An adaptive choice of primal constrains for BDDC domain decomposition algorithms
Bibliographic record
Abstract
An adaptive choice based on parallel sums for the primal space of BDDC [1] deluxe methods [2] is analyzed. The primal constraints of a BDDC algorithm provide the global, coarse part of such a preconditioner and is of crucial importance for obtaining rapid convergence of these preconditioned conjugate gradient methods for the case of many subdomains. For problems in three dimensions, there is a need to develop algorithms and results for equivalence classes with three or more elements, e.g., subdomain edges. For this purpose, parallel sums for general equivalence classes are considered. The use of parallel sums for equivalence classes with two elements (subdomain faces) has proven very successful; see [3]. An upper bound of the square of the norm of a jump operator PD acting on the elements in a product space related to the subdomains is derived; it has been known that such a bound provides an estimate of the condition number of the BDDC algorithm; see [4]. This bound is given in terms of parallel sums of single Schur complements and sums of other Schur complements. Hence, generalized eigenvalue problems with parallel sums related to the faces and edges of the subdomains are formulated. A few eigenvectors associated with the smallest eigenvalues are selected and they generate a primal constraint. These generalized eigenvalue problems are defined in terms of the relevant Schur complements and Schur complements of these Schur complements associated with a minimal energy extension, e.g., from a subdomain edge of a three-dimensional finite element problem. Numerical results for elliptic problems verify the performance of the algorithm, using a series of experiments with regular subdomains as well as subdomains generated by a METIS mesh partitioner. There is also fast convergence for problems with a quite irregular coefficient inside the subdomains.
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".