Computational complexity of numerical solutions of initial value problems for differential algebraic equations (abstract only)
Bibliographic record
Abstract
We investigate the cost of solving initial value problems for differential algebraic equations depending on the number of digits of accuracy requested. A recent result showed that the cost of solving initial value problems (IVP) for ordinary differential equations (ODE) is polynomial in the number of digits of accuracy. This improves on the classical result of information-based complexity, which predicts exponential cost. The new theory is based on more realistic assumptions. The algorithm analyzed in this thesis is based on a previously published Taylor series method for solving a general class of differential algebraic equations. We consider DAE of constant index to which the method applies. The DAE is allowed to be of arbitrary index, fully implicit and have derivatives of order higher than one. Similarly, by considering a realistic model, we show that the cost of computing the solution of IVP for DAE with the algorithm adopted and by using automatic differentiation is polynomial in the number of digits of accuracy. We also show that non-adaptation is more expensive than adaptation, giving thus a theoretical justification of the success of adaptivity in practice. A particular case frequently arising in practical applications, the index-1 DAE, is treated separately, in more depth. On the other hand, an analysis of the higher-index DAE is significantly more complicated and applies to a wider class of problems. In both cases, continuous output is also given. These results apply to many important problems arising in practice. We present an interesting theoretical application to polynomial system solving.
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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 | 0.003 | 0.016 |
| 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 | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".