High-contrast homogenization: Entry.Response.Kernel.CentredVarianceTailRoute #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The centred variance bound and the adjoint tail route #
At each depth, the flat average over the depth-n subcells of the recentred transported cell
averages is compared with the same average of the uncentred ones, using the identity that the
mean square distance from the mean is the second moment minus the squared mean; since the flat
average of the subcell averages is the parent average, this dominates the centred scale term by
the uncentred one. This file records the resulting per-scale older-scale bound at the estimate's
own carriers, proves the adjoint twin of the two older-scale tail branches of
e.response.weak.estimate, and composes the two branches with the finite-window head through the
branch-free selector to obtain the diagonal weak-norm estimate for the transposed recentred
coefficient, the adjoint twin of l.weaknorms.moreproto.
The centred scale term is dominated by the uncentred one #
At each depth the flat average over the depth-n subcells of the recentred transported cell
averages is compared with the same average of the uncentred ones. Since the flat average of the
subcell averages is the parent average (an equal-volume partition), this is the elementary
identity "the mean square distance from the mean is the second moment minus the square of the
mean". This file proves the purely algebraic inequality over an arbitrary finite index set, and
then its instance at the estimate's carriers, where the metric transport R is applied first.
Bilinearity helpers #
The algebraic inequality #
Variance is at most the second moment. For a finite nonempty index set Z, a family of
doubled vectors y and a doubled vector m whose two components are the corresponding flat
averages of the family, the flat average of the squared distance from m is at most the flat
average of the squared norm. This is
E‖y - m‖² = E‖y‖² - ‖m‖² ≤ E‖y‖².
The instance at the estimate's carriers #
The centred transported scale term is dominated by the uncentred one. Applying the
algebraic inequality to y w = blockMatVecMul R ((X)_{V_w}) and
m = blockMatVecMul R ((X)_U), with the parent-mean identity supplied as a hypothesis and the
metric transport R distributed over the difference by linearity.
The per-scale older-scale bound at the estimate's own carriers #
The cell-average estimate e.response.weak.estimate controls, at each depth n, the normalized
L² average over the aligned depth-n subcells of the recentred transported subcell averages of
the doubled optimizer field, by √2 · K · B · ℰ. Here K is the square root of the spectral
bound of the normalized block, B = 1 + √M · 3^{ρ n / 2} combines the all-scale maximum M with
the geometric window weight, and ℰ is the pathwise optimizer energy of the parent adapted cell.
This module specializes the generic composed bound scaleTail_of_inputs to the estimate's own
carriers: the selected grid respGrid, the parent cell respCell, the metric transport
blockSqrt (respM0 F), the coefficient field respCoeffMinus F a, and the optimizer field of the
parent. The three analytic inputs are supplied generically: the variance step
scaleTerm_centred_le fed with the parent-mean identity
cellAverage_parent_eq_avg_subcells, the per-cell quadratic bound hcell (a hypothesis), and
the parent energy partition parent_energy_partition_cell.
The per-scale older-scale bound at the estimate's own carriers. For an invertible selected
grid, a generation t, a depth n and a parent harmonic optimizer, if every aligned depth-n
subcell bounds the quadratic metric form of its transported cell average by (K B)^2 times twice
its energy average, then the normalized L² average of the recentred transported subcell averages
is at most √2 · K · B · ℰ, where K = √(‖(Ehat_t^-)_+‖) and B = 1 + √M · 3^{ρ n / 2}. The
parent-mean identity and the parent energy partition are discharged internally from IsUnit q; the
integrability and nonnegativity of the parent energy density remain explicit hypotheses.
The two tail branches of the cell-average estimate, adjoint sign #
The cell-average estimate e.response.weak.estimate bounds the normalized scale-average seminorm
of the recentred, metric-transported subcell averages of the doubled optimizer field, in two
branches according to the size of the all-scale maximum M = respAllScaleMax P γ jStar F t a.
This module is the adjoint twin of tailBranches_minus: the parent optimizer is attached to
the transposed recentred response coefficient respCoeffPlus F a, and the reference sample is the
adjoint normalised block respEhatPlus.
Every input of the older-scale branch is composed here once at the estimate's own carriers: the
per-cell bound tailCell_of_bridge_plus, the generic per-scale assembly scaleTail_of_inputs
instantiated at the adjoint coefficient, the summability summable_centred, and the
integrability and nonnegativity inputs. The abstract branch splitting primal_branches then
turns the per-scale bound into the two printed branch bounds.
The per-scale family is M_0^{1/2} · ((X)_{z + U_{t-n}} − (X)_{U_t}), the transported recentring of
the parent optimizer field X = (∇v, b ∇v). On 1 < M the whole series is absorbed by the
older-scale term; on M ≤ 1 only the scales beyond a window H are absorbed and the first H + 1
scales remain as an explicit head sum.
The two older-scale branches of the cell-average estimate, adjoint sign. For an
invertible selected grid, a generation t, a parent harmonic optimizer attached to the transposed
recentred response coefficient, and the pointwise elliptic data of the parent and normalized
blocks, the normalized seminorm of the recentred transported optimizer field obeys the bad-branch
bound 16 / (1 - Quenched.contrastRho γ) · K · √M · ℰ when 1 < M and the good-branch bound
∑_{n ≤ H} 3 ^ (-(n / 2)) S n + 16 / (1 - Quenched.contrastRho γ) · K · 3 ^ (-(Quenched.contrastAlpha γ H)) · ℰ when
M ≤ 1, where K = √(‖(Ehat_t^+)_+‖), ℰ is the weak optimizer energy, and S n is the
depth-n scale average.
The cell-average estimate for the transposed recentred coefficient #
The adjoint twin of the cell-average estimate l.weaknorms.moreproto: the two older-scale
branches at the estimate's own carriers and the finite-window head compose through the branch-free
selector, exactly as for the recentred coefficient itself. The statement is the route's, carried
here so that the route file stays short.
The cell-average estimate, adjoint sign. The scale-average seminorm of the recentred
doubled optimizer state of the transposed recentred coefficient is bounded by the two finite
recent-scale sums plus the older-scale energy term, with the printed cutoff selector. See
e.response.weak.estimate.