High-contrast homogenization: Geometry.Canonical #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The canonical metric of a positive block #
For a positive doubled block E the reference text sets, in
e.scale.selection.canonical.metric, M(E) := E # E^♯ and
𝔡(E) := |M(E)^{-1/2} E M(E)^{-1/2}|^2. The facts about the canonical metric
used in p.global.selection are that
M(E) is self-dual, that it has a unique factorization
M(E) = G_{-g}^t diag(m, m^{-1}) G_{-g} with m positive and g skew, that
𝔡(E) = |(E^♯)^{-1/2} E (E^♯)^{-1/2}|, and — when E^♯ ≤ E — the balance chain
E^♯ ≤ M(E) ≤ E ≤ 𝔡(E)^{1/2} M(E) together with det M(E) = 1.
Self-duality is the inverse and congruence clauses of the geometric-mean
toolkit applied to the reflection. The factorization is self-duality read
through the Schur data: the sharp map sends (s, s_*, k) to (s_*, s, -k^t),
so a self-dual block has s = s_* and k skew. The imbalance identity is the
observation that the generalized eigenvalues of E relative to E^♯ are those
of C := M(E)^{-1/2} E M(E)^{-1/2} relative to C^{-1}, that is the
eigenvalues of C^2.
The pair (m, g) is introduced by the well-definedness-first route: the
uniqueness theorem existsUnique_canonFactor comes first, the definition
canonFactor is made by choice, and canonFactor_spec and canonFactor_eq
then characterize it, so that nothing downstream depends on the choice.
The canonical metric block M(E) = E # E^♯ of
e.scale.selection.canonical.metric.
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The imbalance 𝔡(E) = |M(E)^{-1/2} E M(E)^{-1/2}|^2 of
e.scale.selection.canonical.metric.
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Self-duality #
The canonical metric is self-dual.
The canonical factorization #
The canonical factorization (e.scale.selection.canonical.metric).
There is a unique pair consisting of a positive m and a skew g with
M(E) = G_{-g}^t diag(m, m^{-1}) G_{-g}.
The hedged existential behind the canonical pair: on positive data it is
existsUnique_canonFactor, elsewhere it is a junk witness.
The canonical pair (m(E), g(E)), by choice from
existsUnique_canonFactor; the junk value on non-positive data is never
used.
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The canonical metric m(E) of e.scale.selection.canonical.metric.
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The canonical shear g(E) of the canonical factorization.
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Specification of the canonical pair. m(E) is positive, g(E) is
skew, and they factor M(E) as in e.scale.selection.canonical.metric.
The canonical pair is characterized, not merely chosen: any positive
m and skew g factoring M(E) are m(E) and g(E).
The lower-right block of the canonical metric is m(E)^{-1}. This is
what makes the reference text's step "taking the lower-right principal blocks"
in e.global.selection.metric.loss and e.global.selection.metric.comparison read
as a statement about m.
The canonical metric is positive definite, hence so is its inverse.
The imbalance identity #
The imbalance identity e.response.canonical.imbalance:
𝔡(E) = |(E^♯)^{-1/2} E (E^♯)^{-1/2}|.
The balance chain and the determinant #
The scalar in the balance chain is the relative size of E against M(E).
The canonical balance chain: the canonical metric lies between a block and its dual.
The canonical metric has unit determinant.