Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation
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Bibliographic record
Abstract
Summary In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p ‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.
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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 | 0.002 | 0.008 |
| 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 | 0.001 |
| Open science | 0.001 | 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 it