High-contrast homogenization: Entry.Response.Kernel.HeadCellDeficitInputs #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The adjoint head-cell hypothesis and its algebraic inputs #
For the adjoint sign, this file supplies the head's per-cell hypothesis at the diagonal weak-norm
estimate's own carriers — the estimate of p.response.transfer — by carrying a pointwise elliptic
representative of the recentred coefficient. It records that the normalized block commutes with a
finite average of cell defects, that the averaged response deficit and the response loads
respLsq^∓
are block quadratic forms linear in finite sums, and that the depth-n triadic subcells of an
adapted cell are pairwise disjoint and cover it up to a Lebesgue-null set.
The head's per-cell hypothesis at the estimate's own carriers, adjoint sign #
headCell_of_bridge supplies the per-cell hypothesis of the recent difference by carrying a
pointwise elliptic representative of the recentred coefficient: IsEllipticFieldOn is
pointwise, while a point of CoeffSpace d is only an a.e. class, so the recentred field cannot be
fed to the port directly. This module performs the same representative transport for the adjoint
recentred coefficient a_+ = aᵀ + g.
The port is recentEnergyMap_port_plus, whose data are the parent optimizer restricted to each
aligned subcell and a child optimizer attached to the same coefficient there. The difference of
the two optimizer fields is a single doubled state on the subcell, and on the subcell the
pointwise block identity X · 𝐀 X = 2 ξ · a ξ converts the port's Chapter-2 difference energy
into the estimate's 2 * volumeAverage shape. Both the cell average and the energy average are
a.e. invariant under the representative, so the conclusion returns to respCoeffPlus F a.
The per-cell hypothesis of the head's recent difference, adjoint sign. For an invertible
selected grid, a generation t, a depth n, a parent harmonic optimizer attached to the
transposed recentred response coefficient and a child optimizer on every aligned depth-n subcell,
the transported cell-average metric square of the difference of their optimizer fields is at most
the printed geometric square times twice the difference energy average there.
The bound is the recent-difference port recentEnergyMap_port_plus at a pointwise elliptic
representative of respCoeffPlus F a, transported back along the a.e. equality of the
representative with respCoeffPlus F a. The port's right-hand side, the Chapter-2 average of the
doubled block energy density of the difference, is converted to the estimate's 2 * volumeAverage
through the pointwise identity for a doubled state (ξ, a ξ); both the cell average and the energy
average are a.e. invariant, so the conclusion is stated at respCoeffPlus F a.
The normalized block commutes with finite averages #
normalizedBlock H F = F^{-1/2} H F^{-1/2} is linear in its first argument, so the flat
average of the normalized cell defects is the normalized block of the flat average of the
defects. This file records that identity — in scalar, finite-set and recent-carrier form —
and then reads the resulting quadratic bound off the already proved Loewner comparison
quadratic_le_norm_normalizedBlock.
This is the algebraic half of the difference-energy step of the cell-average lemma: the normalizing congruence is applied to the averaged defect rather than to each defect separately. Nothing here assumes a sign or a symmetry of the defect block, only that the normalizing block is positive definite at the quadratic-bound consumer.
The normalization distributes over a finite weighted average: conjugating each summand by
the fixed root and then averaging is the same as conjugating the averaged defect. The scalar
c is applied only to the output, so the identity holds for every finite set Z, including
the empty set and c = 0.
The recent averaged normalized defect of weakAverageDefect is the normalized block of
the flat average of the recent cell defects. This is
normalizedBlock_finset_average at the triadic index box, the weight
(card)⁻¹ and the recent coarse-block difference.
The quadratic form of the flat recent average of the cell defects, read against a positive
definite metric E, is controlled by the operator norm of its E-normalized block times the
E-quadratic form of the test vector. This is the difference-energy bound of the cell-average
lemma, obtained from quadratic_le_norm_normalizedBlock by identifying the normalized
block of the averaged defect with weakAverageDefect.
The doubled quadratic form of a finite sum of blocks is the sum of the quadratic forms: the
pairing (X, Y) ↦ X · A Y is linear in the flat representative A, and ofFullBlockMat
respects finite sums. The identity holds for every finite index set, including the empty one.
The doubled quadratic form is homogeneous in the flat representative A: dilating a block by
a scalar dilates the quadratic form by the same scalar.
The averaged response deficit as a block quadratic form #
The averaged difference of the response functional between the child cells of a triadic box and
the parent cell is the block quadratic form of the response load against the averaged
coarse-block difference. The two cell values enter only through the hypotheses hJ and hJ₀,
so the identity is purely algebraic in the coarse blocks; identifying those values with the
response functional is a separate step.
The identity holds for every triadic index box, including the empty one. On each child the
difference of the two quadratic forms is the quadratic form of the block defect
blockSub (child block) (parent block), so the parent term cancels and the outer normalization
never needs the box to be nonempty.
The averaged response deficit is the block quadratic form of the averaged coarse-block
defect. With J w the response value on the child cell adaptedCellAtCenter q (t - n) w and J₀
the response value on the parent cell adaptedCell q t, each written as the quadratic form of
its coarse block against the load x, the normalized average of the child-to-parent differences
is the quadratic form of the normalized average of the block defects
blockSub (child block) (parent block).
The depth-n subcells cover their parent adapted cell up to a null set #
The cells adaptedCellAtCenter q (t - n) w over the index box triadicIndexBox d n are 3^{nd}
pairwise disjoint open subcells of the parent HighContrast.adaptedCell q t. They omit the
interior seams, so they are not literally a cover, but the omitted set is Lebesgue-null.
The three facts proved here are the volume bookkeeping (card times a subcell volume equals
the parent volume), the equality of the union with the parent at the level of measures, and
the nullity of the difference.
The volume of any depth-n subcell, multiplied by the number of subcells, is the volume
of the parent cell. The identity is purely translation invariance of Lebesgue measure plus
the scale arithmetic 3^{nd} · 3^{(t-n)d} = 3^{td}.
The union of the depth-n subcells has the same volume as the parent cell. The union is
an a.e. cover, and the equality follows from finite additivity over the pairwise disjoint
measurable subcells together with the per-cell volume identity.
The parent cell differs from the union of its depth-n subcells by a null set. This is
the a.e. covering statement: the union is measurable, contained in the parent, and has the
same (finite) measure, so the difference is null.