Superconvergent interpolants for collocation methods applied to mixed-order BVODEs
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Bibliographic record
Abstract
Continuous approximations to boundary value problems in ordinary differential equations (BVODEs), constructed using collocation at Gauss points, are more accurate at the mesh points than at off-mesh points. From these approximations, it is possible to construct improved continuous approximations by extending the high accuracy that is available at the mesh points to off-mesh points. One possibility is the bootstrap approach, which improves the accuracy of the approximate solution at the off-mesh points in a sequence of steps until the accuracy at the mesh points and off-mesh points is consistent. A bootstrap approach for systems of mixed-order BVODEs is developed to improve approximate solutions produced by COLNEW, a Gauss-collocation-based software package. An implementation of this approach is discussed and numerical results presented which confirm that the improved approximations satisfy the predicted error bounds and are relatively inexpensive to construct.
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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) | 0.000 | 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) | 0.004 | 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 it