LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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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.

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Schwarz local matrices and exact solves are well defined

Statement

For the data of Finite-dimensional Schwarz subspace-correction data, every local matrix $A_{i} = R_{i}^{T} A R_{i}$ with $i \in I$ is symmetric positive definite and therefore invertible. If the coarse index $0$ is present, then $A_{0} = P^{T} A P$ is the SPD Galerkin coarse matrix. Hence every exact local solve in the definition exists uniquely.

Facts & Assumptions

Given: The real SPD matrix $A$ , the full-column-rank prolongations $R_{i}$ , and the index set $I$ of Finite-dimensional Schwarz subspace-correction data.

[F1]

The definition gives $A_{i} = R_{i}^{T} A R_{i}$ and requires each $R_{i}$ to have full column rank (Finite-dimensional Schwarz subspace-correction data).

[F2]

For SPD $A$ and full-column-rank $P$ , the Galerkin matrix $P^{T} A P$ is SPD and the exact Galerkin coarse correction is defined (The Galerkin coarse correction is an energy-orthogonal projector).

[F3]

A real SPD matrix has a unique solution to $M x = b$ for every right-hand side, so it is invertible (For Hermitian positive-definite $A$ , the exact solution of $A x = b$ is the unique minimizer of the quadratic energy).

[F4]

Transposition reverses matrix products, and $A^{T} = A$ (Transpose is linear and involutive, and $(A B)^{T} = B^{T} A^{T}$ ).

Proof

1.1F1F4givenalgebra

Fix any $i \in I$ with $i \neq 0$ and any nonzero $z \in R^{n_{i}}$ . Full column rank in [F1] gives $R_{i} z \neq 0$ , so $z^{T} A_{i} z = (R_{i} z)^{T} A (R_{i} z) > 0.$ Also [F4] gives $A_{i}^{T} = R_{i}^{T} A^{T} R_{i} = A_{i}$ . Thus every such $A_{i}$ is SPD.

1.2F2given

If $0 \in I$ , then $R_{0} = P$ and [F2] gives that $A_{0} = P^{T} A P$ is SPD and the exact coarse solve is defined. If $0 \notin I$ , no separate coarse matrix is part of the one-level definition.

2.1F3step 1.1step 1.2∎

By [F3], every SPD $A_{i}$ in step 1.1 or 1.2 has a unique solution for each local right-hand side. Therefore all exact local solves in Finite-dimensional Schwarz subspace-correction data are well-defined.

Depends on

Used by

Cited to discharge well-definedness by Finite-dimensional Schwarz subspace-correction data.

Dependency tree · two levels

12 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

Jinchao Xu, Iterative Methods by Space Decomposition and Subspace Correction, SIAM Review 34(4) (1992) (standard reference, not scraped)
Peter Bastian, Lecture Notes on Parallel Solution of Large Sparse Linear Systems, University of Heidelberg (2015) (standard reference, not scraped)