Topics in quasi-Newton and space-time spectral methods
Bibliographic record
Abstract
The first part of this thesis focuses on quasi-Newton methods. Broyden's method is a quasi-Newton method which is used to solve a system of nonlinear equations. Almost all convergence theory in the literature assumes existence of a root and bounds on the nonlinear function and its derivative in some neighbourhood of the root. All these conditions cannot be checked in practice. The motivation of this work is to derive a convergence theory where all assumptions can be verified, and the existence of a root and its superlinear rate of convergence are consequences of the theory. The theory is simple in the sense that it contains as few constants as possible. The method of Broyden-Fletcher-Goldfarb-Shanno (BFGS) is also a quasi-Newton method for unconstrained minimization. We generalize a convergence theory where all assumptions are verifiable and existence of a minimizer and superlinear convergence of the iteration are conclusions. In a continuation of this part, we consider Perry nonlinear conjugate gradient (NCG) method and scaled memoryless BFGS method. In the second part, space-time spectral methods are considered. For time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods which converge spectrally in both space and time have appeared recently. In this thesis, it is shown that a Chebyshev spectral collocation method of Tang and Xu for the heat equation converges exponentially when the solution is analytic. We also derive a condition number estimate of the method. Another space-time Chebyshev collocation scheme which is easier to implement is proposed and analyzed. We also present space-time spectral collocation methods for the Schrodinger, wave, Airy and beam equations. In particular, fully spectral convergence and a condition number estimate are shown for Schrodinger and wave equations. Numerical results verify the theoretical results, and demonstrate that the space-time methods also work for some common nonlinear PDEs (Allen-Cahn, viscous Burgers', Sine-Gordon, KdV, Kuramoto--Sivashinsky and Cahn-Hilliard equations).
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".