Adjoint of Least Squares Shadowing: Existence, Uniqueness and Coarse Domain Discretization
Bibliographic record
Abstract
Chaotic dynamical systems are characterized by the sensitive dependence of trajectories on initial conditions. Conventional sensitivity analysis of time-averaged functionals yields unbounded sensitivities when the simulation is chaotic. The least squares shadowing (LSS) is a popular approach to computing bounded sensitivities in the presence of chaotic dynamical systems. The current paper proves the existence, uniqueness, and boundedness of the adjoint of the LSS equations. In particular, the analysis yields a sharper bound on the condition number of the LSS equations than currently demonstrated in existing literature and shows that the condition number is bounded for large integration times. The derived bound on condition number also shows a relation between the conditioning of the LSS and the time dilation factor which is consistent with the trend numerically observed in the previous LSS literature. Furthermore, using the boundedness of the condition number for large integration times, we provide an alternate proof to (Chater et al., 2017) of the convergence of the LSS sensitivity to the true sensitivity at the rate of $\mathcal{O}\left(\frac{1}{\sqrt{T}}\right)$ regardless of the boundary conditions imposed on the adjoint, as long as the adjoint boundary conditions are bounded. Existence and uniqueness of the solution to the continuous-in-time adjoint LSS equation ensure that the LSS equation can be discretized independently of the primal equation and that the true LSS adjoint solution is recovered as the time step is refined. This allows for the adjoint LSS equation to be discretized on a coarser time domain than that of the primal governing equation to reduce the cost of solving the linear space-time system.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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".