Fast Sparse Matrix Reordering on GPU for Cholesky Based Solvers
Bibliographic record
Abstract
Cholesky-based linear solvers are widely employed to solve large sparse positive semi-definite linear systems. Within the reordering, analyzing, factorizing, and solving pipeline, the reorder- ing of sparse matrices is a critical step in reducing non-zero fill-ins, thereby decreasing runtime and memory consumption. However, this stage remains the most time-consuming component of the linear solving process. There is currently a lack of GPU implementations specifically designed for matrix reordering in linear solving. This thesis proposes, to our knowledge, the first GPU-based nested dissection reordering algorithm. Our approach aims to achieve signif- icantly faster reordering times compared to traditional CPU-based methods while maintaining comparable quality in terms of non-zero fill-ins. We have implemented the proposed algorithm and conducted comprehensive performance comparisons with established CPU-based Nested Dissection implementations on various triangle mesh inputs. Our results demonstrate that the GPU-based reordering algorithm can achieve more than a 5 times speedup on average when applied to triangle mesh inputs. However, we produce an average of 6 times more non-zero ele- ments after Cholesky factorization compared to METIS, a widely-used graph partitioning soft- ware, based on our tests. Future work focuses on refining our partitioning strategy to achieve better fill-in reduction without sacrificing the significant speed advantages. Finally, we discuss the insights gained from our current implementation and outline future directions for further optimization and analysis.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.004 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.004 |
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 teacher head, 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".