Convergence properties of the primal and dual forms of variational data assimilation
Bibliographic record
Abstract
Abstract The variational data assimilation problem can be solved in either its primal (3D/4D‐Var) or dual (3D/4D‐PSAS) form. The methods are equivalent at convergence but the dual method exhibits a spurious behaviour at the beginning of the minimization which leads to less probable states than the background state. This is a serious concern when using the dual method in operational implementations when only a finite number of iterations can be afforded. Two classes of minimization algorithms are examined in this article: the conjugate gradient (CG) and the minimum residual (MINRES) methods. While the CG algorithms ensure a monotonic reduction of the cost function, those based on the MINRES enforce instead a monotonic decrease of the norm of the gradient. In this article, it is shown that when applied to the minimization of the dual problem, the MINRES algorithms also lead to iterates for which their ‘image’ in physical space leads to a monotonic decrease of the primal cost function. A relationship is established showing that the primal objective function is related to the value of the dual cost function and the norm of its gradient. This holds for the incremental forms of both the three‐ and four‐dimensional cases. A new convergence criterion is introduced based on the error norm in model space to make sure that, for the dual problem, the same accuracy is obtained in the analysis when only a finite number of iterations is completed. Copyright © 2010 Royal Meteorological Society
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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 | 0.005 | 0.017 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| 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".