Accelerated Bidirectional Modifications of the Steepest Descent Method for Ill-posed Linear Algebraic Systems with Dynamics-theoretical and Optimization Interpretation
Bibliographic record
Abstract
It is well known that the classical numerical algorithm of the steepest descent method (SDM) is effective for well-posed linear systems, but performs poorly for ill-posed ones. In this paper we propose accelerated and/or bidirectional modifications of SDM, namely the accelerated steepest descent method (ASDM), the bidirectional method (2DM), and the accelerated bidirectional method (A2DM). The starting point is a manifold defined in terms of a minimum functional and a fictitious time variable; nevertheless, in the end the fictitious time variable disappears so that we arrive at purely iterative algorithms. The proposed algorithms are justified by dynamics-theoretical and optimization interpretation. The accelerator plays a prominent role of switching from the situation of slow convergence to a new situation that the functional tends to decrease stepwise in an intermittent and ceaseless manner. Three examples of solving ill-posed systems are examined and comparisons are made with exact solutions and with the existing algorithms of the SDM, the Barzilai-Borwein method, and the random SDM, revealing that the new algorithms of ASDM and A2DM have better computational efficiency and accuracy even for the highly ill-posed systems.
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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) | 0.001 | 0.001 |
| 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 | 0.001 | 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".