Bibliographic record
Abstract
This thesis concerns the numerical solution of time-dependent partial differential equations (PDEs) on general surfaces using a recent technique known as the Closest Point Method. The Closest Point Method represents surfaces with a closest point representation which leads to great flexibility with respect to surface geometry, among other advantages. The computation itself alternates between two steps: first, an explicit time step is performed using standard finite difference techniques on a narrow band of grid points surrounding the surface embedded in a higher dimension; second, a closest point extension is used to maintain consistency with the original surface PDE. The Closest Point Method is applied to the important problem of interface motion on surfaces by using level set equations posed on surfaces. New weighted essentially non-oscillatory (WENO) interpolation schemes are derived to perform the necessary closest point extensions. This approach, in combination with standard Hamilton--Jacobi WENO finite difference schemes and explicit time stepping, gives high-order results (up to fifth-order) on a variety of test problems. Example computations are performed on a sphere, torus, triangulated human hand and Klein bottle to demonstrate the flexibility of the method. A new implicit Closest Point Method is presented for surface PDEs which are stiff, for example, because of diffusion terms. The method uses implicit time-stepping schemes to allow large steps but retains the flexibility with respect to surface geometry of the original explicit Closest Point Method. Numerical convergence studies on the heat equation and a fourth-order biharmonic problem demonstrate the accuracy of the method and a variety of example computations demonstrate its effectiveness. These include an image processing example of blurring on triangulated surfaces, heat diffusion on a surface consisting of multiple components connected by a thin filament and Turing pattern formation on surfaces using implicit--explicit (IMEX) time stepping. A class of time-stepping methods known as diagonally split Runge--Kutta (DSRK) methods is investigated. These methods appear promising because they offer both high-order convergence and unconditional contractivity (a nonlinear stability property). However, numerical computations and analysis of stage-order demonstrates that unconditionally contractive DSRK methods suffer from order reduction which severely limits their practical application.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.013 | 0.007 |
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".