High-contrast homogenization: Entry.Response.Rows.OptimizerMeanRowCarriers #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The cutoff-mean row on the response's own data, and the optimizer-mean identity #
The cutoff-mean row of the cutoff estimate is the sum of a cell part, bounded by
(L_s^∓ 2τ^∓)^{1/2}, and an oscillation part, bounded by c₂ 3^{-H}(E[J_t^∓] L_s^∓)^{1/2}; this
file instantiates that two-piece bound on the response problem's own cell and oscillation data. It
also identifies the annealed mean Y^∓ of AK.HC (2.32) — defined through the block formula
Y = (I + R Ehat_t^∓) x^∓ — with the expectation of the cell average of the doubled optimizer
state X = (∇v, a_∓ ∇v) on the terminal cell, E[(X_{u_t})_{U_t}] = Y^∓, by an affine argument. A
further lemma records that every aligned cell of the response grid is an adapted translate at its
own centre. Separately, the source-load series L_s^∓, whose summability at the two annealed
means Y^± already follows from the response load bound, is shown summable at an arbitrary dual
vector, needed once the oscillation half of the cutoff-mean row reads the series once per
coordinate direction. The module serves e.response.cutoff.estimate and
p.response.transfer.
The cutoff-mean row on the carriers from its cell and oscillation halves #
The cutoff-mean row of e.response.cutoff.estimate is the sum of a cell part and an oscillation
part. CutoffPairingRowSplit exhibits those two parts explicitly, and CutoffEstimateRowClosure
assembles
the row from a cell-half bound (L_s^∓ 2 tau^∓)^{1/2} and an oscillation-half bound
c₂ 3^{-H} (E[J_t^∓] L_s^∓)^{1/2}. This file instantiates the assembly at the two functions that
CutoffPairingRowSplit produces, so that a caller holding the standing data of the cutoff-mean
defect,
the cell-half bound and the oscillation-half bound obtains the named obligation
CutoffMeanRow∓ with constant max 1 c₂.
The cutoff-mean row for the negative sign on the carriers. Assume the standing data of
the cutoff-mean defect: Y^- is the annealed cell average of the terminal optimizer state for
respCoeffMinus F, the grid is invertible, the sample functions are integrable on the terminal cell
and on each depth-H subcell, and the two parts of the split are P-integrable. Assume the
cell-half bound for the explicit cell part, the oscillation-half bound c₂ 3^{-H} (E[J^-] L^-)^{1/2}
for the explicit oscillation part, and the nonnegativity of tau^- and L^-. Then the named
obligation CutoffMeanRowMinus holds with constant max 1 c₂. The cell-half and
oscillation-half bounds remain undischarged inputs.
The cutoff-mean row for the positive sign on the carriers. The adjoint twin of
cutoffMeanRowMinus_of_cellSplit: with Y^+ the annealed cell average of the terminal optimizer
state for respCoeffPlus F, the same standing data and the cell-half and oscillation-half bounds
for the explicit parts of CutoffPairingRowSplit give CutoffMeanRowPlus with constant max 1 c₂.
The optimizer-mean identity E[(X_{u_t})_{U_t}] = Y^∓ #
In p.response.transfer the annealed mean Y^∓ of AK.HC (2.32) is defined through the block
formula Y = (I + R Ehat_t^∓) x^∓. This module shows that the block formula really is the
expectation of the cell average of the doubled optimizer state X = (∇v, a_∓ ∇v) of the
terminal cell: E[(X_{u_t})_{U_t}] = Y^∓.
The argument is affine. Pathwise the cell average of the optimizer state is
x + R 𝐀(U_t; a_∓) x, with 𝐀 the set-level coarse block matrix; blockResponseMean is
affine in the entries of the block, so integrating the entrywise affine expression reproduces the
same formula with the annealed block Ehat_t^∓ in place of the pathwise one. The sample
coefficient a_∓ is only almost everywhere elliptic, so the pathwise identity is applied to an
elliptic representative and transported back along the almost everywhere agreement of the
optimizer fields and of the coarse block matrices.
The annealed mean Y^- of AK.HC (2.32) is the expectation of the cell average of the
doubled optimizer state of the terminal cell: E[(X_{u_t})_{U_t}] = Y^-, for any measurable
selection a ↦ u_t(a) of response maximizers. The pathwise identity is applied to an
almost everywhere elliptic representative of a_- = a - g, and the resulting statement is
transported back along the almost everywhere agreement of the two optimizer fields and of the
two coarse block matrices.
The adjoint twin of respYMinus_eq_integral_cellAverage_optimizerField: the annealed mean
Y^+ of AK.HC (2.32) is the expectation of the cell average of the doubled optimizer state of
the terminal cell for the adjoint coefficient a_+ = a^t + g. The pathwise identity is
applied to an almost everywhere elliptic representative of a_+, and the resulting statement is
transported back along the almost everywhere agreement of the optimizer fields and of the coarse
block matrices.
Coarse-block integrability on every aligned cell #
The terminal-optimizer replacement row averages the subcell responses over the aligned cells of a fixed generation. This module records that an aligned cell of the response grid is the adapted translate at its own centre, and transports the standing stationary-law integrability of the adapted coarse block to every such aligned cell.
Coarse-block integrability on every aligned cell of the response grid.
The source-load series converges at every dual vector #
RespLoadBound carries the summability of the generation summands of L_s^∓ only at the two
annealed means Y^∓. The oscillation half of the cutoff-mean row of p.response.transfer reads
the same series once per coordinate direction, at the dual vectors (0, δ_i) and (δ_i, 0), so it
needs the summability at an arbitrary Y.
Nothing in the printed proof of the source-load bound restricts the dual vector: the Y-dependence
enters only through the quadratic form ⟨Y, M_0 Y⟩, which is a finite number for every Y. What
carries the convergence is the scale-wise Loewner envelope
E[A(V_{s-n,z}; a)] ≤ C Π e(m)^2 3^{γ(j_*-(s-n))_+} E_s of annealedBlock_le_adaptedMean
together with the calibration Ehat_s^∓ ≤ c M_0; against the weight 3^{-3n/2} the resulting
series is geometric of ratio 3^{γ-3/2} < 1. This module records that, for both signs.
The source-load series converges at every dual vector, for both signs. Under the standing
data of the response window -- stationarity, coarse ellipticity, the source threshold on j_*, the
invertible rounded grid, the window inclusion and the calibration of Ehat_s^∓ against M_0 -- the
generation summands of the source load are summable at every Y, not only at the annealed means
Y^∓ of RespLoadBound.