Superconvergent interpolants for Gaussian collocation solutions of mixed order BVODE systems
Bibliographic record
Abstract
The high quality COLSYS/COLNEW collocation software package is widely used for the numerical solution of boundary value ODEs (BVODEs), often through interfaces to computing environments such as Scilab, R, and Python. The continuous collocation solution returned by the code is much more accurate at a set of mesh points that partition the problem domain than it is elsewhere; the mesh point values are said to be superconvergent. In order to improve the accuracy of the continuous solution approximation at non-mesh points, when the BVODE is expressed in first order system form, an approach based on continuous Runge-Kutta (CRK) methods has been used to obtain a superconvergent interpolant (SCI) across the problem domain. Based on this approach, recent work has seen the development of a new, more efficient version of COLSYS/COLNEW that returns an error controlled SCI. However, most systems of BVODEs include higher derivatives and a feature of COLSYS/COLNEW is that it can directly treat such mixed order BVODE systems, resulting in improved efficiency, continuity of the approximate solution, and user convenience. In this paper we generalize the approach mentioned above for first order systems to obtain SCIs for collocation solutions of mixed order BVODE systems. The main contribution of this paper is the derivation of generalizations of continuous Runge-Kutta-Nyström methods that form the basis for SCIs for this more general problem class. We provide numerical results that (ⅰ) show that the SCIs are much more accurate than the collocation solutions at non-mesh points, (ⅱ) verify the order of accuracy of these SCIs, and (ⅲ) show that the cost of utilizing the SCIs is a small fraction of the cost of computing the collocation solution upon which they are based.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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 | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".