Closest point methods with polyharmonic spline radial basis functions and local refinement
Bibliographic record
Abstract
Closest point methods are a class of embedding methods that have been used to solve partial differential equations on surfaces with the closest point representation of the surface. Recently, several studies replaced the standard Cartesian grid methods in the original Closest point methods with radial basis function generated finite differences. This reduces the computational cost and allows scattered and unstructured grids as well as locally refined uniform grids. This thesis uses the polyharmonic spline function as the radial basis function in the combined method which is different from the usual choice of Gaussian or multiquadric to avoid the shape parameter. We first perform convergence tests of the combined method. In all cases, the radial basis function closest point method uses fewer points in the embedding space while achieving a similar accuracy and convergence rate as the original closest point method. We then focus on solving partial differential equation problems with irregular grids that match features of the surface or the solution. These include using more points near high curvature regions or using more points near fine scale solution features. This can reduce the computational cost compared to using a uniform fine grid over the entire surface. Lastly, we provide an adaptive version of the combined method that is able to solve partial differential equation problems on surfaces when either or both of the surface features and problem features are changing in time.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".