Speed up simplicial selected inversion

[timweiland]

Many sparse problems in statistics have Cholesky factors with only a handful of nonzeros per column: tree-structured models, nested random effects, random walks and other latent Gaussian models. CHOLMOD factorizes these with its simplicial method, so selinv takes the simplicial path. That path was noticeably less optimized than the supernodal one, and the SuiteSparse benchmarks never exercised it.

Changes

• Merge instead of search. The column update looked up each previously computed entry through SparseMatrixCSC indexing, which is a binary search, and it allocated a temporary vector per column. Both row lists involved are sorted, so the lookups are now a single merge. When the merge has to skip far ahead in a long column, a galloping search keeps the cost logarithmic.
• Direct factor access. Simplicial CHOLMOD factors are read directly from their column arrays instead of going through sparse(F.L), which took about as long as the inversion itself on low-fill problems. Unpacked factors, such as those left by lowrankupdate, are handled.
• Fused LL → LDL conversion. This now happens column by column during the sweep instead of in a separate pass.
• selinv_diag. The diagonal is read directly from the result instead of through generic diag on a Symmetric sparse matrix, and is depermuted with a single scatter. Before, selinv_diag was slower than the full selinv it calls.

The supernodal code is untouched.

Results

M1 Max, single-threaded. Low-fill problems are generated by the new benchmark/lowfill_problems.jl:

problem	n	selinv before → after	selinv_diag before → after
tree-structured field	200k	12.2 → 2.5 ms	35.9 → 2.9 ms
tree, 3 correlated fields	60k	6.8 → 1.7 ms	12.6 → 1.7 ms
nested random effects	105k	27.8 → 11.1 ms	38.7 → 11.5 ms
second-order random walk	1M	79.4 → 21.8 ms	146 → 24.8 ms
2D grid (supernodal control)	90k	70.9 → 70.7 ms	72.2 → 71.8 ms

On these problems, a selected inversion now costs 5–25% of the Cholesky factorization.

Higher-fill matrices pushed through the simplicial path, either by forcing CHOLMOD's simplicial mode or through the LDLFactorizations extension, also benefit. For example, crystm03 drops from 19.5 s to 1.6 s with forced simplicial CHOLMOD. The SuiteSparse benchmark set, which is entirely supernodal, is unchanged within noise.

Testing

New tests in test/test_simplicial.jl check the results against a dense inverse for low-fill problems with LL and LDL factors, unpacked factors after a rank update, a short column pointing into a long one, and a supernodal factor passed to selinv_simplicial. The full suite passes on Julia 1.10, 1.11 and 1.12.