ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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 dense coarse solve can dominate local work

Example

Assume all hypotheses of The two-dimensional BDDC and FETI-DP polylogarithmic bound: the two-dimensional scalar model, matching shape-regular quasiuniform meshes with fixed polygon-shape bounds, positive coefficients constant on subdomains, all subdomain vertices primal, coefficient-scaled averaging, and nonzero assembled and multiplier spaces. Consider a dense direct-solve cost model for the resulting positive definite local dual and coarse blocks with $N$ subdomains, local matrix dimension $m$ fixed, and a coarse dimension $n_{c}$ proportional to $N$ . If local matrices are factored concurrently by dense Cholesky, the local arithmetic is $O (N m^{3})$ in total and $O (m^{3})$ ideal parallel time. The coarse factorization uses $O (n_{c}^{3})$ arithmetic and $O (n_{c}^{2})$ storage. For fixed $H / h$ , The two-dimensional BDDC and FETI-DP polylogarithmic bound keeps the iteration condition bound independent of $N$ , while the dense coarse solve can dominate total work.

Facts & Assumptions

Given: A family satisfying all hypotheses of [F1], with $H / h$ and its geometric constants fixed, $N$ positive definite local dual blocks of equal fixed positive dimension $m$ , a positive definite coarse block of dimension $n_{c}$ proportional to $N$ , concurrent local factorizations, and a dense direct Cholesky model. The coarse dimension counts the shared primal variables; it does not require exactly one primal variable per subdomain.

[F1]

The two-dimensional BDDC and FETI-DP polylogarithmic bound: under its fixed geometric and coefficient assumptions, the condition-number bound contains $H / h$ but no subdomain-count factor.

Verification

1.1givenalgebra

A dense Cholesky factorization of an $m \times m$ SPD matrix uses a constant multiple of $m^{3}$ arithmetic operations and stores $O (m^{2})$ entries. Repeating on $N$ local matrices gives total $O (N m^{3})$ work, and perfect concurrency reduces the local phase to $O (m^{3})$ time.

2.1givenF1step 1.1algebra∎

Since $n_{c}$ is proportional to $N$ , dense coarse Cholesky uses $O (N^{3})$ arithmetic and $O (N^{2})$ storage. If $H / h$ stays fixed while the number of subdomains grows, [F1] leaves the iteration condition bound unchanged, but the coarse direct-solve work grows cubically.

Depends on

The two-dimensional BDDC and FETI-DP polylogarithmic bound

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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