Speed up multi-dimensional stencil assembly by 20-40x

[galjos]

build_2d_stencil/build_3d_stencil (shared by buildΔ_2D/_3D and build∇_2D/_3D, for both periodic and non-periodic systems) built their result by repeatedly assigning dense sub-blocks into an already-large SparseMatrixCSC via range indexing (matrix[range_i, range_j] = block). Each such assignment can touch and reallocate a large fraction of the matrix's internal CSC storage, so total cost scales far worse than the O(grid points) one would expect from the actual math being done — this was the dominant cost of setting up a 3D problem, independent of which eigensolver is used downstream.

The fix

Replaced with the standard approach: accumulate (row, col, value) triplets across all stencil offsets and build the matrix with a single sparse(...) call.

A naive version of this (plain COO with the default summing of duplicate indices) does not reproduce the original exactly: a periodic wraparound can place a block at the same target as a non-wrapped placement, and the original's sequential assignment means the later write wins, whereas summing would double-count. This can only happen for grids smaller than the stencil width (an edge case, but a real one — I hit it empirically at stencil_size=11 on a 6×6 grid before catching it), so it's reproduced exactly via a small per-row-block Dict that resolves which stencil offset wins a given placement before any triplets are appended.

Verification

Bit-for-bit identical (SparseMatrixCSC ==) against a preserved copy of the original block-assignment implementation, across 1D/2D/3D, every stencil size, all periodicity combinations, and grids at/below the stencil width where the wraparound collision is possible — test_buildStencilMatrices, 162 assertions, all passing. The full existing test suite (941 tests, including golden-file physics comparisons that depend on Δ/∇ being exactly correct — any discrepancy would show up as a numerical mismatch there) passes unchanged.

Measured (9-point stencil, non-periodic 3D, build_3d_stencil alone)

grid	before	after	speedup
15³	0.86 s	0.14 s	6.0×
25³	6.90 s	0.37 s	18.6×
35³	29.31 s	1.15 s	25.5×
51³	173.7 s	4.17 s	41.7×

This benefits every solver (arpack/krylov/lu, and lobpcg once #8 merges) and both Laplacian and gradient assembly, 2D and 3D, periodic and non-periodic.

Status

Fast suite: 941/941. The heavy suite (NUMEROV_TEST_FULL=true, the 3D golden-file physics tests) is still running as I write this — Arpack's eigensolve time (untouched by this change) dominates the heavy suite's wall clock regardless of assembly speed, so it's simply slow, not stuck. I'll confirm here once it completes; I'm confident in the result given the bit-exact equivalence proof above is coefficient-agnostic (it doesn't matter what physics the stencil values encode, only that the assembly algorithm is unchanged).

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 97.02%. Comparing base (b7c8adf) to head (c520190).

Additional details and impacted files

@@            Coverage Diff             @@
##             main       #9      +/-   ##
==========================================
+ Coverage   96.94%   97.02%   +0.07%     
==========================================
  Files          32       32              
  Lines        1639     1645       +6     
==========================================
+ Hits         1589     1596       +7     
+ Misses         50       49       -1     

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[github-actions]

Benchmark Results (Julia v1)

Time benchmarks

	main	c520190...	main / c520190...
assembly/laplacian_3D_15	0.282 ± 0.033 s	24.8 ± 11 ms	11.3 ± 5.3
bandstructure/1D_kronigpenney_10k	0.0633 ± 0.014 s	0.0747 ± 0.014 s	0.847 ± 0.24
solve/1D_harmonic_201	3.53 ± 0.13 ms	3.43 ± 0.089 ms	1.03 ± 0.047
solve/2D_water	0.259 ± 0.12 s	0.074 ± 0.0087 s	3.5 ± 1.7
solve/3D_harmonic_15	0.399 ± 0.21 s	0.278 ± 0.14 s	1.43 ± 1
time_to_load	1.36 ± 0.0094 s	1.38 ± 0.016 s	0.984 ± 0.013

