An Adaptive Multilevel Wavelet Solver for Elliptic Equations on an Optimal Spherical Geodesic Grid
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Bibliographic record
Abstract
An adaptive multilevel wavelet solver for elliptic equations on an optimal spherical geodesic grid is developed. The method is based on second-generation spherical wavelets on almost uniform optimal spherical geodesic grids. It is an extension of the adaptive multilevel wavelet solver [O. V. Vasilyev and N. K.-R. Kevlahan, J. Comput. Phys., 206 (2005), pp. 412–431] to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. A hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate the Laplace–Beltrami operator. The optimal spherical geodesic grid [Internat. J. Comput. Geom. Appl., 16 (2006), pp. 75–93] is convergent in terms of local mean curvature and has lower truncation error than conventional spherical geodesic grids. The overall computational complexity of the solver is $O(\mathcal{N})$, where $\mathcal{N}$ is the number of grid points after adaptivity. The accuracy and efficiency of the method is demonstrated for the spherical Poisson equation. Although the present paper considers the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by choosing an appropriate coarse approximation and using recursive surface subdivision.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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