Modified Parareal Algorithm for Solving Time-Dependent Differential Equations
Bibliographic record
Abstract
Parallel algorithms are implemented to compute the solutions of partial differential equations and ordinary differential equations of complex dynamical systems to achieve near real-time solutions. One of the parallel algorithms widely implemented is the Parareal algorithm to solve time-dependent differential equations for various scientific applications. Parareal algorithm has shown promising speedups in achieving near real-time solutions using accelerators. However, it has been observed that the sequential predictorcorrector step of the Parareal algorithm impacts the computational performance. This paper analyses the Parareal algorithm and proposes modification to the predictor-corrector step of the Parareal algorithm to exploit data parallelism more and reduce the computation time. The modified algorithm is implemented to solve two systems of interdependent ODEs. The numerical accuracy and performance analysis of the modified algorithm is shown to be same as the original Parareal. The performance analysis of the modified algorithm on two accelerator computing architectures: Intel Xeon Phi CPU and Graphical processing units with OpenMP, OpenACC, and CUDA programming models are presented. The modified algorithm demonstrates performance improvement ranging from 1.2x-2x with respect to the original Parareal algorithm.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 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) | 0.000 | 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".