High-contrast homogenization: Entry.Response.Cutoff.CutoffQuadraticReadout #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The cutoff-weighted quadratic readout #
On a response cell U with cutoff weight φ, the cutoff-weighted quadratic readout sends a
doubled Hilbert block field z to the volume average
(|U|)^{-1} ∫_U φ(x) ⟪z_1(x), z_2(x)⟫ dx of its potential and flux slots; this file constructs
that readout as a continuous functional on the Hilbert space of block fields, evaluates it on
the diagonal of the cutoff-weighted energy bilinear form of an elliptic coefficient, and proves
the almost-sure strong measurability of the cutoff pairing for an arbitrary response maximizer
by reducing it, almost everywhere on the response cell, to the readout applied to the Chapter-2
canonical optimizer state. The labelled result this file serves is the cutoff estimate
e.response.cutoff.estimate.
Measurability of the cutoff pairing in the sample #
The cutoff pairing of the response estimate e.response.cutoff.estimate is the volume average of
the cutoff-weighted product of the gradient defect and the flux defect of the doubled optimizer
state of a response maximizer. The doubled optimizer state of an arbitrary response maximizer is
determined almost everywhere on the response cell by the Chapter-2 canonical maximizer, by
a.e. gradient uniqueness for response maximizers (AK.HC (2.9)); the canonical selection is the one
whose cell averages depend measurably on the coefficient sample.
The reduction below is the pairing analogue of the canonical identification carried out for the
scale-average seminorm of e.response.weak.estimate: for each coefficient sample the two doubled
optimizer fields agree almost everywhere on the cell, so the two cutoff pairings agree, and the
measurability of the pairing transfers from the canonical coefficient family to the arbitrary family
of maximizers. The measurability of the canonical quadratic readout itself, whose input is the
canonical coefficient object rather than the arbitrary family, is exactly the missing input recorded
as the two hypotheses hcanonMinus and hcanonPlus.
The cutoff pairing of e.response.cutoff.estimate is almost-everywhere strongly measurable in
the coefficient sample for an arbitrary family of response maximizers, provided the corresponding
quadratic readout of the canonical Chapter-2 optimizer state of the two recentred coefficient
families is measurable. The arbitrary maximizer is identified with the canonical one almost
everywhere on the cell by a.e. gradient uniqueness (AK.HC (2.9)), so the pairing depends on the
sample only through the canonical selection.
Measurability of the cutoff pairing for an arbitrary maximizer family #
The cutoff pairing of the response estimate e.response.cutoff.estimate is the volume average,
over the response cell U_t, of the cutoff-weighted Euclidean pairing of the potential defect and
the flux defect of the doubled optimizer state of a response maximizer. This file records that the
absolute value of that pairing is almost-everywhere strongly measurable in the coefficient
sample for an arbitrary family of response maximizers.
The doubled optimizer state of an arbitrary response maximizer agrees almost everywhere on the
response cell with the Chapter-2 canonical maximizer, by a.e. gradient uniqueness for response
maximizers (AK.HC (2.9)); the pairing therefore agrees almost everywhere with the canonical pairing.
The canonical pairing is then reduced to its genuinely quadratic term by the pointwise expansion of
the integrand, whose three remaining linear readouts of the canonical optimizer state are
measurable. Only that quadratic canonical readout is carried as an explicit hypothesis, in the two
recentrings of e.response.cutoff.estimate (AK.HC Lemma A.1, (A.4)).
A volume average over a set V is unchanged by an almost-everywhere equality of the integrand
on a larger set U ⊇ V. This is the scalar analogue of cellAverage_congr_ae.
The absolute value of the cutoff pairing of e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)) is almost-everywhere strongly measurable in the coefficient sample for an
arbitrary family of response maximizers of the two recentred response coefficient families, provided
the cutoff-weighted volume average of the quadratic self-pairing of the canonical Chapter-2
optimizer
state is measurable in each case.
The arbitrary maximizer is identified with the canonical one almost everywhere on the response cell
by a.e. gradient uniqueness for response maximizers (AK.HC (2.9)), so the pairing depends on the
sample only through the canonically selected optimizer state. The canonical pairing is expanded
into its quadratic term and the three linear readouts whose measurability is recorded in
aestronglyMeasurable_abs_canonical_cutoff_pairing_minus_of_quadratic and its plus twin.
The cutoff-weighted quadratic readout as a continuous Hilbert functional #
On a positive-volume response cell U with a cutoff weight φ, the cutoff-weighted quadratic
readout sends a doubled Hilbert block field z to the volume average
(|U|)⁻¹ ∫_U φ(x) ⟪z₁(x), z₂(x)⟫ dx,
where z₁ and z₂ are the potential and flux components of z. Because φ is bounded, this is
the diagonal of a bounded bilinear form on HilbertBlockL2 U, hence continuous; this is the
analytic ingredient behind e.response.cutoff.estimate.
The pointwise linear map (p, q) ↦ (q, 0) on doubled Hilbert vectors, transported from the
algebraic block carrier.
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The pointwise action of blockFluxProjL: the potential slot is discarded and the flux slot is
paired with zero.
The pointwise readout of blockFluxProjL against X is the pairing of the potential and flux
components.
The L² realization of blockFluxProjL.
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The L² realization of blockFluxProjL acts pointwise.
Pointwise multiplication by an a.e. bounded scalar field preserves L².
Pointwise multiplication by an a.e. bounded scalar field, as a linear map on the doubled
L² space.
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The weighted multiplication map acts pointwise.
Pointwise multiplication by an a.e. bounded scalar field, as a continuous linear map on the
doubled L² space.
