High-contrast homogenization: Entry.Response.Pairing.SubcellDualPairingIntegrability #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Subcell Dual Pairing and Its Integrability #
Applying the direct full-dual Fenchel pairing of AK.HC, (A.4) to the difference between the
terminal optimizer restricted to a scale-s cell and that cell's own optimizer bounds the
pairing of a state Y = (P, Q) against the cell-average difference by the subcell's coarse-block
energy of Y times the square root of twice the subcell's response deficit; transported to an
almost-everywhere elliptic representative, this is the subcell dual pairing consumed by the cell
half of the cutoff-mean row. The pairing needs, coordinatewise in L^1(P), the integrability of
the terminal optimizer's doubled cell mean on an aligned subcell — obtained from the Fenchel
probe at a single basis direction, since it is not a functional of the pathwise coarse block —
together with that of the subcell maximizer's own cell mean. The module serves the cell half of the
cutoff-mean row of p.response.transfer.
The full-dual pairing of the subcell difference field #
The direct full-dual pairing AK.HC, display (A.4), applied to the field that the cutoff-mean rows
produce: the difference between the terminal optimizer restricted to a scale-s cell and that
cell's own optimizer. The state Y = (P, Q) is paired with the two slots of the cell-average
difference. The right-hand side is the cell's own coarse-block energy of Y — read through the
two diagonal blocks from which the source load is built — times the square root of the scalar cell
deficit. Because the difference energy of the two states is twice the response deficit, no
subcell optimizer survives in the statement, so no measurable selection is needed downstream.
The cell average of the difference between the terminal optimizer field restricted to the
cell V and the cell's own optimizer obeys the direct full-dual pairing bound: paired against a
state Y = (P, Q), it is controlled by the total of the square roots of the two diagonal
coarse-block forms of Y times the square root of twice the scalar response deficit of the
restricted terminal field.
The subcell dual pairing at an almost-everywhere elliptic coefficient #
The cell half of the cutoff-mean row of p.response.transfer pairs, on each coarse subcell of the
terminal cell, the dual variable Y against the difference between the cell mean of the terminal
optimizer's doubled state and the cell mean of that subcell's own optimizer. The direct full-dual
pairing of AK.HC (A.4) bounds it by the subcell's two-term head times the square root of twice the
subcell deficit. That estimate is stated for a pointwise elliptic coefficient; the response
coefficients respCoeff∓ F a are elliptic only almost everywhere, so this module runs it at a
pointwise elliptic representative and transports every quantity back across the a.e. replacement,
which leaves each of them invariant.
The cell half of the cutoff-mean row of p.response.transfer. Let f be pointwise elliptic
on the terminal cell HighContrast.adaptedCell q t and let b agree with f almost everywhere
there. For a b-harmonic u on the terminal cell, a subcell maximizer v on the aligned subcell
adaptedCellAtCenter q (t - n) w, and a state Y = (P, Q), the pairing of Y with the
difference of
the two cell averages is bounded by the sum of the square roots of the two diagonal coarse-block
forms of Y, times the square root of twice the subcell response deficit of u. The bound holds
for b itself although only the representative f is pointwise elliptic, because every quantity
in the estimate is invariant under the a.e. replacement.
Sample integrability of the two optimizer cell means on an aligned subcell #
The cell half of the cutoff-mean row of p.response.transfer pairs the deterministic dual variable
with the difference of two doubled optimizer cell means on each aligned subcell of the terminal
cell: the mean of the TERMINAL optimizer and the mean of the SUBCELL maximizer. Both are needed
coordinatewise in L^1 of the law.
The subcell maximizer mean is the block response mean of the pathwise coarse block, hence affine
in the entries of that block and integrable by HasIntegrableCoarseBlock. The terminal optimizer
mean is not a functional of the block, but its difference from the subcell mean is controlled
coordinatewise by the Fenchel probe of AK Lemma A.1 read at a single basis direction: the probe
Y = (0, δ_i) bounds the i-th gradient coordinate by the square root of the i-th diagonal
entry of the lower-right sub-block times the square root of twice the subcell deficit, and the
probe Y = (δ_i, 0) bounds the i-th flux coordinate by the upper-left analogue. Both envelopes
are integrable, and the terminal mean is measurable in the sample through the canonical selection,
so the terminal mean is integrable as well.
The subcell maximizer cell mean is integrable, minus sign. On an aligned adapted cell the
doubled optimizer cell mean of a response maximizer for the recentred coefficient a_- = a - g is
the block response mean of the pathwise coarse block at the load (-p, r), hence affine in the
entries of that block; HasIntegrableCoarseBlock on the cell makes every coordinate
P-integrable.
The subcell maximizer cell mean is integrable, plus sign. The adjoint twin of
integrable_cellAverage_subcellOptimizer_respCoeffMinus, for a_+ = aᵀ + g.
Sample integrability of the terminal optimizer cell mean on an aligned subcell #
The doubled optimizer cell mean of the TERMINAL maximizer, read on an aligned subcell of the
terminal cell, is not a functional of the pathwise coarse block, so its P-integrability cannot be
read off HasIntegrableCoarseBlock. It follows instead from the Fenchel probe of AK Lemma A.1,
read at a single basis direction: the probe Y = (0, δ_i) bounds the i-th gradient coordinate of
the difference between the terminal mean and the subcell maximizer mean by
√(σ_{*,V}^{-1})_{ii} √(2 D_V), and the probe Y = (δ_i, 0) bounds the i-th flux coordinate by
√(𝐛_{V,ii}) √(2 D_V). Both envelopes are integrable — the first factor by the entrywise
integrability of the block, the second by the integrability of the subcell deficit — and the
terminal mean is measurable in the sample through the canonical selection, so the difference, and
with the subcell mean also the terminal mean, is integrable.
The terminal optimizer cell mean is integrable on every aligned subcell, minus sign. With
the subcell maximizer family v and the integrability of the subcell deficits, every coordinate of
the cell mean of the doubled optimizer field of the terminal maximizer family uM for
a_- = a - g is P-integrable on every depth-H aligned subcell of the terminal cell.
The terminal optimizer cell mean is integrable on every aligned subcell, plus sign. The
adjoint twin of integrable_cellAverage_terminalOptimizer_respCoeffMinus, for the adjoint
recentred coefficient a_+ = aᵀ + g, the dual load q^+ and the terminal maximizer family
uP.