High-contrast homogenization: Provider.Transport.WhitneyRows #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The rows and the residual of the maximal filling #
The quantitative half of l.two.grid.whitney(i), specialized to
the cross-grid setting of part (iv). A row below the starting scale consists of
cells whose parents escape the target, and the residual left after cutting the
filling off at a scale J consists of points in a scale-J cell that escapes
the target; both therefore sit in the boundary layer and obey
e.source.whitney.volumes, in the cross-grid form
e.two.grid.whitney.volumes and the residual volume of the Whitney selection,
with the printed C_d = 6 d^{3/2} and
the cross-grid factor |p^{-1}q| that K(q,q') dominates.
Letting the cutoff go to -∞ turns the residual bound into the statement that
the selected cells of all scales a ≤ n cover the target up to a null set: the
exhaustion is a countable partition up to null, so no triangulation of the
residual is needed.
Volumes of the target and of the cells #
The volume of a centered triadic cube is 3^{jd}.
The volume of a translated adapted cell is |det p| 3^{jd}. With the
identification of an aligned cell as a translate this gives every weight
|V| / |W| of the filling in closed form.
A translated adapted cell has finite volume.
A translated adapted cell has positive volume.
A translated adapted cell is bounded.
The rows of the filling #
A row of the maximal filling below the starting scale, the cross-grid row
e.two.grid.whitney.volumes: the selected cells of a scale a < n have
total relative volume at most C_d |p^{-1}q| 3^{a-j} with C_d = 6 d^{3/2}.
A row of the maximal filling never exceeds the target, the selected cells of one scale being disjoint subsets of it. This is the bound the square-weight estimate for the bulk cells uses.
The cross-grid row of the maximal filling, e.source.whitney.volumes and
e.two.grid.whitney.volumes in the printed relative-volume form: the
selected cells of a scale a < n satisfy Σ |V| / |W| ≤ C_d |p^{-1}q| 3^{a-j}
with C_d = 6 d^{3/2}.
A row of the maximal filling has total relative volume at most one. This is the input the square weight for the bulk cells uses at the starting scale, where the row bound carries no decay.
The cross-grid factor against the grid ratio #
The grid ratio dominates the cross-grid factor. K(q,q') is a power at
least one of a quantity exceeding each of |q^{-1}q'| and |(q')^{-1}q|, so it
dominates both; this is the step that turns the rows of the filling into the
printed cross-grid rows with constant C_d K_hop.
The cross-grid row of the maximal filling with the printed constant,
e.two.grid.whitney.volumes: Σ |V| / |W| ≤ C_d K_hop 3^{a-j} for
a < n, whenever the grid ratio of the two grids is at most K_hop.
The residual of the filling at a finite cutoff #
The residual volume of the maximal filling, the residual volume of the
Whitney selection: after cutting the filling off at the scale J, the uncovered
part of the target has relative volume at most
2 d^{3/2} |p^{-1}q| 3^{J-j}.
The limit exhaustion #
The maximal filling exhausts the target up to a null set. The residual
volume tends to zero as the cutoff descends, so the selected cells of all scales
a ≤ n are a countable disjoint family covering the target almost everywhere:
this is the countable Whitney partition in the form that
e.fixed.geometry.parent.child reads.
The filling as a family of finite rows #
The maximal filling of a target cell by the cells of a second grid, the
family 𝒱_a(W;q) of l.two.grid.whitney(i) and (iv) presented as
finite rows. Each row is a finite set of aligned q-cells contained in the
target and mutually disjoint across all rows, each cell is a bounded open convex
domain of the printed volume |det q| 3^{ad}, a row below the starting scale has
total volume at most C_d |p^{-1}q| 3^{a-j} times the target's and no row exceeds
the target, and all the rows together cover the target up to a null set.