Direct Solver: Recursive Skeletonization

[alexfikl]

Attempting to break up the direct solver MR into smaller, more reviewable pieces. As a rundown

• [Done] Add proxy generation (already in linalg.proxy).
• [Done] Add single-level skeletonization (Direct Solver: Skeletonization #30).
• [WIP] Add multi-level skeletonization (Direct Solver: Recursive Skeletonization #164).
• [WIP] Add the actual multi-level direct solver (Direct Solver: Multilevel Proxy-based Solver #174).

The code for this is mostly in the deprecated https://gitlab.tiker.net/inducer/pytential/-/merge_requests/137.

[alexfikl]

@inducer This should be ready for a look. It doesn't do much:

• adds a data structure to hold the hierarchical cluster info from boxtree.Tree and merge cluster indices.
• adds some helper functions to do the recursive skeletonization + simple tests.

[inducer]

Noticed a pending review. Figured it'd be better to submit it. :) Don't quite know how old this is.

[inducer]

Is this avoidable?

[alexfikl]

Reworked this a while back, so not quite sure if your comment still applies?

[inducer]

Is this two things rather than one?

[alexfikl]

Yeah, separated into a ClusterTree and ClusterLevel with a ClusterTree.levels() iterator to go through them for the recursive skeletonization.

Naming could still be better probably?

[inducer]

Update the test docstring to describe precisely what's being tested here.

[alexfikl]

Updated the blurb in
https://github.com/inducer/pytential/compare/f71a21b1a54791b02b35712225b9cfaca4be60cf..9a8db7ff410051a6b7d86cd8d5b934d02d094fef

Let me know if it's clear enough!

[Copilot]

Pull request overview

This PR advances the direct solver work by introducing hierarchical clustering infrastructure and wiring it into proxy-based skeletonization to enable recursive (multi-level) skeletonization, along with the associated test and documentation updates.

Changes:

• Add a new pytential.linalg.cluster module providing cluster trees/levels, clustering helpers, and partition_by_nodes.
• Extend skeletonization to produce richer results (including diagonal blocks) and add rec_skeletonize_by_proxy for multi-level skeletonization.
• Update proxy and skeletonization tests (and docs) to match the new clustering/tree and neighbor-gathering APIs.

Reviewed changes

Copilot reviewed 11 out of 11 changed files in this pull request and generated 5 comments.

Show a summary per file

File	Description
pytential/linalg/cluster.py	New clustering module (cluster tree/levels, partitioning, visualization utilities).
pytential/linalg/skeletonization.py	Adds recursive skeletonization and augments skeletonization results/weighting/logging.
pytential/linalg/proxy.py	Removes partition_by_nodes (moved to cluster module) and updates neighbor gathering API.
pytential/linalg/utils.py	Documents/exports helper utilities used by skeletonization (make_flat_cluster_diag, interp_decomp).
test/extra_matrix_data.py	Updates test fixtures to return (index, tree) and import partitioning from linalg.cluster.
test/test_linalg_cluster.py	New tests for cluster tree construction and (un)clustering behavior.
test/test_linalg_proxy.py	Adapts tests to updated cluster index return type and neighbor gathering signature.
test/test_linalg_skeletonization.py	Updates tests to exercise multilevel skeletonization flow.
test/test_matrix.py	Adapts callers to updated get_cluster_index / get_tgt_src_cluster_index return type.
doc/linalg.rst	Includes pytential.linalg.cluster in the linalg documentation.
doc/conf.py	Adds/adjusts intersphinx mappings for new referenced types.

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