High-contrast homogenization: Entry.Response.Direct.SubcellDeficitIdentity #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Subcell Deficit of the First Error Row #
On each coarse subcell of the terminal cell, the terminal optimizer restricted there is an
admissible competitor for the subcell's own response problem, so the subcell deficit — response
minus the average of the terminal optimizer's response integrand — is nonnegative, pathwise, at
a pointwise elliptic representative of respCoeff∓ F a. Exact partition averaging collapses
the flat average of the subcell averages of that integrand to its average over the terminal cell,
so the flat average of the deficits equals the flat average of the subcell responses minus the
terminal response, an identity between P-integrable quantities. The response integrand of a
b-harmonic field expands pointwise as -(1/2) ∇v·b∇v - p·(b∇v) + q·∇v, so its
average over a subset is minus half the symmetric energy plus the pairing of (-p, q) with the
slot-swapped cell average; the pathwise response on every subcell is likewise P-integrable.
These are the pathwise and averaged subcell-deficit facts that the cell half of
p.response.transfer consumes.
Entrywise integrability of the coarse block response on every aligned subcell #
The stationarity input of the terminal-optimizer replacement of p.response.transfer needs the
annealed response on each aligned cell of a fixed scale, so the pathwise response must be
P-integrable there for every aligned cell and not only for the centred one. On an aligned cell
the pathwise response is a quadratic form in the entries of the recentred coarse block, and the
entries of that block are fixed real linear combinations of the entries of the coarse block of
the sample; HasIntegrableCoarseBlock therefore transfers to the recentred block and the
response is integrable. This module records that consequence for the two recentred families
a_- and a_+.
Entrywise integrability of the response on an aligned subcell, minus sign. If every
entry of the coarse block of the sample is P-integrable on the aligned cell
adaptedCellAtCenter q j w of the scale 3^j q ℤ^d, then the pathwise response
ResponseJ (adaptedCellAtCenter q j w) p r a_- of the recentred coefficient a_- = a - g is
P-integrable.
Entrywise integrability of the response on an aligned subcell, plus sign. If every
entry of the coarse block of the sample is P-integrable on the aligned cell
adaptedCellAtCenter q j w of the scale 3^j q ℤ^d, then the pathwise response
ResponseJ (adaptedCellAtCenter q j w) p r a_+ of the transposed recentred coefficient
a_+ = aᵀ + g is P-integrable.
The flat sum of the subcell deficits of the first error row #
The subcell deficit of the first error row of p.response.transfer is the amount by which the
terminal optimizer fails to maximize the response on an aligned subcell of the coarse scale.
Exact partition averaging collapses the flat average of the subcell averages of the response
integrand of the terminal optimizer to its average over the terminal cell, which at a maximizer
is the terminal response. Hence the flat average of the subcell deficits is the flat average of
the subcell responses minus the terminal response, an identity between quantities each of which
is a fixed linear functional of the coarse blocks of the sample and is therefore P-integrable.
The flat sum of the subcell deficits is the flat sum of the subcell responses minus the
terminal response, minus sign. On each aligned subcell of the coarse scale the first error row
of p.response.transfer measures the response of the sample against the subcell average of the
response integrand of the terminal optimizer. The subcells partition the terminal cell and all
have the same volume, so the normalized sum of those averages is the terminal cell average, which
at a maximizer is the terminal response J_t^-. Therefore the normalized sum of the subcell
deficits equals the normalized sum of the subcell responses minus J_t^-.
The flat sum of the subcell deficits is the flat sum of the subcell responses minus the
terminal response, plus sign. The transposed twin of avsum_subcellDeficit_respCoeffMinus_eq:
the coefficient family a_+ = aᵀ + g, the dual load q^+ and the terminal optimizer family uP
replace their minus-sign counterparts, and the terminal response is J_t^+.
