CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Uniform state accuracy alone does not bound parameter-gradient error

Statement refuted

Uniform convergence of approximate states to an exact state, by itself, implies convergence or a uniform bound for their parameter derivatives.

Facts & Assumptions

Given: The displayed $C^{1}$ scalar maps and the uniform state norm on $(- 1, 1)$ .

[F1]

The discrete-continuous gradient estimate requires independent state, adjoint, derivative-data, endpoint, and quadrature bounds (Discrete and continuous ODE gradients agree to order p under augmented error bounds).

Counterexample

For $0 < h \leq 1$ and $θ \in (- 1, 1)$ , let the exact state be $u (θ) = 0$ and the approximate state be $u_{h} (θ) = h^{2} \sin (θ / h^{3})$ . The exact and approximate reduced objectives are these states themselves.

1.1givenalgebra

For every $θ$ and $h$ , $∣ u_{h} (θ) - u (θ) ∣ = h^{2} ∣ \sin (θ / h^{3}) ∣ \leq h^{2}$ . Thus the state error tends uniformly to zero as $h ↓ 0$ .

2.1F1step 1.1algebra∎

Differentiating gives $u_{h}^{'} (θ) = h^{- 1} \cos (θ / h^{3})$ and $u^{'} (θ) = 0$ . At $θ = 0$ , the gradient error is $∣ u_{h}^{'} (0) - u^{'} (0) ∣ = h^{- 1}$ , which is unbounded as $h ↓ 0$ . This refutes the stated implication; the example lacks a uniform derivative-data bound of the type required by [F1].

Depends on

Discrete and continuous ODE gradients agree to order p under augmented error bounds

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Michael B. Giles and Niles A. Pierce, An Introduction to the Adjoint Approach to Design, §4 (standard reference, not scraped)
William W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, §§3–4 (standard reference, not scraped)