High-contrast homogenization: Entry.Response.Rows.TerminalHalfEnergyIntegrability #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Sample Integrability of the Terminal and Half-Energy Readouts #
For each aligned subcell of the coarse scale, this file shows the subcell average of the terminal
optimizer's energy density and the (φ - 1)-weighted terminal energy are both P-integrable,
by domination: the subcell energies are nonnegative with flat average the nonnegative terminal
energy, while a response cutoff's weight lies in [0, 2] so the cutoff-weighted half-energy is
dominated by the terminal energy itself. It also records, on an aligned cell, the
almost-everywhere strong measurability and the P-integrability of the individual entries of the
pathwise coarse block of the recentred coefficients a_- = a - g and a_+ = a^T + g, read
through the block response mean of the subcell maximizer and the two diagonal quadratic forms
against the dual variable.
Paper: p.response.transfer.
Sample integrability of the subcell and weighted optimizer energies #
The terminal-optimizer replacement row of p.response.transfer needs, for each aligned subcell of
the coarse scale, the sample integrability of the subcell average of the terminal optimizer's
energy density, and the sample integrability of the (φ - 1)-weighted terminal energy. Both are
obtained by domination. The subcell energies are nonnegative and their flat average is the
terminal cell energy, which at a maximizer is twice the terminal response; and the cutoff satisfies
|φ - 1| ≤ 1, so the weighted terminal energy is dominated by the terminal energy, itself twice the
response. The four statements below are the two signs of each fact, for the recentred coefficient
a_- = a - g and its adjoint twin a_+ = aᵗ + g.
Paper: p.response.transfer.
Sample integrability of the subcell energy, minus sign. On every aligned subcell of the
coarse scale the average of the terminal optimizer energy density is P-integrable, because that
average is nonnegative and the flat sum of the subcell averages equals the terminal cell average,
which at a maximizer is twice the terminal response J_t^-; the response is P-integrable and each
subcell average is P-a.e.-strongly measurable.
Sample integrability of the weighted terminal energy, minus sign. For a response cutoff
φ with 0 ≤ φ ≤ 2, the (φ - 1)-weighted terminal optimizer energy is P-integrable: its
absolute value is at most the terminal energy average, which at a maximizer is twice the terminal
response J_t^-, and the response is P-integrable.
Sample integrability of the subcell energy, plus sign. The twin of
integrable_volumeAverage_energy_adaptedCellAtCenter_respCoeffMinus for the adjoint recentred
coefficient a_+ = aᵗ + g and its terminal optimizer.
Sample integrability of the weighted terminal energy, plus sign. The twin of
integrable_volumeAverage_weighted_energy_respCoeffMinus for the adjoint recentred coefficient
a_+ = aᵗ + g.
Sample integrability of the cutoff half-energy defect #
The centred cutoff decomposition of p.response.transfer integrates, against the law P of
coefficients, the difference between the cutoff-weighted half-energy of the terminal optimizer and
the terminal response. A response cutoff keeps its weight between zero and two, so the
cutoff-weighted half-energy is dominated by the terminal energy, which at a maximizer is twice the
terminal response; the response is P-integrable and the weighted energy readout is measurable in
the sample, so that difference is P-integrable.
Both signs of the recentred coefficient, a_- = a - g and its adjoint twin a_+ = aᵗ + g, are
covered.
Paper: p.response.transfer.
Sample integrability of the cutoff half-energy defect, minus sign. For a response cutoff
φ with 0 ≤ φ ≤ 2 and a terminal optimizer family for the recentred coefficient a_-, the
difference between the cutoff-weighted half-energy of the optimizer and the terminal response
J_t^- is P-integrable. The half-energy is one half of the φ-weighted average of the terminal
energy density; its absolute value is at most twice the terminal response, since the terminal energy
average is twice the response and |φ| ≤ 2, so integrability follows by domination from the
P-integrability of the response.
Sample integrability of the cutoff half-energy defect, plus sign. The twin of
integrable_cutoffHalfEnergyAux_sub_respJ_respCoeffMinus for the adjoint recentred coefficient
a_+ = aᵗ + g and its terminal optimizer.
The full entry family of the recentred coarse block on an aligned cell #
The cell half of the cutoff-mean row of p.response.transfer reads the pathwise coarse block of
the recentred coefficients a_- = a - g and a_+ = aᵀ + g through three different functionals:
the two diagonal quadratic forms against the dual variable, the block response mean of the subcell
maximizer, which uses ALL four sub-blocks, and the stationarity collapse at the centred cell,
which asks only for measurability. CarrierIntegrabilityConditions records the two diagonal
sub-blocks; this
module records the whole entry family, in both the integrable and the measurable form, from the
same congruence bridge.
Recentring by a constant skew matrix field is a block congruence on an aligned adapted cell, so
every entry of the recentred block is a fixed real linear combination of the entries of the coarse
block of the sample; integrability is then HasIntegrableCoarseBlock and measurability is
unconditional.
Every entry of the recentred coarse block is integrable on an aligned cell, minus sign.
The pathwise coarse block of a_- = a - g on an aligned adapted cell is the constant congruence
Gᵀ 𝐀(V; a) G of the coarse block of the sample, so each of its entries is a fixed real linear
combination of integrable functions.
Every entry of the recentred coarse block is integrable on an aligned cell, plus sign.
The adjoint twin of integrable_blockMatEntry_respCoeffMinus_adaptedCellAtCenter, with the
congruence
G_+ = G D of the adjoint recentring.
Every entry of the recentred coarse block is measurable on an aligned cell, minus sign. No integrability is needed: the congruence bridge writes the entry as a fixed linear combination of the entries of the coarse block of the sample, which are measurable outright.
Every entry of the recentred coarse block is measurable on an aligned cell, plus sign.
The adjoint twin of aestronglyMeasurable_blockMatEntry_respCoeffMinus_adaptedCellAtCenter.
Measurability of the recentred coarse block at the centred cell, minus sign. The centred
cell is the aligned cell at index 0, so the aligned statement applies verbatim. This is the
hmeasBlk premise of the cell half of the cutoff-mean row.
Measurability of the recentred coarse block at the centred cell, plus sign. The adjoint
twin of aestronglyMeasurable_blockMatEntry_respCoeffMinus_adaptedCell.