High-contrast homogenization: Provider.Quenched.SmallContrastCanonRecenter #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The one-time recentering of the reference: canonical data under a shear #
Section 2.5 of HC records that
subtracting a constant skew matrix h from the coefficient field leaves the
solution space unchanged and conjugates every coarse block by the shear
G_h, leaving the two diagonal Schur coordinates untouched and subtracting
h from the Schur skew coordinate (HC (2.65)). The paper uses this
to assume, without loss of generality, that the relevant homogenized skew
vanishes.
This file records what the same recentering does to the canonical data of
the reference block: the canonical metric m(E) is invariant and the
canonical shear g(E) drops by the recentering skew,
m(G_h^t E G_h) = m(E), g(G_h^t E G_h) = g(E) - h.
Consequently the one-time recentering by h := g(E) produces a reference
whose canonical shear vanishes, which is the normalization under which the
weak-norm metric factor of the small-contrast estimate is controlled by the
reference's imbalance alone.
The proof is the congruence equivariance of the geometric mean together with
the shear covariance of the sharp involution (fullBlockSharp_shear_conj),
which is available exactly because the recentering matrix is skew.
The canonical metric block under a skew shear congruence #
The canonical metric block is shear-covariant. For a skew h,
M(G_h^t E G_h) = G_h^t M(E) G_h. This is matGeomMean_conj composed with
the recentering identity fullBlockSharp_shear_conj, the covariance of the
primal-adjoint involution under a constant skew shear.
The canonical pair under a skew shear congruence: the canonical metric is unchanged and the canonical shear drops by the recentering skew.
The block-level statements #
The canonical metric of the recentered reference is the canonical metric of the reference.
The canonical shear of the recentered reference drops by the recentering skew — the canonical-data form of HC (2.65).
The canonical shear of a positive block is skew.
The one-time global recentering. Recentering the reference by its own canonical shear produces a reference whose canonical shear vanishes, while its canonical metric — the matrix defining the adapted cubes — is unchanged. This is the normalization the printed proof installs once and for all.
The canonical block as a sheared diagonal #
The canonical metric block is the shear congruence of the split metric
diag(m(E), m(E)^{-1}) by the canonical shear. This is the factorization
e.scale.selection.canonical.metric written structurally.