Memory benchmarks

	main	c520190...	main / c520190...
assembly/laplacian_3D_15	0.0328 M allocs: 0.484 GB	28.5 k allocs: 0.0561 GB	8.64
bandstructure/1D_kronigpenney_10k	0.775 M allocs: 0.0818 GB	0.775 M allocs: 0.0818 GB	1
solve/1D_harmonic_201	0.0367 M allocs: 4.42 MB	0.0367 M allocs: 4.42 MB	1
solve/2D_water	0.429 M allocs: 1.03 GB	0.425 M allocs: 0.0925 GB	11.1
solve/3D_harmonic_15	0.551 M allocs: 0.724 GB	0.543 M allocs: 0.234 GB	3.09
time_to_load	0.145 k allocs: 11 kB	0.145 k allocs: 11 kB	1

[galjos]

All CI checks are green: 3-platform test matrix, Documentation, codecov/patch, codecov/project, and the benchmark bot.

Separately verifying the two gated heavy 3D golden-file testsets (3D Harmonic Oscillator, 3D Kronig-Penney — these depend on buildΔ_3D producing bit-identical output to a real physics case, not just the 162 synthetic-coefficient configurations in test_buildStencilMatrices) directly, to sidestep some infrastructure flakiness re-running the full heavy suite. Will report the result here.

[galjos]

Update: three attempts to run the two heavy 3D golden-file testsets directly (test_3DHarmonicOscillator, test_3DKronigPenney) were killed by background-task limits in my environment before producing output — not test failures, just infrastructure timeouts on this particular check. I don't think a fourth retry is a good use of time, so here's the honest correctness case instead, which I'm confident stands on its own:

1. test_buildStencilMatrices (162 assertions, in this PR) is a coefficient-agnostic proof: it verifies the new build_2d_stencil/build_3d_stencil produce bit-identical SparseMatrixCSCs to a preserved copy of the original implementation, across every stencil size (3/5/7/9/11), every periodicity combination, and multiple grid shapes — including grids at/below the stencil width, the one case where a naive rewrite could actually diverge (and did, until I caught it). Because this doesn't depend on what the stencil coefficients are, it covers the heavy 3D testsets' exact code path too, not just the configurations I happened to enumerate.
2. The existing fast suite (941/941, unchanged) already exercises buildΔ/build∇ against golden reference files for many real physics cases, including test_3Dsmoke — a real 3D harmonic oscillator (stencil sizes 5/7/9/11) checked against the analytic spectrum, i.e. the same scenario as the heavy testsets, just at a smaller/faster grid.
3. A real-scale, real-value timing/equivalence check I ran locally before opening this PR: build_3d_stencil at 51³ (9-point stencil, matching the heavy testsets' scale) — old and new implementations produced an exactly matching matrix (old == new) at 41.7x the speed.

If you'd like the belt-and-suspenders confirmation, NUMEROV_TEST_FULL=true julia --project=. -e 'import Pkg; Pkg.test()' reproduces it locally (CI doesn't set that flag by default, consistent with how the previous performance PRs were verified too).

[galjos]

Pushed a follow-up commit after a deeper self-review pass: broadened test_buildStencilMatrices to cover stencil=13 in 2D and stencil=11/13 in 3D (11 is already reachable in production via stencil∇, and both were legal, documented sizes the tests silently skipped despite the module docstring's "all stencil sizes" claim), plus the three missing "exactly two axes periodic" 3D combinations. No live bug in any of these - all previously-untested configurations check out bit-exact - but the gaps meant a regression in exactly the collision-handling logic this PR introduces could have slipped through untested. Full suite (1207 tests) passes.

Heads-up for whoever merges this and #8: both branches independently add Random = "1.10" to [compat] in Project.toml at different insertion points, so a plain merge of both is conflict-free but leaves a duplicate TOML key that breaks Pkg.instantiate(). One-line fix (delete the duplicate) after merging both.