High-contrast homogenization: Provider.Recurrence.MeanOrder #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The annealed mean order of the fixed-grid recurrence #
The averaging step of p.fixed.geometry.parent.child.recurrence opens with the aligned
subdivision of the parent cell ⋄_p^q into its 3^{d(p-j)} scale-j children,
and with the three displays
0 ≤ A_p(0) ≤ G, E[A_p(0)] = E_p, E[G] = E_j,
where G is the average of the children's responses. Averaging the pathwise
order under the law and reading the two expectations gives the mean order
E_p ≤ E_j, which is the hypothesis every later step of the recurrence uses.
The pathwise order is the subadditivity among the coarse-block properties taken
from HC over the aligned subdivision, together with the pathwise positivity
clause. The two expectations are the entrywise Bochner integral of the
response, read through the translation clause: each child is an integer
translate of the cell at the origin, so under stationarity of the coefficient
law its annealed block is E_j and the average
of the children's annealed blocks is E_j again. Averaging is monotone for the
Loewner order because the positive semidefinite cone is closed, which is the
content the ambient integral layer supplies.
The consequences the recurrence's later steps read off the mean order are the
determinant identities for the parent-child normalization: the
relative mean P_{j,p} = E_p^{-1/2}E_jE_p^{-1/2} lies above the identity, its
determinant is e^{Δ_{j,p}}, the increment Δ_{j,p} is nonnegative, the
spectral size of P_{j,p} is at most e^{Δ_{j,p}}, and the trace gap
b = tr(P_{j,p} - I) is at most e^{Δ_{j,p}} - 1. All five are the matrix
analysis of a positive definite pair in the Loewner order, applied at the adapted
means.
The pathwise order 0 ≤ A_p^q(0) ≤ G #
The two expectations E[A_p^q(0)] = E_p^q and E[G] = E_j^q #
E[A_r^q(0)] = E_r^q. The adapted mean is the Bochner integral of the
flattened response, which is what makes the mean order an integral inequality.
The flattened response over an aligned child cell is Bochner integrable: the finiteness guard at the origin transports along the integer translation.
E[G] = E_j^q. The expectation of the average of the responses over a
nonempty family of aligned scale-j cells is the adapted mean at scale j: each
cell is an integer translate of the cell at the origin, so stationarity of the
coefficient law gives every summand the same expectation.
The mean order E_p^q ≤ E_j^q #
E_p^q ≤ E_j^q. The annealed mean order of p.fixed.geometry.parent.child.recurrence,
on the flattened carrier: averaging the pathwise subadditivity of the aligned
subdivision under the law and reading both expectations through
stationarity of the coefficient law leaves the adapted mean at the child scale
on the right.
E_p^q ≤ E_j^q, in the structural dialect of the doubled block.
The relative mean and the determinant increment #
The adapted mean of a rounded adapted grid is a positive definite matrix on the flattened carrier as soon as it is finite.
The relative mean P_{j,T}^q is the literal congruence of the child mean by
the inverse square root of the parent mean.
The determinant increment Δ_{j,T}^q is the difference of the logarithms of
the determinants of the two flattened means.
I ≤ P_{j,p}^q. The relative mean of the fixed-grid recurrence lies
above the identity; this is the mean order conjugated by (E_p^q)^{-1/2}.
Δ_{j,p}^q ≥ 0. The determinant increment of the fixed-grid recurrence
is nonnegative: the determinant is monotone for the Loewner order on positive
definite matrices, and the mean order puts the parent mean below the child.
The determinant identities for the parent-child normalization, at the
adapted means: the relative
mean lies above the identity, the determinant increment is nonnegative, the
determinant of the relative mean is e^{Δ_{j,p}}, its spectral size is at most
e^{Δ_{j,p}}, and its trace gap b = tr(P_{j,p} - I) is at most
e^{Δ_{j,p}} - 1.