Exploiting mesh structure to improve multigrid performance for saddle-point problems
Bibliographic record
Abstract
In recent years, solvers for finite-element discretizations of linear or linearized saddle-point problems, like the Stokes and Oseen equations, have become well established. There are two main classes of preconditioners for such systems: those based on a block-factorization approach and those based on monolithic multigrid. Both classes of preconditioners have several critical choices to be made in their composition, such as the selection of a suitable relaxation scheme for monolithic multigrid. From existing studies, some insight can be gained as to what options are preferable in low-performance computing settings, but there are very few fair comparisons of these approaches in the literature, particularly for modern architectures, such as GPUs. In this paper, we perform a comparison between a Block-Triangular preconditioner and monolithic multigrid methods with the three most common choices of relaxation scheme – Braess-Sarazin, Vanka, and Schur-Uzawa. We develop a performant Vanka relaxation algorithm for structured-grid discretizations, which takes advantage of memory efficiencies in this setting. We detail the behavior of the various CUDA kernels for the multigrid relaxation schemes and evaluate their individual arithmetic intensity, performance, and runtime. Running a preconditioned FGMRES solver for the Stokes equations with these preconditioners allows us to compare their efficiency in a practical setting. We show that monolithic multigrid can outperform Block-Triangular preconditioning, and that using Vanka or Braess-Sarazin relaxation is most efficient. Even though multigrid with Vanka relaxation exhibits reduced performance on the CPU (up to 100% slower than Braess-Sarazin), it is able to outperform Braess-Sarazin by more than 20% on the GPU, making it a competitive algorithm, especially given the high amount of algorithmic tuning needed for effective Braess-Sarazin relaxation.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 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 | 0.001 |
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".