ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A two-observation 4DVAR objective has an exact reverse gradient

Example

Take the scalar discrete model $x_{0} (θ) = θ$ , $x_{1} = 2 x_{0}$ , and $x_{2} = 3 x_{1}$ . Instantiate the strong-constraint objective of Strong- and weak-constraint 4DVAR with a finite gradient run with $T = 2$ , $d_{0} = d_{1} = d_{2} = p_{1} = p_{2} = 1$ , $U_{0} = U_{1} = U_{2} = R$ , model maps $Ψ_{0} (x) = 2 x$ and $Ψ_{1} (x) = 3 x$ , and observation maps $G_{1} = G_{2} = {id}_{R}$ . Choose $m_{b} = y_{1} = y_{2} = 0$ and the scalar positive-definite covariance matrices $B = Γ_{1} = Γ_{2} = [1]$ . The decision variable is the initial state $x_{0} = θ$ ; there are no weak-constraint model-error penalties. The objective is exactly $J (θ) = \frac{1}{2} x_{0}^{2} + \frac{1}{2} x_{1}^{2} + \frac{1}{2} x_{2}^{2} .$

Facts & Assumptions

Given: The finite scalar state-step graph, identity observation maps, unit positive-definite covariances, strong-constraint objective, and exact-real arithmetic.

[F1]

The reverse recursion for a finite $C^{1}$ state-step objective is its exact gradient (The reverse recursion is the exact gradient of a finite state-step objective).

[F2]

The strong-constraint variational assimilation objective contains a background penalty and observation-misfit terms along the generated model trajectory (Strong- and weak-constraint 4DVAR with a finite gradient run). The stated linear maps are $C^{1}$ on their open domains and the selected scalar covariances are positive definite, so that objective is admissible.

Verification

1.1givenF2algebra

Direct substitution gives $x_{1} = 2 θ$ and $x_{2} = 6 θ$ , so $J (θ) = \frac{1}{2} (1 + 4 + 36) θ^{2} = \frac{41}{2} θ^{2}$ and $J^{'} (θ) = 41 θ$ .

2.1F1step 1.1algebra∎

The terminal adjoint seed is $p_{2} = x_{2} = 6 θ$ . The reverse recursion [F1] gives $p_{1} = x_{1} + 3 p_{2} = 20 θ$ and $p_{0} = x_{0} + 2 p_{1} = 41 θ$ . Since $D x_{0} / D θ = 1$ , the adjoint gradient is $p_{0} = 41 θ$ , agreeing with the direct derivative.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

William W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, §3 (standard reference, not scraped)
Michael B. Giles and Niles A. Pierce, An Introduction to the Adjoint Approach to Design, §2.4 (standard reference, not scraped)