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The additive Schwarz inverse minimizes summed local energy

Statement

For the data of Finite-dimensional Schwarz subspace-correction data, assume $\sum_{i \in I} V_{i} = V$ . Let $R = [R_{i}]_{i \in I}$ and $D = diag (A_{i})_{i \in I}$ , so $B = R D^{- 1} R^{T}$ . Then $B$ is SPD and, for every $x \in V$ , $x^{T} B^{- 1} x = \min_{\begin{array}{c} z = (z_{i})_{i \in I} \\ R z = x \end{array}} \sum_{i \in I} z_{i}^{T} A_{i} z_{i} .$

Facts & Assumptions

Given: The local SPD matrices $A_{i}$ , the prolongations $R_{i}$ , and the spanning condition $\sum_{i \in I} Ran (R_{i}) = V$ .

[F1]

The Schwarz definition gives $A_{i} = R_{i}^{T} A R_{i}$ and $B = \sum_{i \in I} R_{i} A_{i}^{- 1} R_{i}^{T}$ (Finite-dimensional Schwarz subspace-correction data).

[F2]

Every local matrix $A_{i}$ is SPD and invertible (Schwarz local matrices and exact solves are well defined).

Proof

1.1F1F2givenalgebra

The block matrix $D$ is SPD by [F2]. The spanning assumption makes the concatenated map $R$ surjective. If $x^{T} B x = 0$ , then $0 = (R^{T} x)^{T} D^{- 1} (R^{T} x),$ so $R^{T} x = 0$ because $D^{- 1}$ is SPD. Surjectivity of $R$ then gives $x = 0$ . Thus $B$ is SPD and has an inverse.

2.1F1F2step 1.1algebra∎

For fixed $x$ , define $z_{*} = D^{- 1} R^{T} B^{- 1} x$ . Then $R z_{*} = B B^{- 1} x = x$ , so it is feasible. For any other feasible $z$ , write $w = z - z_{*}$ . Then $R w = 0$ and $z_{*}^{T} D w = x^{T} B^{- 1} R w = 0$ . Therefore $z^{T} D z = z_{*}^{T} D z_{*} + w^{T} D w \geq z_{*}^{T} D z_{*}$ . Direct multiplication gives $z_{*}^{T} D z_{*} = x^{T} B^{- 1} x$ . This proves the minimum identity.

Depends on

Used by

Additive Schwarz condition bound from stable decomposition and synthesis Theorem

Dependency tree · two levels

8 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)