High-contrast homogenization: Provider.Transport.TransportMeanDomination #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The mean of the filling, and its bridge normalization #
e.two.grid.whitney.average is a pathwise statement: the coarse response of
a target cell of the new grid is below the weighted sum of the responses of the
old grid's cells that fill it, plus the below-start series. Every estimate that
follows it reads the annealed blocks instead, and this file makes that passage
and the normalization that comes with it.
Three steps, in the order the proof uses them.
The grouping map. A cell of the filling at a scale at or below the checkpoint
lies in a unique cell of the checkpoint scale, and that cell meets the target.
This is the index map under which the inherited rows of
e.two.grid.old.history.factor are grouped, and it is nothing but the
iterated parent taken the right number of times.
The mean. Averaging the pathwise domination is monotone for the Loewner order, so the annealed block of the target is below the same weighted sum of the annealed blocks of the filling's cells plus the mean of the below-start term. The only hypotheses are integrability of the three families, which the bounded-window multiplier supplies.
The normalization. Conjugating that mean domination by the inverse square root
of the new terminal block turns it into the printed
K_W = S^tP_bar_WS - (1-m_W)S^tS + R_W of e.two.grid.whitney.mean.bound, because
the bridge congruence S = (E_t^q)^{1/2}(E_n^{q'})^{-1/2} carries the old
relative mean P_{r,t}^q exactly onto the normalization of E_r^q by
E_n^{q'}. That identity is the reason the bridge error enters additively and
never multiplies a cell response.
The total relative volume of any finite set of rows is at most one, because the
cells of the filling are pairwise disjoint and lie in the target; that is the
convexity budget m_W ≤ 1 of the convexity bound for the mean penalty.
The grouping map #
The convexity budget of the filling #
The total relative volume of a finite set of rows of the maximal filling is
at most one. The cells of the filling are pairwise disjoint and contained in
the target, so the sum of their volumes is at most the target's. This is the
budget m_W ≤ 1 that the convexity bound for the mean penalty consumes.
The mean of the exhaustion #
The annealed form of e.two.grid.whitney.average. Averaging the
pathwise domination of the target's coarse response by the weighted responses of
the filling plus the below-start term gives the same domination between the
annealed blocks, with the mean of the below-start term as the remainder.
Averaging is monotone for the Loewner order, and the integral of the finite weighted sum is the weighted sum of the integrals; the only hypotheses are the integrability of the target, of each selected cell, and of the below-start term.
The bridge normalization of the mean domination #
The bridge congruence carries the old relative mean onto the new
normalization. With S = (E_t^q)^{1/2}(E_n^{q'})^{-1/2}, the congruence
S^tP_{r,t}^qS is the normalization of E_r^q by the new terminal block, the
two square roots of the old terminal block cancelling. This is the identity
behind e.two.grid.whitney.mean.bound: the bridge never sees an individual
cell response, only the terminal normalization.
The normalized mean domination, in the exact shape the cell gap of
e.two.grid.whitney.mean.bound and the nonlinear row consume:
P_{j,n}^{q'} ≤ S^tP_bar_WS - (1-m_W)S^tS + R_W.
Conjugating the annealed domination by the inverse square root of the new
terminal block is monotone, the bridge carries each old relative mean onto the
new normalization, and the unfilled mass is exactly the identity part of the
convex combination P_bar_W.