High-contrast homogenization: Entry.Response.Kernel.AdjointRecentHeadEstimate #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The adjoint recent head, and the optimizer-energy partition it needs #
This file completes, for the adjoint response coefficient, the recent head of the diagonal
weak-norm estimate: the two-halved assembly and the resulting bound at the estimate's own
carriers. It also proves that the pathwise optimizer energy of a parent adapted cell recombines,
in general and then on the adapted-cell tree, into the flat average of the depth-n subcell
energies, and establishes the integrability on the estimate's own cell of the doubled optimizer
field for both the recentred and adjoint coefficients — inputs the energy bound and the
cell-average steps both require. It serves the cell-average estimate l.weaknorms.moreproto.
The recent head of the cell-average estimate, adjoint sign #
The weak-norm estimate for the response controls the recent head of the cell-average defect for
two reflected coefficients: the recentred field a_- = a - g with response matrix
E_-^t = respEhatMinus, and the adjoint field a_+ = a^t + g with adjoint response matrix
E_+^t = respEhatPlus. This file assembles the adjoint head: the one-sided weakCellSum half,
the weakAverageSum half, their composition into the printed summand, and the reduction of the
per-scale analytic input through the two energy facts. Every statement is the adjoint-sign
counterpart of its recentred-sign original; only the carrier changes, and no constant or binder
order moves.
The weakCellSum half of the recent-head cell-average estimate for the adjoint sample
a_+ = respCoeffPlus F a, stated against the metric factor the estimate prints. The spectral
bound sqrt (blockSpecBound (normalizedBlock E_+ M_0)) replaces the operator norm delivered by
the reflected-defect comparison; on positive definite E_+ and M_0 the two coincide. This is
the cell half of the response weak-norm estimate.
The two-halves composition for the adjoint sample with the child-maximizer family produced
before the per-scale analytic constraint: the same maximizer family then serves every admissible
per-scale family A. The maximizers are chosen from the cell half alone, so moving exists V
outside forall A is a binder reorder and strictly strengthens the statement.
The recent head of the cell-average estimate for the adjoint sample, against the actual parent optimizer rather than a family of child maximizers. The child maximizers are eliminated by the recent-scale decomposition and the per-scale family is restricted to the explicit average-defect family, leaving the analytic per-scale input as the only remaining hypothesis.
The recent head for the adjoint sample reduced to the two analytic energy facts. The per-scale
input is discharged through scaleInput_of_analytic_plus: a per-cell energy family G bounds
the quadratic defect cellwise and its cell average is controlled by the response load
respLsqPlus times the averaged recent-defect size. The side condition 0 <= respLsqPlus is
discharged from the positive definiteness of the adjoint response matrix. This is the adjoint
counterpart of the reduction of the recent-head estimate to the energy-map and difference-energy
bounds.
The finite-window head of the cell-average estimate at the carriers, adjoint sign #
The recent head of l.weaknorms.moreproto is bounded by the two finite defect sums once two
analytic facts are supplied at each depth: a per-cell metric bound for the recent difference of
the parent optimizer and a child optimizer, and an averaged bound for the difference energies.
Both are now available at the estimate's own grid, sample, metric and maximizer, so the head
carries no analytic hypothesis of its own: the coefficient's positive definiteness comes from the
law-integrability of the coarse block on the cell, and every geometric input from invertibility of
the selected grid.
The finite-window head of the cell-average estimate, at the carriers. On the branch
where the all-scale maximum does not exceed 1, the 3^{-n/2}-weighted sum over the window
n ≤ H of the normalized L² averages of the recentred transported subcell averages is bounded
by 16 K L times the sum of the two finite defect sums, where K is the square root of the
printed spectral bound of the normalized reference block and L the square root of the response
load.
The parent optimizer energy recombines over the depth-n subcells #
The pathwise normalized optimizer energy weakOptimizerEnergy U b u is the square root of the
average of the doubled-optimizer energy density x ↦ (X x).1 · (X x).2, where X = optimizerField b u is the doubled optimizer field. The first statement below squares that definition: on a
region where the average is nonnegative, squaring the energy recovers the average.
