TheoremStatement: AI-adaptedProof: 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.

An additive error ledger for inexact Parareal sweeps

Statement

Use the notation and Lipschitz hypotheses of A conditional binomial error bound for Parareal, with $A = \max {1, a}$ . Let computed iterates $V_{n}^{k} \in D$ satisfy $V_{0}^{k} = x_{0}$ , $∥ V_{n}^{0} - x_{n} ∥ \leq M$ , and, for each requested correction, $∥ V_{n + 1}^{k + 1} - [G_{n} (V_{n}^{k + 1}) + (F_{n} - G_{n}) (V_{n}^{k})] ∥ \leq η$ for a constant $η \geq 0$ . Define $P_{n}^{k} = M (\binom{n}{k}) A^{n} b^{k}$ and a nonnegative forcing array by $R_{0}^{k} = 0$ , $R_{n}^{0} = 0$ , and $R_{n + 1}^{k + 1} = A R_{n}^{k + 1} + b R_{n}^{k} + η .$ Then $∥ V_{n}^{k} - x_{n} ∥ \leq P_{n}^{k} + R_{n}^{k}$ for every completed node and sweep.

The base envelope $M$ includes coarse-initialization error relative to the fine trajectory. The per-update defect $η$ must bound the actual combined effect of fine/coarse evaluation errors and correction-solve error; a residual tolerance alone supplies no forward-error bound.

Facts & Assumptions

Given: The Parareal Lipschitz hypotheses, the stated computed recurrence defect, and the definitions of $P_{n}^{k}$ and $R_{n}^{k}$ .

[F1]

A conditional binomial error bound for Parareal: with $A = \max {1, a}$ , the exact-arithmetic error is bounded by $P_{n}^{k} = M (\binom{n}{k}) A^{n} b^{k}$ .

[A1]

Every computed comparison point lies in the common set on which the two Lipschitz bounds hold, and the combined update defect has norm at most $η$ .

Proof

1.1givenA1algebra

Set $e_{n}^{k} = ∥ V_{n}^{k} - x_{n} ∥$ . Subtracting $x_{n + 1} = F_{n} (x_{n})$ from the computed recurrence, splitting off the update defect, and applying the two Lipschitz inequalities gives $e_{n + 1}^{k + 1} \leq a e_{n}^{k + 1} + b e_{n}^{k} + η \leq A e_{n}^{k + 1} + b e_{n}^{k} + η$ .

1.2givenF1A1algebra

At node $0$ , $e_{0}^{k} = 0$ , $P_{0}^{k} = 0$ for $k > 0$ , and $R_{0}^{k} = 0$ ; at $k = 0$ , the base coarse-error assumption gives $e_{n}^{0} \leq M \leq P_{n}^{0} + R_{n}^{0}$ .

2.1F1A1step 1.1step 1.2inductionalgebra∎

Assume $e_{n}^{k} \leq P_{n}^{k} + R_{n}^{k}$ at node $n$ for every completed sweep. For $k \geq 0$ , step 1.1 gives $e_{n + 1}^{k + 1} \leq A (P_{n}^{k + 1} + R_{n}^{k + 1}) + b (P_{n}^{k} + R_{n}^{k}) + η$ . The $P$ terms satisfy $A P_{n}^{k + 1} + b P_{n}^{k} = M A^{n} b^{k + 1} (A (\binom{n}{k + 1}) + (\binom{n}{k})) \leq M A^{n + 1} b^{k + 1} ((\binom{n}{k + 1}) + (\binom{n}{k})) = P_{n + 1}^{k + 1}$ because $A \geq 1$ and Pascal's identity holds. The remaining terms equal $R_{n + 1}^{k + 1}$ by definition. Therefore the bound holds at node $n + 1$ ; induction proves it for all nodes and sweeps.

Depends on

A conditional binomial error bound for Parareal

Used by

Separate Parareal iteration error from fine-method error Corollary

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

Martin J. Gander, 50 Years of Time Parallel Time Integration (standard reference, not scraped)
Ben S. Southworth, Necessary Conditions and Tight Two-Level Convergence Bounds for Parareal and Multigrid Reduction in Time (standard reference, not scraped)