A posteriori error analysis for a space‐time parallel discretization of parabolic partial differential equations
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Bibliographic record
Abstract
Abstract We construct a space‐time parallel method for solving parabolic partial differential equations by coupling the parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of “local” problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint‐based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete parareal iterations and incomplete iterations of the domain decomposition solver. We first develop an error analysis for the parareal method applied to parabolic partial differential equations, and then refine this analysis to the case where the associated spatial problems are solved using overlapping domain decomposition. These constitute our time parallel algorithm and space‐time parallel algorithm respectively. Numerical experiments demonstrate the accuracy of the estimator for both algorithms and the iterations between distinct components of the error.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.003 |
| 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)
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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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