High-contrast homogenization: Provider.Transport.WindowCellBounds #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The window multiplier on one cell, in both orientations #
e.source.adapted.bound opens with a pathwise bound that every later
estimate of the source-control subsection reads: on an arbitrary translate
V = y + β_a^{\mathbf s} of an adapted cell of a deterministic rounded grid
inside the window, the coarse response is below B_{\mathbf s}Y_P3^{g(j_*-a)_+}
times the reference block, and the sharp adjoint response is below the same
multiple of the dual reference block. Both halves are fields of the window
multiplier, and this file puts them in the form the estimates consume.
The two orientations are one statement. The sharp adjoint response
π_*^{-1}(V), the dual of the coarse block, is the block reflection of the
response, the dual reference block π_*^{-1} is the block reflection of the
reference block,
and the reflection is a congruence by an involution, so it preserves the Loewner
order in both directions and commutes with scalar dilation. The adjoint bound
is therefore the reflection of the primal bound, and the scalar normalizations
attached to the two are the same number.
Two arithmetic facts are recorded because the shape of the printed constants
needs them. The boundary constant B_{\mathbf q} = C_dπ’_{\mathbf q}ΞΆ_g is
nonnegative whenever the dimensional constant is, the geometric series ΞΆ_g
being positive for g < 1; and the burn discount 3^{g(j_*-a)_+} is positive
always and equal to one at or above the alignment scale. Without the first, the
ββ₯0β coefficients of the moment clauses would collapse to zero and the printed
estimates would be false as written.
The reflection dictionary #
The sharp adjoint response, the dual of the coarse block, is the block reflection of the coarse response.
The block reflection commutes with scalar dilation.
The block reflection preserves the Loewner order. It is a congruence by an involution, so the quadratic form of the reflection at a doubled vector is the quadratic form of the block at the swapped vector.
The block reflection is an order isomorphism. It is its own inverse, so the order it preserves it also reflects.
The block reflection preserves positive definiteness: swapping the halves of a doubled vector detects vanishing.
The sharp adjoint response is symmetric at every sample.
An arbitrary translate of an adapted cell is a domain #
A translate of an adapted cell is a bounded open convex domain. The translation is a homeomorphism, an affine map and a bounded displacement.
A translate of an adapted cell is nonempty.
The coarse response on a translate of an adapted cell is positive definite, pathwise in the coefficient field.
A translate of an adapted cell has the volume of the cell it translates.
The coarse response on a translate of an adapted cell is measurable in the coefficient field, for every law at once.
The two printed scalar factors #
The boundary constant B_{\mathbf q} = C_dπ’_{\mathbf q}ΞΆ_g of the moment
bound for the coarse block on an adapted cell is nonnegative: the witness
eccentricity is a square root and the geometric series is positive.
The burn discount 3^{g(j_*-a)_+} of e.source.adapted.bound is
positive.
The burn discount is one at or above the alignment scale.
The pathwise cell bounds #
The primal cell bound, e.source.adapted.bound on an arbitrary
translate of an adapted cell inside the window, with the two printed factors
collected in front of the multiplier.
The adjoint cell bound, the same display for the sharp adjoint response
against the dual reference block π_*^{-1}.
The pathwise scalar bounds #
A nonnegative Loewner bound against a positive reference block is a bound on the scalar size, the lower Loewner constraint being vacuous on a positive semidefinite numerator.
The primal cell moment, pathwise: the scalar normalization
|π^{-1/2}π(V)π^{-1/2}| of a cell of the window is below
B_{\mathbf s}Y_P3^{g(j_*-a)_+}.
The adjoint cell moment, pathwise: the same bound for
|π_*^{1/2}π_*^{-1}(V)π_*^{1/2}|, the sharp adjoint normalization in the
moment bound for the dual block on an adapted cell.