High Performance Parallel Block Supernode Direct Solver for Stiffness Equation
Bibliographic record
Abstract
The process of direct method to solve large sparse linear equation mainly includes reordering, symbolic factorization, numerical factorization and triangular solving. Traditional symbolic factorization predicts the pattern of $L$ based on single column and single row index. We propose to directly partition supernodes based on characteristics of matrix reordered by METIS, and then perform parallel symbolic factorization based on supernodes and row index fragments. A parallel block supernode numerical factorization strategy is proposed based on the concept of task pool here. In triangular solving stage, unlike traditional algorithms based on DAXPY and DDOT operations, we propose a new parallel triangular solving algorithm based on DGEMM and DTRSM operations. We name the parallel solver as finite element analysis direct solver (FEADS) and compare it with the advanced MKL PARDISO and MUMPS. The stiffness equations of 394770 and 719871 dimensions are solved using the solvers on two different computers. On the first computer, the solving efficiency of FEADS and MKL PARDISO is comparable, while MUMPS is relatively backward. On the second computer, FEADS performs especially well. For solving the case with 394770 dimensions, FEADS leads MKL PARDISO and MUMPS by 21.92% and 42.35%, respectively. For solving the case with 719871 dimensions, FEADS leads 34.75% and 38.38% respectively.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".