Convergence rates of adaptive methods, Besov spaces, and multilevel\n approximation
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Bibliographic record
Abstract
This paper concerns characterizations of approximation classes associated to\nadaptive finite element methods with isotropic h-refinements. It is known from\nthe seminal work of Binev, Dahmen, DeVore and Petrushev that such classes are\nrelated to Besov spaces. The range of parameters for which the inverse\nembedding results hold is rather limited, and recently, Gaspoz and Morin have\nshown, among other things, that this limitation disappears if we replace Besov\nspaces by suitable approximation spaces associated to finite element\napproximation from uniformly refined triangulations. We call the latter spaces\n*multievel approximation spaces*, and argue that these spaces are placed\nnaturally halfway between adaptive approximation classes and Besov spaces, in\nthe sense that it is more natural to relate multilevel approximation spaces\nwith either Besov spaces or adaptive approximation classes, than to go directly\nfrom adaptive approximation classes to Besov spaces. In particular, we prove\nembeddings of multilevel approximation spaces into adaptive approximation\nclasses, complementing the inverse embedding theorems of Gaspoz and Morin.\n Furthermore, in the present paper, we initiate a theoretical study of\nadaptive approximation classes that are defined using a modified notion of\nerror, the so-called *total error*, which is the energy error plus an\noscillation term. Such approximation classes have recently been shown to arise\nnaturally in the analysis of adaptive algorithms. We first develop a\nsufficiently general approximation theory framework to handle such\nmodifications, and then apply the abstract theory to second order elliptic\nproblems discretized by Lagrange finite elements, resulting in\ncharacterizations of modified approximation classes in terms of memberships of\nthe problem solution and data into certain approximation spaces, which are in\nturn related to Besov spaces.\n
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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