High-contrast homogenization: Provider.Transport.FillingExhaustion #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The moving finite exhaustion of a target cell #
e.two.grid.whitney.average is the display
0 โค ๐(W) โค ฮฃ_{r=j_*}^{j-ฮป_j} ฮฃ_{Vโ๐ฑ_r(W;q)} (|V|/|W|)๐(V) + G_W^{<j_*},
together with its adjoint version, for a target W = z + โ_j^{q'} filled by the
cells of the second grid q from the scale n = j - ฮป_j downwards. The printed
proof continues the filling to a finite cutoff J < j_*, triangulates the
remaining set, and lets J โ -โ; the right side is then the increasing limit of
the finite partial sums, the residual integral vanishing by absolute continuity.
Here the triangulation is bypassed, as it is for the geometry. The residual of
the filling at the cutoff J is not decomposed: the sub-partition estimate of
FillingSubadditivity charges it to one constant, and volume_residual_le --
the residual volume of the Whitney selection, with its rate
2d^{3/2}K_{hop}3^{J-j} -- sends the weight in front of that constant to zero.
Two consequences are recorded.
The finite exhaustion. At every cutoff J โค n the coarse response of the
target lies below the finite double sum over the rows J โค r โค n of the filling,
plus the printed residual rate times one constant independent of J. This is
subadditivity over a finite Whitney exhaustion, with the residual left whole.
The limit. Consequently the coarse response of the target lies below any
bound valid for every finite partial double sum. That is the form
e.two.grid.whitney.average is consumed in: its right side is a sum of
nonnegative terms -- the rows from j_* to n, and the below-start series
G_W^{<j_*} of e.two.grid.whitney.source -- so it
dominates every finite partial sum, and no series has to be summed to apply it.
The rows of the truncated filling are gathered into one finite index set of scale-centre pairs, the disjoint union over the scales of the rows themselves; no new object is introduced for it.
The rows of a truncated filling as one finite family #
The finite exhaustion at a cutoff #
Subadditivity over a finite Whitney exhaustion, with the residual left
whole. At every cutoff J โค n the quadratic form of the coarse response of the
target is below the finite double sum over the rows of the filling, plus the
printed residual rate 2d^{3/2}|p^{-1}q|3^{J-j} of the residual volume of the
Whitney selection times a single nonnegative constant that does not depend on the
cutoff.
That constant is the elliptic majorant of the response integrand together with the affine term the incomplete weights leave behind; it is not exhibited, because the rate annihilates it.
e.two.grid.whitney.average, the upper clause. If a real number
dominates every finite partial double sum of the filling's rows, then it dominates
the quadratic form of the coarse response of the target.
This is the limiting form of the exhaustion: the printed right side
ฮฃ_{r=j_*}^{n}ฮฃ_V(|V|/|W|)๐(V) + G_W^{<j_*} is a sum of nonnegative terms and
therefore dominates every finite partial sum, so it is admissible here. No series
is summed and no triangulation of the residual is used; the residual is paid by
the residual volume of the Whitney selection alone.
The adjoint half of e.two.grid.whitney.average. The sharp adjoint
response is the block reflection of the coarse response, and the reflection swaps
the halves of the doubled vector, so the same limiting exhaustion holds for
๐_*^{-1} with G_{*,W}^{<j_*} in place of G_W^{<j_*}.
The Loewner form, and the two clauses together #
The coarse response of a target is positive semidefinite as a doubled
matrix, the first clause of e.two.grid.whitney.average in the currency
the transport's matrix algebra works in.
e.two.grid.whitney.average, both clauses, in both orientations and
in the Loewner order of the doubled matrices: the coarse response of the target is
positive, and it is below every symmetric block whose quadratic form dominates all
the finite partial double sums of the filling's rows.
The printed right side ฮฃ_{r=j_*}^{n}ฮฃ_V(|V|/|W|)๐(V) + G_W^{<j_*} is such a
block: every one of its terms is a positive multiple of a coarse response, hence
positive, so its quadratic form dominates each partial sum.