The second statement is the partition identity that lets the per-cell energy map recombine into
the parent energy. When the parent adapted cell is the almost-everywhere disjoint union of its
3^{nd} aligned depth-n subcells, all of them of equal volume, the unweighted flat average of
the subcell energy averages is the parent average. The equal-volume hypothesis is what makes the
unweighted mean correct; without it the recombination would have to be the volume-weighted sum
∑_w (|V w| / |U|) ⨍_{V w} g. It is stated through .toReal, and together with the positivity of
the parent volume it excludes a zero-volume subcell: a zero-volume cell would force the left side
of hvolw to vanish while the right side is positive. The almost-everywhere partition step itself
is average_over_aePartition.
Squaring the pathwise normalized optimizer energy recovers the average of the doubled field's energy density, as soon as that average is nonnegative.
The parent optimizer energy recombines over the depth-n subcells. Under the
almost-everywhere partition hypotheses, the flat average over the 3^{nd} aligned depth-n
subcells of their energy averages is the squared pathwise optimizer energy of the parent adapted
cell.
The parent optimizer energy partition at the adapted cells #
The partition identity recombines the flat average of the depth-n subcell energy averages
into the squared pathwise optimizer energy of the parent adapted cell. In its general form
that identity carries nine geometric and integrability hypotheses. For the aligned adapted
cells every geometric one of them is supplied by IsUnit q and the tree's cell lemmas: the
subcells are open hence measurable, pairwise disjoint, contained in the parent, omit only the
null grid seams, and have equal volume, and the parent cell has positive finite volume. This
file discharges those hypotheses and lands the clean statement.
The nonnegativity of the averaged doubled-optimizer density and its integrability are not
available from IsUnit q alone: the coefficient field is an arbitrary matrix-valued field, so
x ↦ (X x).1 · (X x).2 need not be nonnegative and need not be integrable. They remain
explicit hypotheses rather than being replaced by a stronger ellipticity assumption.
The parent optimizer energy partition at the adapted cells. For an invertible grid
q, generation t and depth n, the flat average over the 3^{nd} aligned depth-n
subcells of their averaged doubled-optimizer energy densities equals the squared pathwise
normalized optimizer energy of the parent adapted cell HighContrast.adaptedCell q t. Every
geometric and finiteness hypothesis of the general partition identity is discharged from
IsUnit q; only the nonnegativity and the integrability of the averaged density remain as
hypotheses.
Integrability of the doubled optimizer field on the estimate's own cell #
The cell-average steps of the diagonal weak-norm estimate move a matrix through the cell average
of the doubled optimizer field optimizerField b u = (∇v, b ∇v). Integrability of both slots and
of the scalar self-pairing x ↦ blockVecDot X(x) X(x) is needed on the estimate's own adapted
cell, not only on its aligned subcells.
The two slots are handled as in the subcell statements. The gradient slot is the weak gradient of
an H¹ function and is L² with no ellipticity hypothesis. The flux slot x ↦ b(x) ∇u(x) is
L² because the tree supplies an a.e.-equal representative that is pointwise elliptic on the cell
(exists_elliptic_representative_respCoeffMinus, …Plus); the two fields agree almost everywhere,
so integrability transfers. The self-pairing is L¹ because the product of two L² scalar
components is L¹ (Cauchy-Schwarz), and finite-measure L² is L¹.
The recentred coefficient a_- = respCoeffMinus F a #
Both slots of the doubled optimizer field are L² on the adapted cell, for
respCoeffMinus F a. The gradient slot is the weak gradient of the H¹ function carried by
the AHarmonicFunction, and the flux slot is L² through the a.e.-equal pointwise elliptic
representative of respCoeffMinus F a on the cell.
Both slots of the doubled optimizer field are integrable on the adapted cell, for
respCoeffMinus F a. Square-integrability on the finite-measure adapted cell gives
integrability of each component.
The transposed recentred coefficient a_+ = respCoeffPlus F a #
Both slots of the doubled optimizer field are L² on the adapted cell, for
respCoeffPlus F a. The transposed twin of
memVectorL2_optimizerField_respCoeffMinus_cell.
Both slots of the doubled optimizer field are integrable on the adapted cell, for
respCoeffPlus F a. The transposed twin of
integrableOn_optimizerField_respCoeffMinus_cell.