Optimizing multigrid reduction‐in‐time and Parareal coarse‐grid operators for linear advection
Bibliographic record
Abstract
Abstract Parallel‐in‐time methods, such as multigrid reduction‐in‐time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time‐dependent partial differential equations (PDEs) in modern high‐performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection‐dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant‐coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse‐grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. One of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse‐grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge–Kutta methods combined with upwind‐finite‐difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed‐ups over sequential time‐stepping.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 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".