High-contrast homogenization: Entry.Response.Rows.ScaleDefectIdentity #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Scale Defect as an Annealed Mean Deficit #
Subdividing the terminal cell U_t into the aligned cells of the coarse scale s, the
restriction of the terminal optimizer to each subcell is an admissible competitor there, so its
response value falls short of that subcell's own response by a nonnegative deficit. Exact
partition averaging identifies the flat average of the restricted response values with the
terminal response itself, and stationarity identifies the annealed subcell response with the
annealed response at the coarse scale on every subcell; taking expectations, the flat average of
the deficits equals exactly the scale defect tau^± = E[J(U_s)] - E[J(U_t)]. Because the
quadratic-response identity measures the energy of a difference field by twice the deficit, the
row assembly consumes this defect with a factor two: the flat average, over the depth-H aligned
subcells, of twice each subcell's deficit integrates to 2 tau^∓. The identity is the mean-deficit
row of p.response.transfer.
The mean deficit of the subdivision is the scale defect #
The terminal-optimizer replacement of p.response.transfer subdivides the terminal cell U_t
into the 3 ^ (H * d) aligned cells of the coarse scale s. On each of them the restriction of
the terminal optimizer is an admissible competitor, so its response value falls short of that
subcell's own response by a nonnegative deficit. Exact partition averaging says that the flat
average of the restricted response values is the terminal response value itself, and stationarity
says that the annealed subcell response is the same on every subcell, hence equal to the annealed
response at the coarse scale. Therefore the expectation of the flat average of the deficits is
exactly the scale defect
tau^± = E[J(U_s, p, q^±; a_±)] - E[J(U_t, p, q^±; a_±)].
No subcell optimizer is selected: the deficit is defined directly from the restricted terminal optimizer, so no measurability of a subcell optimizer family is needed.
The mean subcell deficit of the subdivision is the minus scale defect
(p.response.transfer). For a stationary law P, the flat average over the depth-H aligned
subcells of the coarse scale s of the gap between each subcell's own response and the restricted
terminal response integrates to the defect respTauMinus between the coarse and terminal scales.
The restricted-response half identifies the averaged pathwise response with the terminal response
by exact partition averaging, and the subcell-response half replaces the averaged subcell responses
by the common annealed coarse-scale response by stationarity.
The mean subcell deficit of the subdivision is the plus scale defect
(p.response.transfer). The adjoint twin of integral_avsum_subcellDeficit_eq_respTauMinus:
for a stationary law P, the flat average over the depth-H aligned subcells of the coarse scale
s of the gap between each subcell's own response and the restricted terminal response integrates
to the adjoint defect respTauPlus. The restricted-response half is exact partition averaging and
the subcell-response half is the stationarity of the adjoint annealed response.
Twice the annealed mean subcell deficit is twice the scale defect #
The row assembly of p.response.transfer consumes the scale defect with a factor two, because the
quadratic-response identity measures the energy of the difference field by twice the deficit. This
module records that rescaling of the mean subcell deficit: the flat average, over the depth-H
aligned subcells of the coarse scale s, of twice each subcell's deficit integrates to twice the
scale defect tau^∓.
Twice the mean subcell deficit is twice the minus scale defect (p.response.transfer).
The rescaling by the constant 2 of integral_avsum_subcellDeficit_eq_respTauMinus: for a
stationary law P, the flat average over the depth-H aligned subcells of the coarse scale s of
twice the gap between each subcell's own response and the restricted terminal response integrates
to twice the defect respTauMinus between the coarse and terminal scales.
Twice the mean subcell deficit is twice the plus scale defect (p.response.transfer).
The rescaling by the constant 2 of integral_avsum_subcellDeficit_eq_respTauPlus: for a
stationary law P, the flat average over the depth-H aligned subcells of the coarse scale s of
twice the gap between each subcell's own response and the restricted terminal response integrates
to twice the adjoint defect respTauPlus between the coarse and terminal scales.