Krylov. This would of course be very cool to have, optimally I believe that we'd find heuristics to choose between MAP, Schwarz, and Diagonal preconditioning.