A GPU-Based MIS Aggregation Strategy: Algorithms, Comparisons, and Applications within AMG
Bibliographic record
Abstract
The algebraic multigrid (AMG) method is often used as a preconditioner in Krylov subspace solvers such as the conjugate gradient method. An AMG preconditioner hierarchically aggregates the degrees of freedom during the coarsening phase in order to efficiently account for lower-frequency errors. Each degree of freedom in the coarser level corresponds to one of the aggregates in the finer level. The aggregation in each level in the hierarchy has a significant impact on the effectiveness of AMG as a preconditioner. The aggregation can be formulated as a partitioning problem on the graph induced from the matrix representation of a linear system. We present a GPU implementation of a "bottom-up" partitioning scheme based on maximal independent sets (MIS). We also present some novel topology-informed metrics that measure the quality of a partition. To test our implementation and the metrics, we use an existing AMG preconditioned conjugate gradient (PCG-AMG) solver and show that our metrics are correlated with the time and the number of iterations needed for the linear system to converge to a solution. For comparable coarsening ratios, we show that the MIS-based aggregation methods outperform Metis-based "top-down" aggregation method for the PCG-AMG method. Our results also indicate that MIS-based aggregation methods provide aggregates that are evaluated more favorably by our metrics than the aggregates provided by the Metis-based method.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".