High-contrast homogenization: Entry.Response.Rows.CutoffMeanRowCarriers #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The cutoff-mean row on the response's own data #
The cutoff-mean row of p.response.transfer is the sum of a cell half, bounded from the
response's own cell data by (L_s^∓ 2 τ^∓)^{1/2}, and an oscillation half, read at the dual
variable Y^∓ and bounded by its depth-H and descendant contributions against
c₂ 3^{-H}(E[J_t^∓] L_s^∓)^{1/2}; this file combines the two into the row's final concrete form
for both signs. The oscillation bound is stated through a single constant, here shown to be
nonnegative and to satisfy the elementary arithmetic identity the combination needs, and a short
identity for the cutoff's own fluctuation average closes the remaining bookkeeping. Together these
give the cutoff-mean row exactly in the form the cutoff estimate consumes, with no further
hypothesis on the coefficient sample.
The cutoff-mean row at the carriers #
The cutoff-mean row of p.response.transfer is the sum of a cell half and an oscillation half.
CellHalfFinalBound bounds the cell half by (L_s^∓ 2 tau^∓)^{1/2} from the carriers' own
annealed data. The oscillation half of CutoffOscillationHalfClosure, read at the dual variable
Y^∓,
bounds the scalar annealed functional ⨍_{U_t}(φ-1)(⟨Y₂, ∇u⟩ + ⟨Y₁, a∇u⟩) against its depth-H
cell part; CellHalfFinalBound identifies the crossed pairing of the vector oscillation part
with exactly that difference, once the within-cell fluctuation is split by
volumeAverage_fluct_eq_osc_add_cell. Together with the coordinate integrabilities of
CutoffMeanDefectIntegrability, the row assembly of OptimizerMeanRowCarriers then gives the
named obligation
CutoffMeanRow∓ with no sample-side premise left.
The constant of the oscillation half of the cutoff-mean row: the cutoff Lipschitz constant
32 d² Θ, the square root of the geometric series ∑ 3^{-n/2} of the descendant weights, the
generation shift of the source-load layers and the √8 of the doubled cell energy.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The oscillation-half constant is nonnegative.
The cutoff-mean row at the carriers, minus sign. Under the annealed data of the response
window — stationarity, coarse ellipticity, the invertible rounded grid, the maximizer family, the
two terminal responses and the convergence of the source-load series at every dual vector — the
named obligation CutoffMeanRowMinus holds with constant max 1 (respCutoffOscConst d).
The cutoff-mean row at the carriers, minus sign. Under the annealed data of the response
window — stationarity, coarse ellipticity, the invertible rounded grid, the maximizer family, the
two terminal responses and the convergence of the source-load series at every dual vector — the
named obligation CutoffMeanRowPlus holds with constant max 1 (respCutoffOscConst d).