Validated Continuation for a simple two-dimensional elastic solid
Bibliographic record
Abstract
In this thesis, we validate solutions of a fourth order boundary value problem arising from the study of microstructure in shape memory alloys using the computer-assisted ’Validated Continuation’ method described in Gameiro & Lessard (2010). The problem, as formulated in Healey&Miller (2007), considers a simplified two-dimensional model of an elastic solid undergoing forced anti-plane shear in the presence of interfacial energy.The resulting parameter-dependent differential equation can be subdivided into a fourth order linear part andtwo nonlinearities, one of which, for a particular loading parameter, vanishes. This poses an important special case for the equation.Using the reciprocal of the capillarity coefficient as continuation parameter, several paths of equilibrium solutions bifurcating from the trivial solution are followed. Each numerical solution is then validated by a constructive analytical proof yielding a so-called validation radius. Existence of this radius allows to conclude that in the vicinity of the approximated numerical zero of the finite dimensional approximated equation there has to be a truesolution of the partial differential equation. We achieve validation results for the special case for several numerical solutions at the beginning of different paths. However, due to computational limits, we can not pursue the paths to higher values of the continuation parameter, i.e. to solutions corresponding to a small capillarity coefficient and, thus, to small interfacial energy. We also analyze the difficulties of treating the general case with the same method and conclude with some ideas on different approaches for that problem.
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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 | 0.006 |
| 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 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".