LemmaStatement: Literature-sourcedProof: 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.

Parareal reproduces the first k fine-grid nodes after k sweeps

Statement

For the exact-arithmetic recurrence in Finite-budget Parareal correction sweeps, let $x_{0}$ be the initial state and $x_{n + 1} = F_{n} (x_{n})$ its sequential fine trajectory. For every completed sweep $k$ and every $0 \leq n \leq \min {k, N}$ , $U_{n}^{k} = x_{n}$ . Consequently, if the requested sweep budget $K \geq N$ is fully completed, the returned table contains the complete sequential fine trajectory in row $K$ .

Facts & Assumptions

Given: The exact-arithmetic recurrence is defined on the rows considered; each row called completed is formed from successful evaluations.

[F1]

Fine and coarse propagators on a common macro grid: the sequential fine trajectory is $x_{n + 1} = F_{n} (x_{n})$ from $x_{0}$ .

[F2]

Finite-budget Parareal correction sweeps: $U_{0}^{k} = x_{0}$ for every sweep, the initial row is coarse-propagated, and each correction is $U_{n + 1}^{k + 1} = G_{n} (U_{n}^{k + 1}) + F_{n} (U_{n}^{k}) - G_{n} (U_{n}^{k})$ . The algorithm returns every fully completed row and may stop early after a propagation failure or time-budget expiry.

Proof

1.1givenF1F2algebra

At sweep $k = 0$ , the only claimed node is $n = 0$ , and $U_{0}^{0} = x_{0} = x_{0}$ by initialization and the definition of the fine trajectory [F1].

1.2givenF1F2inductionalgebra

Suppose the claim holds at sweep $k$ . At sweep $k + 1$ , $U_{0}^{k + 1} = x_{0}$ by [F2]. Induct on $n$ from $0$ to $\min {k, N - 1}$ ; if $U_{n}^{k + 1} = x_{n}$ , then $n \leq k$ also gives $U_{n}^{k} = x_{n}$ , so [F2] yields $U_{n + 1}^{k + 1} = G_{n} (x_{n}) + F_{n} (x_{n}) - G_{n} (x_{n}) = F_{n} (x_{n}) = x_{n + 1}$ by [F1]. Thus the claim holds through node $\min {k + 1, N}$ .

2.1F2step 1.2inductionalgebra∎

Induction on $k$ proves the first assertion. If the requested rows $0, \dots, K$ all complete and $K \geq N$ , applying it at $k = K$ gives $U_{n}^{K} = x_{n}$ for every $0 \leq n \leq N$ ; by [F2], row $K$ is in the returned table.

Depends on

Used by

A coarse phase mismatch can increase the Parareal error Example

Dependency tree · two levels

4 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

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, in Multiple Shooting and Time Domain Decomposition Methods (standard reference, not scraped)