High-contrast homogenization: Entry.Response.Direct.SubcellEnergyIntegrability #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Sample Integrability of the Subcell Deficits and Energies #
The subcell deficit of the first error row — the amount by which the terminal optimizer fails
to maximize the response on an aligned subcell of the coarse scale — is nonnegative, and its
flat sum over a generation equals the flat sum of the subcell responses minus the terminal
response, both P-integrable; decomposing the subcell average of the response integrand into the
subcell energy and the subcell cell average of the doubled optimizer field, each measurable,
makes the subcell deficit itself P-integrable. The weighted energy readout of the terminal
optimizer is measurable in the sample for every weight bounded by two; taking the weight to be
the indicator of an aligned subcell turns the terminal-cell average into the subcell average up
to the constant ratio of volumes, so the subcell average of the terminal optimizer's energy
density is itself measurable. These results serve the terminal-optimizer replacement row of
p.response.transfer.
Sample integrability of the subcell deficits #
The subcell deficit of the terminal-optimizer replacement 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. It is nonnegative, and its flat sum over the generation is the flat sum of the
subcell responses minus the terminal response, both P-integrable; its own measurability comes
from the decomposition of the subcell average of the response integrand into the subcell energy
and the subcell cell average of the doubled optimizer field, each measurable in the sample. Hence
each deficit is P-integrable.
Integrability of the subcell deficits, minus sign. On each aligned subcell of the coarse
scale, the amount by which the terminal optimizer for a_- = a - g fails to maximize the subcell
response is P-integrable. The deficit is nonnegative, its flat sum over the triadic index box
is the flat sum of the subcell responses minus the terminal response J_t^-, each P-integrable,
and it is measurable in the sample through the subcell energy and the subcell average of the
doubled optimizer field; domination by the flat sum then gives integrability of each term.
Integrability of the subcell deficits, plus sign. The transposed twin of
integrable_subcellDeficit_respCoeffMinus: the coefficient family a_+ = aᵀ + g, the dual load
q^+ and the terminal optimizer family uP replace their minus-sign counterparts, and
nonnegativity, the flat-sum identity and sample measurability transport verbatim.
The subcell energy of the terminal optimizer is measurable in the sample #
The weighted energy readout of the terminal optimizer of p.response.transfer is measurable in the
coefficient sample for every weight bounded by two. Taking the weight to be the indicator of an
aligned subcell turns the terminal-cell average into the subcell average, up to the constant ratio
of the two volumes; hence the subcell average of the energy density of the terminal optimizer is
measurable in the sample, for an arbitrary family of terminal maximizers.
The subcell energy of the terminal optimizer is measurable in the sample, minus sign. For
every aligned subcell of the coarse scale, the subcell average of the energy density of the
terminal optimizer of the recentred coefficient a_- = a - g is a measurable function of the
coefficient sample, for an arbitrary family of terminal maximizers.
The subcell energy of the terminal optimizer is measurable in the sample, plus sign. The
adjoint twin of measurable_volumeAverage_energy_adaptedCellAtCenter_respCoeffMinus.