Parallel in Time Algorithm with Spectral-Subdomain Enhancement for Volterra Integral Equations
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Bibliographic record
Abstract
This paper proposes a parallel in time (also called time parareal) method to solve Volterra integral equations of the second kind. The parallel in time approach follows the spirit of the domain decomposition that consists of breaking the domain of computation into subdomains and solving iteratively the subproblems in a parallel way. To obtain a high order of accuracy, a spectral collocation accuracy enhancement in subdomains will be employed. Our main contributions in this work are twofold: (i) A time parareal method is designed for the integral equations, which to our knowledge is the first of its kind. The new method is an iterative process combining a coarse prediction in the whole domain with fine corrections in subdomains by using spectral approximation, leading to an algorithm of very high accuracy. (ii) A rigorous convergence analysis of the overall method is provided. The numerical experiment confirms that the overall computational cost is considerably reduced while the desired spectral rate of convergence can be obtained.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| 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) | 0.004 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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