Multiple Scale Methods For Optimization Of Discretized Continuous Functions
Bibliographic record
Abstract
Discretized versions of optimization problems over continuous arguments are routinely solved at a single fine resolution, incurring a per-iteration cost that grows, often superlinearly, with the number of grid points. This paper analyzes a multiscale method that instead solves a hierarchy of increasingly fine dyadic discretizations. Linear interpolation of each coarse solution warm starts the next finer scale using any q-linearly convergent update rule as the inner solver. Each coarse problem is a consistent discretization of the continuous problem. Structural properties such as convexity and smoothness are preserved. For problems with Lipschitz-continuous solutions, two variants of the method converge to the fine-scale solution with explicit error bounds. The fine-scale solution in turn approximates the continuous solution once the grid is sufficiently fine, with quantified constants. The total cost to reach a fixed accuracy is provably lower than that of single-scale optimization whenever the cost of one update grows at least linearly in the problem size. Numerical experiments on probability density demixing problems, including geological survey data, show four- to sevenfold speedups while using a fraction of the memory.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".