Finite element bounds for linear functionals of exact solutions to the three-dimensional Poisson's equation
Bibliographic record
Abstract
A method for obtaining rigorous upper and lower bounds to an output of the exact solution of the three dimensional Poisson problem is described. More recently bounds to the exact outputs of interest have been obtained for both the Poisson equation and the advection-diffusion-reaction equation. In this work, the objective is to reduce the cost of these calculations in the context of three dimensional problems. The new ingredients are two-fold. First, the finite element tearing and interconnecting (FETI) procedure is invoked in order to calculate the inter-subdomain continuity multipliers (hybrid fluxes) and extend the approach to three space dimensions where bounds to the exact output of interest in a cube geometry are reported. Second, the computational cost is reduced by using the FETI procedure because the primal and adjoint field solutions can be calculated jointly with the hybrid fluxes required for the subdomain problem.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 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".