Least-squares reconstruction of a 3D potential from its measured spatial velocity field
Bibliographic record
Abstract
This paper describes a new method for the least-squares reconstruction of a hypersurface from measured 3D gradient data, for example, the potential function of a measured velocity field. The novel aspect of the proposed algorithm is that the solution relies on representing 3D data with hypermatrices, and obtaining the least-squares solution to the problem as the solution of a hypermatrix equation, i.e., a set of linear equations that equates hypermatrices. The new approach enables the solution of the reconstruction problem over an l x m x n hypersurface in O(n^4) time (with l ~ m ~ n), whereas a straightforward approach of solving the problem with a standard least-squares approach (vectorization) would yield an order O(n^9) algorithm. The new algorithm is therefore five orders of magnitude faster than the state-of-the-art. The new algorithm is tested with synthetic data with synthetic noise. The method is, however, applicable to the problem of reconstructing a potential from velocity data measured, for example, via particle image velocimetry.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| 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) | 0.003 | 0.002 |
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".