Integrability of the flat sum of the subcell responses, minus sign. Each aligned subcell
response of the recentred coefficient a_- is P-integrable when the coarse blocks of the sample
are, so the normalized sum over the triadic index box is P-integrable as well.
Integrability of the flat sum of the subcell responses, plus sign. The transposed twin of
integrable_avsum_responseJ_adaptedCellAtCenter_respCoeffMinus, with the coefficient family a_+
instead of a_-.
Nonnegativity of the subcell deficit at the response coefficients #
On each coarse subcell of the terminal cell, the terminal optimiser restricted there is an
admissible competitor for the subcell's own response problem, so the subcell deficit of
p.response.transfer — the subcell response minus the average over the subcell of the terminal
optimiser's response integrand — is nonnegative, pathwise in the coefficient sample. The response
coefficients respCoeff∓ F a are elliptic only almost everywhere, so the argument runs at the
pointwise elliptic representative supplied by EllipticRepresentativeInputs and transports both
the response
and the averaged integrand back across the almost-everywhere replacement, which leaves each of them
invariant. This is the hypothesis hD of the cell half of the cutoff-mean row of
p.response.transfer, for the minus sign and for its plus twin.
The subcell deficit is nonnegative, minus family. Let a_- = respCoeffMinus F a be the
response coefficient of the terminal cell U_t = respCell jStar F t, let uM a be a a_--harmonic
field on U_t, and suppose that on every aligned subcell U_s(w) the field v w a maximises that
subcell's response for the loads (p, q^-). Then, for every subcell w of the depth-H partition
of U_t (t = s + H) and every sample a, the subcell deficit
J(U_s(w), p, q^-; a_-) − ⨍_{U_s(w)} g_{U_t}(uM a)
is nonnegative: uM a restricted to the subcell is an admissible competitor there, and the
restriction does not change the scalar response integrand. The coefficient is elliptic only almost
everywhere, so the competitor argument is run at the pointwise elliptic representative of a_- and
both terms are carried back, which leaves them unchanged.
The subcell deficit is nonnegative, plus family. The transposed twin of
zero_le_subcellDeficit_respCoeffMinus: with the response coefficient a_+ = respCoeffPlus F a,
the loads (p, q^+) and subcell maximisers v w a for a_+, the subcell deficit
J(U_s(w), p, q^+; a_+) − ⨍_{U_s(w)} g_{U_t}(uM a)
is nonnegative on every subcell of the depth-H partition of the terminal cell, pathwise in the
sample a. The a.e.-elliptic representative of a_+ again makes the competitor argument valid.
The subcell average of the response integrand #
The response integrand of a b-harmonic field expands pointwise as
-(1/2) ∇v·b∇v - p·(b∇v) + q·∇v. Averaging this identity over an arbitrary measurable
subset V of the cell expresses the average of the response integrand as minus one half of
the average pathwise symmetric energy on V, plus the pairing of (-p, q) with the
slot-swapped cell average of the doubled optimizer field on V. This is the algebraic
expansion behind the cell-average half of the response identity, stated on a subcell rather
than on the cell itself, which is what turns the subcell deficit of the first error row of
p.response.transfer into quantities that are measurable in the sample.
The general identity is stated for an arbitrary subcell V ⊆ U and the three integrability
facts it consumes; the two carrier specialisations run it on an aligned adapted subcell and
transport the result across the almost-everywhere elliptic representative of a response
coefficient.
The subcell average of the response integrand of a b-harmonic field on U: the average
over any subset V ⊆ U is minus one half of the average pathwise symmetric energy on V, plus
the pairing of (-p, r) with the slot-swapped average of the doubled optimizer field on V.
The subcell average of the response integrand for the minus response coefficient, on an
aligned adapted subcell of the terminal cell: minus one half of the subcell energy plus the
pairing of (-p, q^-) with the slot-swapped subcell average of the doubled optimizer field.
The coefficient is transported from its almost-everywhere elliptic representative.
The plus twin of the subcell average identity, for the plus response coefficient and the
load q^+.