A Hybrid Algorithm Based on Optimal Quadratic Spline Collocation and Parareal Deferred Correction for Parabolic PDEs
Bibliographic record
Abstract
Parareal is a kind of time parallel numerical methods for time-dependent systems. In this paper, we consider a general linear parabolic PDE, use optimal quadratic spline collocation (QSC) method for the space discretization, and proceed with the parareal technique on the time domain. Meanwhile, deferred correction technique is also used to improve the accuracy during the iterations. In fact, the optimal QSC method is a correction of general QSC method. Along the temporal direction we embed the iterations of deferred correction into parareal to construct a hybrid method, parareal deferred correction (PDC) method. The error estimation is presented and the stability is analyzed. To save computational cost, we find out a simple way to balance the two kinds of iterations as much as possible. We also argue that the hybrid algorithm has better system efficiency and costs less running time. Numerical experiments by multicore computers are attached to exhibit the effectiveness of the hybrid algorithm.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".