ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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 coarse phase mismatch can increase the Parareal error

Example

On $X = R^{2}$ , let the fine propagator be $F = I$ and the coarse propagator be $G = - I$ on each of two macro intervals, with $x_{0} = (1, 0)$ . Both maps preserve the Euclidean norm, and the initial coarse endpoint equals the fine endpoint. Yet after one Parareal sweep the endpoint error has norm $4$ . The second sweep is exact, as required by the first- $k$ -nodes lemma. Norm preservation alone therefore does not ensure monotone error decrease.

Facts & Assumptions

Given: Two real-linear propagators, two macro intervals, and the exact Parareal recurrence.

[F1]

Finite-budget Parareal correction sweeps: $U_{0}^{k} = x_{0}$ and $U_{n + 1}^{k + 1} = G (U_{n}^{k + 1}) + F (U_{n}^{k}) - G (U_{n}^{k})$ .

[F2]

Parareal reproduces the first k fine-grid nodes after k sweeps: after $k$ completed sweeps, each node $n \leq k$ equals the sequential fine node.

[A1]

$N = 2$ , $x_{0} = (1, 0)$ , $F = I$ , and $G = - I$ .

Verification

1.1givenA1algebra

The coarse initialization is $U_{1}^{0} = - x_{0}$ and $U_{2}^{0} = x_{0}$ , while the sequential fine endpoint is $x_{2} = F^{2} x_{0} = x_{0}$ . Both maps preserve the Euclidean norm because $∥ I x ∥_{2} = ∥ - I x ∥_{2} = ∥ x ∥_{2}$ .

2.1givenF1step 1.1algebra

By [F1], $U_{1}^{1} = G x_{0} + (F - G) x_{0} = - x_{0} + 2 x_{0} = x_{0}$ , and $U_{2}^{1} = G x_{0} + (F - G) (- x_{0}) = - x_{0} - 2 x_{0} = - 3 x_{0}$ .

3.1givenA1F2step 2.1algebra∎

Therefore $∥ U_{2}^{1} - x_{2} ∥_{2} = ∥ - 4 x_{0} ∥_{2} = 4$ . Since $K = N = 2$ , [F2] gives $U_{2}^{2} = x_{2}$ . The error growth at the first sweep is transient and does not contradict finite-step exactness.

Depends on

Used by

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Sources

Martin J. Gander, Time Parallel Time Integration, Chapter 2: Methods Based on Multiple Shooting (standard reference, not scraped)
Martin J. Gander, 50 Years of Time Parallel Time Integration (standard reference, not scraped)