Convergent finite difference solvers for viscosity solutions of the\n elliptic Monge-Amp\\`ere equation in dimensions two and higher
Bibliographic record
Abstract
The elliptic Monge-Amp\\`ere equation is a fully nonlinear Partial\nDifferential Equation that originated in geometric surface theory and has been\napplied in dynamic meteorology, elasticity, geometric optics, image processing\nand image registration. Solutions can be singular, in which case standard\nnumerical approaches fail. Novel solution methods are required for stability\nand convergence to the weak (viscosity) solution.\n In this article we build a wide stencil finite difference discretization for\nthe \\MA equation. The scheme is monotone, so the Barles-Souganidis theory\nallows us to prove that the solution of the scheme converges to the unique\nviscosity solution of the equation.\n Solutions of the scheme are found using a damped Newton's method. We prove\nconvergence of Newton's method and provide a systematic method to determine a\nstarting point for the Newton iteration.\n Computational results are presented in two and three dimensions, which\ndemonstrates the speed and accuracy of the method on a number of exact\nsolutions, which range in regularity from smooth to non-differentiable.\n
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| 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".