High-contrast homogenization: Entry.Response.Direct.OptimizerEnergyDefectRow #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The terminal-optimizer replacement row, assembled on the response's own data #
The (φ-1)-weighted optimizer energy on the terminal cell splits, over the subdivision into the
aligned cells of the coarse scale, into an oscillation part — costing the cutoff class's Lipschitz
gain times the terminal response — and a subcell-mean part, the mean-zero weighting of the subcell
half-energies; combining the two gives the first error row of p.response.transfer. This file
instantiates that row on the same carriers the cutoff estimate uses: the difference between the
cutoff half-energy of the terminal optimizer and the terminal response, integrated over the law of
coefficients. The pathwise identity behind that difference, cutoffHalfEnergyAux - respJ equal to
half the (φ-1)-weighted optimizer energy, holds only for a coefficient that is pointwise elliptic
on the cell, so it is transported here across an almost-everywhere equality to the two response
coefficients respCoeffMinus and respCoeffPlus, each elliptic only almost everywhere.
The terminal-optimizer replacement row of p.response.transfer #
The (φ - 1)-weighted optimizer energy on the terminal cell splits, over the subdivision into
the aligned cells of the coarse scale, into its oscillation part and its subcell-mean part. The
oscillation part costs the Lipschitz gain of the cutoff class times the terminal response. The
subcell-mean part is the mean-zero weighting of the subcell half-energies, and the abstract row
bounds its expectation by τ + 2 √(τ cJ), where τ is the scale defect and cJ the common
annealed subcell response. Since cJ = EJ + τ and √(τ (EJ + τ)) ≤ √(τ EJ) + τ, the two halves
combine into the printed shape 6 τ + 4 √(τ EJ) + 2 K EJ.
Paper: p.response.transfer.
The terminal-optimizer replacement row. Suppose the terminal response Wtot differs from
the normalized mean-zero weighting N ∑ c w Ecell w of the subcell half-energies, where
N = |triadicIndexBox d n|⁻¹, by at most K times twice the terminal response Jt. Suppose
further that on each subcell the half-energy (1/2) Ecell and the subcell response J satisfy the
one-parameter comparison |(1/2) Ecell - J| ≤ D + 2 √(J D) with nonnegative J and deficit D,
that the weights c are bounded by 1 and average to 0, that ∫ J w = cJ is the same on every
subcell, and that the annealed flat average of the deficits D is the scale defect τ. When
cJ = EJ + τ with EJ = ∫ Jt, the expectation of the terminal response is at most
6 τ + 4 √(τ EJ) + 2 K EJ. This is the first error row of p.response.transfer assembled from
its oscillation and subcell-mean halves.
The terminal-optimizer replacement row on the carriers of the cutoff estimate #
This assembles the first error row of p.response.transfer on the same carriers the cutoff
estimate uses. The left-hand side is the difference between the cutoff half-energy of the
terminal optimizer and the terminal response, integrated over the law of coefficients. The three
geometric inputs — the pathwise identification of that difference with the (φ - 1)-weighted
optimizer energy, the oscillation split of that energy over the subdivision into the aligned
coarse cells, and the subcell comparison of the half-energy with the subcell response — enter as
hypotheses in the shape in which the modules that prove them produce them. The conclusion is the
printed bound
C (τ^- + (τ^- E[J_t^-])^{1/2} + 3^{-H} E[J_t^-])
with the explicit dimensional constant C = max 3 (32 d^2 * responseCutoffProfileConst).
Paper: p.response.transfer.
The terminal-optimizer replacement row on the cutoff carriers. Assume the pathwise
identification hid of the cutoff half-energy minus the terminal response with the
(φ - 1)-weighted optimizer energy; the oscillation split hosc of that weighted energy over the
aligned cells of the coarse scale, costing 32 d^2 responseCutoffProfileConst 3^{-H} times twice
the terminal response; and the subcell comparison hcmp of half the subcell energy with the
subcell response through its nonnegative deficit. Assume the cutoff fluctuation has mean-zero
normalized subcell means hc0 and is bounded by 1 on each subcell hc1, the subcell responses
and deficits are nonnegative and integrable, the subcell responses have a common annealed value
hJval, and the annealed flat mean of the deficits is the scale defect hDval. Then the
law-integral of the cutoff half-energy 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^-]. This is the first error row of
p.response.transfer on the carriers of the cutoff estimate.
The cutoff energy defect at an almost-everywhere elliptic coefficient #
The response identity cutoffHalfEnergy_sub_respJ_eq identifies the cutoff energy defect
cutoffHalfEnergyAux - respJ with half the (φ - 1)-weighted optimizer energy, but only for a
coefficient that is pointwise elliptic on the cell. The two coefficient fields of the response
problem, respCoeffMinus F a and respCoeffPlus F a, are elliptic only almost everywhere: a point
of CoeffSpace d is an a.e. class, and no representative is elliptic pointwise.
The identity is nevertheless invariant under an a.e. replacement of the coefficient, because
carrying a harmonic function along the replacement leaves its gradient — and hence both the
cutoff half-energy and the pathwise response — unchanged. Running the pointwise identity at the
elliptic representative supplied by exists_elliptic_representative_respCell_respCoeffMinus and
transporting back gives the defect identity verbatim for the carrier coefficient and the given
maximizer:
cutoffHalfEnergyAux_sub_respJ_eq_respCoeffMinus— the minus familya_- = a - g;cutoffHalfEnergyAux_sub_respJ_eq_respCoeffPlus— the adjoint twina_+ = a^t + g.
The cutoff energy defect of the minus family, with no pointwise ellipticity hypothesis.
The adjoint twin.