A Remark on IVP and TVP Non-Smooth Viscosity Solutions to Hamilton-Jacobi Equations
Bibliographic record
Abstract
First-order PDEs arise in many applications in physics, control, image processing and other. Some boundary conditions for the PDE need to be formulated to close the problem. In time dependent problems very often the boundary conditions are specified at the initial time instant, and the problem is called the initial value problem (IVP). Respectively, such conditions may be needed at the terminal time instant, in which case one has the terminal value problem (TVP). For time invariant problems these two types of problems still exist, though the difference between them is not that explicit. For computational convenience one can transform from TVP to IVP or vice versa. In control problem a TVP for HJBI equation naturally arises, while in mechanics (physics) a IVP originally arises. Which type of the problems should be considered depends also on the fact that the sought for function (value function or action) in these problems is introduced as the function of the left or right endpoint of the cost function in integral form. As to first-order PDEs arising in image processing there is no straightforward indication what kind of problem must be considered. This paper discusses the difference between IVP and TVP solutions, the connection between Hamiltonians arising in formulation of IVP and TVP, several illustrative examples are demonstrated, one of them showing a smoothening phenomenon in the consequent solutions to IVP, TVP.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".