High-contrast homogenization: Entry.Response.Direct.DescendantFenchelProbe #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Fenchel Probe on a Descendant Cell and State Measurability #
Each generation increment of the descendant sum pairs the dual variable Y with the cell
average, over a descendant cell V of the terminal cell U, of the doubled state of the
terminal optimizer u; restricting u to V changes neither its gradient nor its flux, so the
Fenchel probe of AK.HC (A.4) applies on V with the pathwise coarse block there, bounding the
crossed pairing — and, at the carriers, each of its two crossed slots separately — by the
cell's own two-term head times the square root of twice the cell energy. Because a response
maximizer's doubled optimizer field is determined, up to a null set, by the coefficient sample
alone, every cutoff-weighted average of that field over a measurable subcell is itself measurable
in the sample, for a_- and a_+ alike. These pathwise bounds on the two crossed slots are
the input of the descendant sum of p.response.transfer, and the sample-measurability of the
weighted averages is the companion entering the weak quantity of e.response.weak.estimate.
The crossed pairing of the terminal optimizer on a descendant cell #
Each generation increment of the descendant sum of p.response.transfer pairs the annealed mean
Y with the cell average, over a descendant cell V of the terminal cell U, of the state of
the terminal optimizer u. Restricting u to V changes neither its gradient nor its flux, so
the Fenchel probe applies on V with the pathwise coarse block of V, and both crossed slots
are controlled by the source-load head of V times the square root of twice the half cell energy
of u on V.
The state of the restricted optimizer is the state of the optimizer. Restricting an
a-harmonic function from U to an open subset V on which the coefficient is elliptic does not
change the doubled optimizer state.
The crossed pairing of the terminal optimizer on a descendant cell. Both crossed slots
of the pairing of the annealed mean Y with the cell average of the terminal optimizer state on
a descendant cell are at most the source-load head of that cell times the square root of twice
the half cell energy. This is the pathwise input of the descendant sum of
p.response.transfer.
The crossed pairing of the terminal optimizer on a descendant cell, at the carriers #
Each generation increment of the descendant sum of p.response.transfer pairs the dual variable
Y with the cell average, over a descendant cell of the terminal cell, of the doubled state
(∇u, a∇u) of the terminal optimizer u. The Fenchel probe bounds each of the two crossed slots
by the two-term head of that cell times the square root of twice the half cell energy, so their sum
is bounded by the head times the square root of twice the doubled cell energy. The probe is stated
for a pointwise elliptic coefficient, while the recentred coefficients a_∓ are elliptic only
almost everywhere; it is therefore run at a pointwise elliptic representative on the terminal cell,
which serves every descendant cell at once, and every quantity is transported back across the
almost-everywhere replacement.
The crossed pairing of the terminal optimizer on a descendant cell, minus sign. On each
depth-m aligned descendant cell of the terminal cell, the cell average of the scalar crossed
pairing ⟨Y₂, ∇u⟩ + ⟨Y₁, a_-∇u⟩ of the terminal optimizer is at most the pathwise two-term head of
that cell times the square root of twice the doubled cell energy. This is the pathwise input of
the descendant sum of p.response.transfer.
The crossed pairing of the terminal optimizer on a descendant cell, plus sign. The adjoint
twin of abs_volumeAverage_cross_optimizerField_respCoeffMinus_adaptedCellAtCenter_le, for the
recentred
coefficient a_+ = aᵀ + g and the dual variable Y^+.
Weighted averages of the doubled optimizer field are measurable in the sample #
The doubled optimizer field of a response maximizer for the recentred coefficients a_- and a_+
is determined, up to a null set of the response cell, by the coefficient sample alone: Chapter-2
almost-everywhere gradient uniqueness identifies every response maximizer with the canonical
selection. Consequently every cutoff-weighted average of a coordinate of the doubled optimizer
field over a measurable subcell is a measurable function of the sample, for an arbitrary family of
maximizers. This is the linear companion of the weighted energy readout: it gives the
sample-measurability of the cutoff-weighted state mean entering the weak quantity of
e.response.weak.estimate.
An arbitrary response maximizer for the recentred coefficient a_- on an invertible adapted
cell has the same doubled optimizer field almost everywhere as the canonical Chapter-2 selection.
An arbitrary response maximizer for the recentred coefficient a_+ on an invertible adapted
cell has the same doubled optimizer field almost everywhere as the canonical Chapter-2 selection.
Every cutoff-weighted average of a coordinate of the doubled optimizer field of an arbitrary
family of response maximizers for the recentred coefficient a_- is measurable in the coefficient
sample. The field need not be measurable in the sample, but its weighted average is determined by
the measurable canonical selection through Chapter-2 a.e. gradient uniqueness, so the weighted
subcell mean entering the weak quantity of e.response.weak.estimate is measurable.
Every cutoff-weighted average of a coordinate of the doubled optimizer field of an arbitrary
family of response maximizers for the recentred coefficient a_+ is measurable in the coefficient
sample. This is the plus twin of
measurable_volumeAverage_weighted_optimizerField_respCoeffMinus.