High-contrast homogenization: Entry.Response.Direct.OptimizerRowSideConditions #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Discharging the side conditions of the first error row #
The split of the cutoff's oscillation against the terminal-optimizer energy on each aligned coarse
subcell, proved for a general coefficient of the response class, is specialized here to the two
concrete response coefficients respCoeffMinus and respCoeffPlus. The first error row of the
cutoff estimate — comparing the cutoff half-energy of the terminal optimizer with the terminal
response, integrated over the law of coefficients — states that comparison with every pathwise
ingredient left as a hypothesis; this file discharges every one of those hypotheses on the
response's own data for the negative sign, leaving no side condition open. The same first error row
is then stated, and proved on the response's own data, for the adjoint sign. Together these are the
terminal-optimizer replacement row and the first error row of p.response.transfer, on the
response's own coefficient carriers.
The cutoff oscillation split at a coefficient of the carrier #
The terminal-optimizer replacement row of p.response.transfer splits the cutoff fluctuation
φ - 1 against the optimizer energy on each aligned subcell of the coarse scale. The variational
identities that produce the split are stated for a coefficient that is elliptic pointwise, while a
sample of the coefficient carrier is elliptic only almost everywhere. These two statements run the
split at a pointwise elliptic representative of a_- = a - g (respectively a_+ = aᵗ + g) on the
terminal cell and transport the result back: the carried-along harmonic function has the same
gradient, every quantity of the split is invariant under the almost-everywhere replacement, and the
pathwise response is unchanged as well.
Paper: p.response.transfer.
The oscillation split of the cutoff fluctuation against the optimizer energy, for the minus
family, with no pointwise ellipticity hypothesis. On the terminal cell respCell jStar F t, for a
cutoff φ of the response class and a response maximizer u of the loads (p, r) for the
recentred coefficient respCoeffMinus F a, the (φ − 1)-weighted optimizer energy differs from the
normalized sum, over the depth-n triadic subdivision, of the products of the subcell means of
φ − 1 and of the energy by at most 32 d² responseCutoffProfileConst 3^{-n} times twice the
pathwise response respJ (respGrid jStar F) t p r (respCoeffMinus F a). The coefficient is
elliptic only almost everywhere, so the estimate is run at a pointwise elliptic representative and
transported back; the response of the representative agrees with the response of the carrier
coefficient, and the carried-along optimizer has the same gradient. This is the scale-gain step of
the cutoff estimate of p.response.transfer, in the form the row assembly consumes.
The adjoint twin of abs_volumeAverage_sub_one_energy_sub_avsum_le_respCoeffMinus: the same
oscillation split of the cutoff fluctuation against the optimizer energy, for the transposed
recentred coefficient respCoeffPlus F a = aᵗ + g and a response maximizer u of the loads
(p, r) for it. The coefficient is elliptic only almost everywhere, so the estimate runs at a
pointwise elliptic representative and is transported back exactly as in the minus case. This is
the scale-gain step of the adjoint cutoff estimate of p.response.transfer.
The first error row of p.response.transfer on the carriers, side conditions discharged #
The terminal-optimizer replacement row of p.response.transfer compares the cutoff half-energy of
the terminal optimizer with the terminal response, integrated over the law of coefficients. The
assembly abs_integral_cutoffHalfEnergy_sub_respJ_le_rowMinus states that comparison with every
pathwise ingredient as a hypothesis. This module supplies those ingredients on the coefficient
carriers: the energy-defect identity, the oscillation split over the coarse subdivision, the subcell
comparison, the two cutoff-weight facts, the three signs, the subcell-response integrabilities, the
common annealed subcell value and the identification of the flat mean deficit with the scale defect.
Only the sample integrabilities and the positivity of the canonical metric remain as hypotheses; no
pointwise ellipticity is assumed, because the a.e.-elliptic representative transport produces every
pathwise statement on the carriers.
The first error row of p.response.transfer on the response carriers. Let P be a
stationary probability law on the coefficient space, F a block matrix with positive definite
canonical metric, jStar a scale with 2 d ≤ 3 ^ jStar, and s ≤ t = s + H two scales with
jStar ≤ s. For a cutoff φ of the response class at the terminal scale and a family uM of
terminal maximizers for the recentred coefficients, assume that every coarse block of the sample is
integrable on every aligned cell, that the terminal and coarse pathwise responses are
P-integrable,
and that the subcell deficit, the subcell energy and the (φ - 1)-weighted terminal energy have
P-integrable flat means. Then the law-integral of the cutoff half-energy of the terminal
optimizer
minus the terminal response is at most
max 3 (32 d^2 responseCutoffProfileConst) times
τ^- + (τ^- E[J_t^-])^{1/2} + 3^{-H} E[J_t^-], the printed first error row of
p.response.transfer. The pathwise side conditions of the row are discharged here; the
integrability of the sample data is the only remaining input.
The terminal-optimizer replacement row on the carriers of the cutoff estimate #
The first error row of p.response.transfer is stated on the concrete objects that the cutoff
estimate produces: the cutoff half-energy of a terminal optimizer minus the terminal response,
integrated against the law. The pathwise identification of the defect with the (φ - 1)-weighted
optimizer energy, the oscillation split of that energy over the aligned cells of the coarse scale,
and the subcell comparison of half-energy with subcell response are the three inputs, taken in the
shape in which the modules that prove them state them. Given those, the expectation of the defect
is bounded by C (τ^- + √(τ^- E[J_t^-]) + 3^{-H} E[J_t^-]) with the explicit constant
C = max 3 (32 d² responseCutoffProfileConst).
Paper: p.response.transfer.
The cutoff half-energy row on the carriers of the estimate. Let uP be a terminal
optimizer, φ a cutoff, and suppose the defect cutoffHalfEnergyAux - respJ is identified
pathwise with the (φ - 1)-weighted optimizer energy; suppose the weighted energy splits over the
aligned cells of the coarse scale into its cell means plus the osculation cost
32 d² responseCutoffProfileConst 3^{-H} times twice the terminal response; and suppose on each
subcell the half-energy and the subcell response satisfy the one-parameter comparison
|(1/2) E - J| ≤ D + 2 √(J D). With the cutoff weights averaging to 0 and bounded by 1, with
nonnegative subcell response and deficit, and with the annealed flat deficit equal to the scale
defect τ^-, the expectation of the defect is at most
max 3 (32 d² responseCutoffProfileConst) (τ^- + √(τ^- E[J_t^-]) + 3^{-H} E[J_t^-]). This is the
terminal-optimizer replacement row of p.response.transfer assembled on the carriers the cutoff
estimate uses.