High-contrast homogenization: Entry.Response.Direct.DescendantEnergyAndFenchelSlots #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Integrability and Closed Forms for the Descendant Sum #
This module serves p.response.transfer. At the response carriers, this file shows every linear
readout of each slot of the terminal
optimizer's doubled state, and their cross term, is L^1(P)-integrable on every descendant cell:
the recentred coefficients are elliptic only almost everywhere, so the argument runs at a single
pointwise elliptic representative on the terminal cell, serving every descendant cell at every
depth. Exact partition averaging identifies the flat average of the descendant cell energies, at
every depth, with the terminal cell energy, twice the terminal response at a maximizer; rescaling
by the constant converting it to the doubled energy the two crossed slots carry gives the row's
closed annealed form. The two cutoff-mean rows carry the crossed pairings ⟨N₁, Y₂⟩ and
⟨Y₁, N₂⟩, each bounded by the direct Fenchel pairing's own annealed coarse-block norm
times the square root of the cell energy.
The integrability of the terminal optimizer state on descendant cells #
The descendant sum of p.response.transfer pairs the dual variable Y against the cell average,
over a descendant cell V of the terminal cell U, of the doubled state X_u = (∇u, a_∓ ∇u) of
the terminal optimizer u. Each such pairing is a genuine integral, so every linear readout of
each state slot must be integrable on V. The recentred coefficients a_∓ = respCoeff∓ F a are
elliptic only almost everywhere, so the integrability is read off a pointwise elliptic
representative on the terminal cell — one representative serves every descendant cell at every
depth, because every descendant cell is contained in the terminal cell — and transported back
across the almost-everywhere replacement, which changes the gradient slot not at all and the flux
slot only on a null set.
The state slots of the terminal optimizer are integrable on every descendant cell, minus
sign. For the recentred coefficient a_- = a - g of p.response.transfer, every linear readout
of the gradient slot and of the flux slot of the doubled state of the terminal optimizer is
integrable on each depth-m aligned descendant cell of the terminal cell.
The state slots of the terminal optimizer are integrable on every descendant cell, plus
sign. The adjoint twin, for the recentred coefficient a_+ = aᵀ + g.
The crossed pairing of the terminal optimizer state is integrable on every descendant cell,
minus sign. The scalar crossed pairing ⟨Y₂, ∇u⟩ + ⟨Y₁, a_-∇u⟩ of the descendant sum of
p.response.transfer, and its modulus, are integrable on each depth-m aligned descendant cell.
The crossed pairing of the terminal optimizer state is integrable on every descendant cell,
plus sign. The adjoint twin, for the recentred coefficient a_+ = aᵀ + g and the dual variable
Y^+.
The annealed flat average of the descendant cell energies #
The descendant sum of p.response.transfer pairs each descendant cell average of the terminal
optimizer state with the annealed mean, and the Fenchel probe leaves behind the cell energy
⨍_V ⟨∇u_t, S ∇u_t⟩. Its flat average over the depth-m triadic subdivision of the terminal
cell is the terminal cell energy (exact partition averaging), and for the maximizer the terminal
cell energy is twice the response:
⨍_{U_t} ⟨∇u_t, S ∇u_t⟩ = 2 J(U_t, p, q^∓; a_∓).
Taking expectations, the annealed flat average of twice the half cell energies is exactly
2 E[J_t^∓], the scalar deficit the descendant row consumes. Paper: p.response.transfer.
The annealed flat average of the descendant cell energies is twice the annealed terminal
response, minus sign. Exact partition averaging collapses the flat average of the depth-m
cell energies to the terminal cell energy, and the quadratic-response identity identifies the
latter with twice J_t^-.
The annealed flat average of the descendant cell energies is twice the annealed terminal
response, plus sign. The adjoint twin of
integral_avsum_two_mul_halfEnergy_eq_respEJMinus.
The doubled descendant cell energy of the terminal optimizer at the carriers #
The descendant sum of p.response.transfer pairs each cell head with the square root of twice the
cell energy of the terminal optimizer, and the sum of the two crossed slots carries the doubled
energy rather than the half energy. Rescaling the annealed flat average of twice the half energy
by the constant 4 gives the doubled form, with the dual energy E[J_t^∓] replaced by
4 E[J_t^∓]. Each cell energy is nonnegative because the symmetric part of an elliptic
coefficient is positive semidefinite; the recentred coefficients are elliptic only almost
everywhere, so this is read at a pointwise elliptic representative and transported back.
The descendant cell energy of the terminal optimizer is nonnegative, minus sign. On every
aligned descendant cell of the terminal cell, the cell average of the variation energy integrand of
the recentred coefficient a_- = a - g is nonnegative.
The descendant cell energy of the terminal optimizer is nonnegative, plus sign. The
adjoint twin, for the recentred coefficient a_+ = aᵀ + g.
The annealed flat average of the doubled descendant cell energies, minus sign. The
expectation of the flat average over the depth-m aligned cells of twice the doubled cell energy
of the terminal optimizer is 2 * (4 E[J_t^-]), the value the descendant sum of
p.response.transfer consumes.
The annealed flat average of the doubled descendant cell energies, plus sign. The adjoint
twin, with 4 E[J_t^+].
The two slots of the full-dual pairing against the source-load head #
The two cutoff-mean rows of p.response.transfer carry the two crossed pairings ⟨N₁, Y₂⟩ and
⟨Y₁, N₂⟩ under separate absolute values, so each slot is bounded on its own. The direct
full-dual Fenchel pairing bounds each slot by its own annealed coarse-block norm times the square
root of the cell energy. Adding the other, nonnegative, annealed norm on the right puts both in
the single shape the descendant row consumes, with the source-load head
G_V = |b_V^{1/2} Y₁| + |S_{*,V}^{-1/2} Y₂| and the scalar deficit
D_V = ½ ⨍_V ⟨∇v, S ∇v⟩.
The gradient slot of the full-dual pairing against the source-load head. The pairing of
the annealed flux Y₂ with the gradient cell average of the optimizer state is at most the
source-load head of the cell times the square root of twice the half cell energy.
The flux slot of the full-dual pairing against the source-load head. The pairing of the
annealed gradient Y₁ with the flux cell average of the optimizer state is at most the
source-load head of the cell times the square root of twice the half cell energy.