An efficient numerical algorithm for the L2 optimal transport problem with periodic densities
Bibliographic record
Abstract
We present an extension of the numerical method of Loeper and Rapetti (2005, Numerical solution of the Monge–Ampère equation by a Newton's algorithm. C.R. Acad. Sci. Paris, I, 319–324) for the Monge–Ampère equation to non-uniform target densities and adopt it to solve the optimal transport problem with quadratic cost. The method employs a damped Newton algorithm to solve the Monge–Ampère equation. We show that the algorithm converges for sufficiently large damping coefficients, for the case where the source and target densities are sufficiently smooth, periodic and bounded away from zero. At each Newton iteration, we solve a non-constant coefficient linear partial differential equation. To improve the efficiency of the procedure, we use an analytically preconditioned fast Fourier transform method coupled with GMRES (Strain, J. (1994) Fast spectrally-accurate solution of variable-coefficients elliptic problems. Proc. Amer. Math. Sci., 122, 843–850) to solve this equation, as opposed to a more straightforward approach based on a second-order finite-difference discretization combined with biconjugate gradient used in the original LOEPER and RAPETTI paper. Finally, we present some numerical experiments in image processing to demonstrate the efficiency of the method.
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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 | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".