A first-kind boundary integral study to solve the Laplace-Beltrami equation on a subsurface of the unit sphere and a multigrid algorithm for the acoustic single layer equation
Bibliographic record
Abstract
This dissertation is a study of two independent problems from the common research area of boundary element methods used to solve elliptic boundary value problems. Both topics focus mainly on a single layer approach. For the first project we consider the Laplace-Beltrami Dirichlet problem on a subsurface of the unit sphere in R^3. We derive and analyze a boundary element method for a first-kind integral equation to solve this boundary value problem. The method can be used to study the motion of point vortices on a sphere with impenetrable walls; we compare our approach with previous methods in this field. We derive rigorous error estimates for approximations of the solution to the integral equation in appropriate Sobolev spaces which yield global error estimates for the solution of the boundary value problem. Moreover, we support the theoretical results with numerical evidence gathered from test cases. The second project is concerned with a multigrid preconditioning strategy for the acoustic single layer equation in two dimensions. As proposed in [6], we reformulate the boundary element method in a weaker base inner product and then use a V-cycle multigrid scheme with a Richardson type smoother. We provide a full convergence analysis for the proposed multigrid algorithm based on an analogous result for the single layer equation corresponding to the Laplace operator. Numerical experiments underline the performance of the algorithm. Moreover, we conduct a numerical study of the effect of the weak inner product on the oscillatory behavior of the corresponding eigenfunctions.
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How this classification was reachedexpand
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 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".