High-contrast homogenization: Entry.Response.Rows.CutoffPairingRowSplit #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The cutoff pairing splits into a half-energy, two mean pairings, and a cutoff-mean defect #
Expanding the two recentred differences inside the cutoff pairing of e.response.cutoff.estimate,
distributing the cutoff over the four resulting pieces, and using that the cutoff averages to one
on the cell exhibits half the pairing as the cutoff-weighted half-energy of the optimizer state
minus the two cutoff-mean pairings against the inserted mean Y^∓, plus half its self-pairing.
Integrating this pathwise identity against the law of the coefficients splits the resulting
cutoff-mean defect, on the terminal cell, into a cell part and an oscillation part of the annealed
(φ - 1)-weighted average of the doubled optimizer state X = (∇v, b∇v). Once the cutoff pairing
row is added with coefficient one, the three rows recombine into the estimate's printed right-hand
side C(τ + (τ E[J])^{1/2} + g E[J]) + C(τ L)^{1/2} + C g (E[J] L)^{1/2} (AK.HC (3.45)-(3.54)). A
further fact about the recentring shear G_g and the flux flip D records that the adjoint of a
sheared block is the sheared adjoint, D(G_g^T A G_g)D = G_{-g}^T(DAD)G_{-g}.
The integrated cutoff-mean defect splits into its cell and oscillation parts #
The centred cutoff decomposition of e.response.cutoff.estimate leaves the defect of the
annealed cutoff-weighted state mean against the annealed mean Y^∓. On the terminal cell the
mean Y^∓ is the annealed cell average of the optimizer state X = (∇v, b ∇v), so that defect
is the annealed (φ - 1)-weighted average of X. Subdividing the terminal cell into its
depth-H triadic subcells splits it, coordinate by coordinate, into a cell part — the subcell
means of φ - 1 times the subcell means of the state — and an oscillation part — the subcell
averages of φ minus their own cell means, weighted by the state. This file integrates the
pathwise split cutoffStateMeanAux_fst_eq_osc_add_cell /
cutoffStateMeanAux_snd_eq_osc_add_cell against the law P, keeping the two parts as separate
functions so that a later assembly can consume each one.
The integrated cutoff-mean defect splits for the recentred response
(e.response.cutoff.estimate). Assume that Y^- is the annealed cell average of the terminal
optimizer state X = (∇v, b ∇v) with b = respCoeffMinus F (the hypothesis hY), that the
grid respGrid F is invertible, and that the sample functions of the split are integrable, both
on the terminal cell and on each depth-H triadic subcell, and P-integrable as functions of the
coefficient. Then the annealed cutoff-mean defect
∫_a (cutoffStateMeanAux U φ (X a)) i - (Y^-) i splits, in each of the two coordinates, into the
annealed CELL part — the flat average over the depth-H subcells of (⨍ φ - 1) times the subcell
mean of the coordinate — plus the annealed OSCILLATION part — the flat average over the subcells
of the subcell mean of (φ - ⨍ φ) times the coordinate. The two parts are kept as separate
functions added with +, as the row assembly consumes them separately.
The integrated cutoff-mean defect splits for the adjoint recentred response
(e.response.cutoff.estimate). The adjoint twin of
integral_cutoffStateMeanAux_sub_respYMinus_eq_cell_add_osc: with Y^+ the annealed cell average
of the terminal optimizer state for the adjoint recentred coefficient respCoeffPlus F, the
annealed cutoff-mean defect splits, in each coordinate, into the annealed CELL part — the flat
average over the depth-H subcells of (⨍ φ - 1) times the subcell mean of the coordinate — plus
the annealed OSCILLATION part, the two kept as separate functions.
Assembling the cutoff rows of e.response.cutoff.estimate #
The cutoff energy defect and the centred cutoff-mean row enter the cutoff estimate with the two
bounds C (τ + (τ E[J])^{1/2} + g E[J]) and C (τ L)^{1/2} + C g (E[J] L)^{1/2}, while the
cutoff pairing row enters with coefficient one. Once the response J is split into these three
rows, the printed right-hand side of e.response.cutoff.estimate follows by linear arithmetic.
The response split with half weight on the pairing row: if
|J| ≤ (1 / 2) * pair + row1 + row2 and the two rows obey their printed bounds, then |J| is
bounded by the printed right-hand side of e.response.cutoff.estimate with the pairing row
carried at coefficient one. The half weight is absorbed using the nonnegativity of pair.
The pathwise cutoff decomposition identity on a bare cell #
Expanding the two recentred differences inside the cutoff pairing of e.response.cutoff.estimate,
distributing the cutoff over the four pieces, and using the unit cell average of the cutoff
exhibits half the pairing as the cutoff-weighted half-energy minus the two cutoff-mean pairings
plus half the self-pairing of the inserted mean. This is the pathwise identity on which the
integrated three-row decomposition rests, and it needs no ellipticity, no boundedness of the cell
and no positivity of the cutoff, only the indicated integrabilities.
The pathwise cutoff decomposition identity. On the bare cell U, half the cutoff pairing
of the optimizer state X = (∇v, b ∇v) at the inserted mean Y equals the cutoff-weighted
half-energy cutoffHalfEnergyAux minus half the pairing of the cutoff mean of the first coordinate
with Y.2, minus half the pairing of Y.1 with the cutoff mean of the second coordinate, plus half
the self-pairing of Y. This is the identity consumed by the integrated decomposition of
e.response.cutoff.estimate; the cell and the coefficient are arbitrary and the only hypotheses
are the integrabilities of the cutoff, the sample functions and the coordinates of the optimizer
state.
The adjoint of a sheared block is the sheared adjoint #
The recentring shear G_g = [[Id, 0], [g, Id]] and the flux flip D = diag(Id, -Id) form the
adjoint pair of the response decomposition. Flipping the flux sign commutes with congruence by
G_g up to reversing the shear,
D (G_gᵀ A G_g) D = G_{-g}ᵀ (D A D) G_{-g},
so the adjoint recentred block Ehat_u^+ is the (-g)-shear congruence of the adjoint annealed
block, exactly as Ehat_u^- is the g-shear congruence of the annealed block itself. This is the
algebraic identity that places the adjoint row of e.response.cutoff.estimate on the same footing
as the primal row of p.response.transfer.
Flipping the flux sign commutes with congruence by the recentring shear up to reversing the
shear: D (G_gᵀ A G_g) D = G_{-g}ᵀ (D A D) G_{-g}, where D = blockD d and
G_g = ⟨1, 0, g, 1⟩.
The adjoint recentred block of e.response.cutoff.estimate is the (-g)-shear congruence of
the adjoint annealed block:
Ehat_u^+ = G_{-g}ᵀ (D E_u D) G_{-g}, the adjoint twin of the primal identity
Ehat_u^- = G_gᵀ E_u G_g used by p.response.transfer.