Conjugate Gradient Methods for Spline Collocation Equations
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Bibliographic record
Abstract
We study the parallel computation of linear second order elliptic Partial Differential Equation (PDE) problems in rectangular domains. We discuss the application of Conjugate Gradient (CG) and Preconditioned Conjugate Gradient (PCG) methods to the linear system arising from the discretisation of such problems using quadratic splines and the collocation discretisation methodology. Our experiments show that the number of iterations required for convergence of CG-QSC (Conjugate Gradient applied to Quadratic Spline Collocation equations) grows linearly with the square root of the number of equations. We implemented the CG and PCG methods for the solution of the Quadratic Spline Collocation (QSC) equations on the iPSC/2 hypercube and present performance evaluation results for up to 32 processors configurations. Our experiments show efficiencies of the order of 90%, for both the fixed and scaled speedups.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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 |
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