Underdetermined Fourier extensions for surface partial differential equations
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Bibliographic record
Abstract
Abstract We analyse and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show a general result relating boundedness, solvability and convergence that can be used to find eigenvalues. The method works by extending a solution to a surface PDE into a box-shaped domain so that the differential operators of the extended function agree with the surface differential operators, as in the Closest Point Method. This differs from approaches that require a basis for the surface of interest, which may not be available. Numerical experiments are also provided, demonstrating super-algebraic convergence. Current high-order methods for surface PDEs are often limited to a small class of surfaces or use radial basis functions (RBFs). Our approach offers certain advantages related to conditioning, generality and ease of implementation. The method is meshfree and works on arbitrary surfaces (closed or nonclosed) defined by point clouds with minimal conditions.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| 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 |
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