High-contrast homogenization: Entry.Response.Direct.DescendantHeadBound #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The annealed head of one descendant generation, and its integrands #
Cutting the terminal cell into the 3^{(H+n)d} aligned cells of generation s - n, the pathwise
full-dual pairing on each of them produces a two-term head
√(Y₁ · A(V;a_-)_{11} Y₁) + √(Y₂ · A(V;b)_{22} Y₂), and its annealed flat average over all these
cells is bounded, in nine analytic steps, by the corresponding entry of the annealed coarse block;
the same bound is then read off on the concrete data the response problem actually supplies. The
descendant sum of p.response.transfer uses this bound to read the generation-n layer of the
source load 𝓛_s against the expectation of the flat average of the squared head, for both signs.
The pathwise expansion of the cutoff pairing that this rests on needs six scalar integrands to be
integrable on the terminal cell, each a product of the bounded cutoff with either a coordinate of
the doubled optimizer state, a coordinate pairing of that state with a fixed vector, or a constant,
and this file records that integrability.
The annealed head of the descendant layer of the source load #
Step ζ of the cutoff-mean row of p.response.transfer at descendant depth n. The terminal
cell is cut into the 3^{(H+n)d} aligned cells of generation s - n; on each of them the
pathwise full-dual pairing produces the head √(Y₁ · 𝐀(V; a_-).upperLeft Y₁) + √(Y₂ · 𝐀(V; b).lowerRight Y₂), and its annealed flat average over all 3^{(H+n)d} cells is at
most the depth-n layer of respSourceLoad, i.e. the flat average over the 3^{nd} descendants
of ONE scale-s cell of the same head read on the ANNEALED block.
The estimate is the two-term expectation bound integral_sq_sqrt_add_sqrt_le applied cell by
cell, the identification of each pathwise quadratic form with the annealed one
(integral_vecDot_coarseBlock_upperLeft/…_lowerRight), the grouped-digit decomposition of the
index box sum_triadicIndexBox_add, and the stationarity identity
annealedBlockOf_descendant_respCoeffMinus_eq/…_Plus_eq, which makes the summand independent
of the outer index and collapses the outer sum.
The annealed flat average of the squared pairing head at descendant depth n. Step ζ of
the cutoff-mean row of p.response.transfer at depth n, minus sign: the annealed flat average
over the generation-(s-n) cells of the terminal cell of the squared pathwise head is at most the
depth-n layer of the source load.
The annealed flat average of the squared pairing head at descendant depth n, plus
sign. The adjoint twin of integral_avsum_sq_head_descendant_le_respCoeffMinus, for the
recentred family a_+ = a^t + g.
The annealed head of one descendant generation at the carriers #
The descendant sum of p.response.transfer reads the generation-n layer of the source load
𝓛_s against the expectation of the flat average, over the depth-(H+n) aligned cells of the
terminal cell, of the squared pathwise two-term head
(⟨Y₁, 𝐛_V Y₁⟩^{1/2} + ⟨Y₂, σ_{*,V}^{-1} Y₂⟩^{1/2})². DescendantHeadBound proves that bound
from nine analytic side conditions on the pathwise coarse blocks. This module discharges all nine
at the carriers, from the coarse-block integrability of the sample on every aligned cell of the
response grid and the nonnegativity of the two diagonal forms, so that the head bound carries no
analytic side condition.
The annealed head of a descendant generation is at most the corresponding layer of the
source load, minus sign, at the carriers. The expectation of the flat average over the
depth-(H+n) aligned cells of the squared pathwise two-term head of a_- = a - g is at most the
flat average over the generation-n cells of the squared annealed two-term head, i.e. the
generation-n layer of 𝓛_s^- of p.response.transfer.
The annealed head of a descendant generation is at most the corresponding layer of the
source load, plus sign, at the carriers. The expectation of the flat average over the
depth-(H+n) aligned cells of the squared pathwise two-term head of a_- = a - g is at most the
flat average over the generation-n cells of the squared annealed two-term head, i.e. the
generation-n layer of 𝓛_s^- of p.response.transfer.
Integrability of the cutoff integrands of the pathwise expansion of p.response.transfer #
The pathwise expansion of the cutoff pairing in the cutoff estimate of p.response.transfer
needs six scalar integrands to be integrable on the terminal cell. All six are products of the
bounded cutoff φ with either a coordinate of the doubled optimizer state X = (∇v, b ∇v), a
coordinate pairing of X with a constant vector, or the pairing X₁·X₂. On a set of finite
volume, with both slots of X square integrable and 0 ≤ φ ≤ 2, the bounded-measurable-factor
lemma gives the first three, and the L² × L² → L¹ Hölder pairing gives the last.
A measurable function bounded by Mφ in absolute value, multiplied by an integrable
function, is integrable on any set. The pointwise bound |φ x| ≤ Mφ supplies the
almost-everywhere norm bound required by Integrable.bdd_mul.