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The continuous weighted multiplication map acts pointwise.
The cutoff-weighted quadratic readout: the volume average of φ against the pointwise
potential-flux pairing of a doubled L² block field. This is the quadratic readout of
e.response.cutoff.estimate.
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Under an a.e. bound on φ, the readout is the diagonal of the bounded bilinear form induced by
weighted multiplication.
The cutoff-weighted quadratic readout is continuous in the doubled L² field, for any cutoff
weight bounded a.e. by 2; this is the analytic input to e.response.cutoff.estimate.
On a block field promoted to the doubled L² space, the readout is the volume average of the
pointwise potential-flux pairing.
The cutoff-weighted energy functional of the doubled minimizer #
For a response cell U, a coefficient a lying on a quantitative ellipticity slice, and a cutoff
φ, the Hilbert Mu realization of a carries a continuous bilinear energy form. Multiplying the
first argument pointwise by the cutoff and evaluating the energy on the diagonal gives the
functional z ↦ ⨍_U φ ⟨z, B_a z⟩, the cutoff-weighted energy of a doubled field.
The pointwise self-pairing of the canonical optimizer state is the doubled block energy of the
Mu minimizer plus twice the potential-flux pairing of the minimizer, so the quadratic term of the
expansion of the cutoff pairing e.response.cutoff.estimate (AK.HC Lemma A.1, (A.4)) is the value
at the minimizer of the sum of the cutoff-weighted energy and twice the cutoff-weighted
potential-flux readout. This file records the continuity of that sum in the doubled field, the
analytic half of the measurability of the quadratic term.
The cutoff-weighted energy of a doubled field is continuous in the field. The energy bilinear
form of the Mu realization is continuous, pointwise multiplication by the a.e. bounded cutoff is
continuous linear, and evaluation of a continuous linear functional is continuous. This is the
analytic input to the cutoff-weighted quadratic term of e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)).
The sum of the cutoff-weighted energy of a doubled field and twice its cutoff-weighted
potential-flux readout is continuous in the field. Its value at the Mu minimizer is the
quadratic term of the expansion of the cutoff pairing e.response.cutoff.estimate
(AK.HC Lemma A.1, (A.4)).
The cutoff-weighted quadratic readout of the canonical optimizer state #
The cutoff pairing of the response estimate e.response.cutoff.estimate expands, in the notation of
AK.HC Lemma A.1, (A.4), into one quadratic term and three linear terms. The quadratic term is the
cutoff-weighted volume average of the Euclidean pairing of the two slots of the canonical optimizer
state,
a ↦ ⨍_{adaptedCell} φ · ⟨Z(a).1, Z(a).2⟩,
where Z(a) is the canonical optimizer state of the recentred coefficient. This is exactly the
hquadMinus/hquadPlus input of the family reduction of the cutoff pairing.
The second slot of the canonical optimizer state is the coefficient applied to the first, so this
readout is one half of the cutoff-weighted block self-pairing of Z(a). The block self-pairing of
Z(a) is not the block self-pairing of the Chapter-2 doubled minimizer X(a): the a.e.
extraction
identity ae_toFullBlockVec_canonicalOptimizerBlockState reads
Z(a).α = X(a).α + (B_a X(a)).α.swap,
so the two slots of Z(a) are the two slots of X(a) shifted by the swapped block image of X(a).
This file records the exact algebraic link between the two self-pairings: pointwise,
⟨Z(a).1, Z(a).2⟩ = ⟨X(a), B_a X(a)⟩ + 2 ⟨X(a).1, X(a).2⟩.
Consequently the cutoff-weighted quadratic readout is not the (manifestly continuous) functional
z ↦ ⨍ φ ⟨z.1, z.2⟩ of the Hilbert minimizer; the correct functional of the minimizer additionally
carries the weighted energy ⨍ φ ⟨z, B z⟩, whose coefficients depend on the sample. The
measurability of the cutoff-weighted quadratic readout therefore does not follow from the
Hilbert-space continuity argument applied to ⨍ φ ⟨z.1, z.2⟩, and the identity below is the missing
algebraic link in that route.
For an elliptic matrix A, the two slots of the block image B A X satisfy
⟨(B X).2, (B X).1⟩ = ⟨X.1, X.2⟩. Writing s, k for the symmetric and skew parts of A and
(B X).1 = s p + k s⁻¹ r, (B X).2 = s⁻¹ r with r = q - k p, this is the cancellation of the
skew form on s⁻¹ r together with the symmetry of s. It is the exact reason the block
self-pairing of the canonical optimizer state differs from that of the doubled minimizer by a cross
pairing.
Pointwise link between the self-pairing of the canonical optimizer state and the doubled
minimizer energy. If Z = X + swap (B X) is the state whose slots are the shifted slots of the
doubled minimizer X, then
⟨Z.1, Z.2⟩ = ⟨X, B X⟩ + 2 ⟨X.1, X.2⟩.
The cross term 2 ⟨X.1, X.2⟩ and the block energy ⟨X, B X⟩ are exactly what is lost by reading
the canonical state pairing as the pairing of the minimizer itself.
The same pointwise link, phrased for the coordinatewise extraction of the canonical optimizer
state of e.response.cutoff.estimate: almost everywhere on the domain, the pairing of the two slots
of the canonical optimizer state equals the doubled block energy of the minimizer plus twice the
cross pairing of the minimizer slots. This is the algebraic content of the gap between the
cutoff-weighted quadratic readout and the pairing of the Hilbert minimizer (AK.HC Lemma A.1,
(A.4)).