High-contrast homogenization: Geometry.ReferenceRadius #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The canonical metric of a reference block #
The reference-radius estimate in e.initial.geometry.bounds reads the canonical
balance chain on the lower-right principal blocks. If the Schur
data of a self-dominating block E are (s, s_*, k), the balance chain
E^♯ ≤ M(E) ≤ E becomes s^{-1} ≤ m(E)^{-1} ≤ s_*^{-1}, that is
s_* ≤ m(E) ≤ s; the aspect bounds λ_0 I ≤ s_* and s ≤ Λ_0 I of
e.reference.aspect.ratio then place m(E) between λ_0 I and Λ_0 I and bound
its projective distance to the identity by (1/2) log Π.
Both statements here are about the matrix core of the corollary and carry no
probabilistic hypothesis: the reference block enters through its Schur data and
through the hypothesis E^♯ ≤ E, which for the coefficient law of
e.coarse.ellipticity is the primal-adjoint order for the reference block.
The canonical metric of a self-dominating block sits between its Schur blocks, the principal blocks of the reference metric.
The reference radius. The canonical
metric of a self-dominating block lies between the two scalar aspect bounds of
its Schur data, and its projective distance to the identity is at most half the
logarithm of their quotient — the reference aspect ratio Π of
e.reference.aspect.ratio.