Smoothing Analysis of Two Robust Multigrid Methods for Elliptic Optimal Control Problems
Bibliographic record
Abstract
Abstract. In this paper we study and compare two multigrid relaxation schemes with coarsening by two, three, and four for solving elliptic sparse optimal control problems with control constraints and combined [Formula: see text] and [Formula: see text] cost functional. First, we perform a detailed local Fourier analysis (LFA) of a well-known collective Jacobi relaxation (CJR) scheme for the unconstrained case with only [Formula: see text] cost functional, where the optimal smoothing factors are derived. This insightful analysis reveals that the optimal relaxation parameters depend on both the mesh step size [Formula: see text] and the regularization parameter [Formula: see text], which was not investigated in literature. Second, we propose and analyze a new mass-based Braess–Sarazin relaxation (BSR) scheme, which is proven to provide smaller smoothing factors than the CJR scheme when [Formula: see text] for a small constant [Formula: see text]. Finally, these schemes are successfully extended to control-constrained cases through the semismooth Newton method, where the corresponding Jacobian systems are treated by the proposed multigrid schemes. The nonstandard coarsening by three or four with BSR is competitive with the standard coarsening by two. Numerical examples are presented to validate our theoretical outcomes. The proposed inexact BSR (IBSR) scheme, where two preconditioned conjugate gradients (PCG) iterations are applied to solve the Schur complement system, yields a better computational efficiency than the CJR scheme in the conducted numerical comparison.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".