High-contrast homogenization: Entry.Response.Direct.TriadicRefinementIncrement #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Refining the cutoff's oscillation split by one triadic generation #
A flat average, over the depth-m triadic subcells of the terminal cell, of a weight c·w against
the cell average of an integrable function, is unchanged when each depth-m cell is refined into
its own triadic subcells one generation further; this is the bookkeeping that the descendant
telescoping of p.response.transfer sums generation by generation. Exact partition averaging turns
the (φ-1)-weighted average of a density D on a cell into the normalized average, over its own
triadic subcells, of the per-subcell means of φ-1 and of D, so replacing each subcell's mean of
φ-1 by that of the coarser cell containing it leaves only a 3^{-n} remainder. Because a response
cutoff of scale 3^t is Lipschitz with constant 32 d^2 Θ 3^{-t}, the increment of the cutoff's
cell average between a cell and the coarser cell of the previous generation containing it is bounded
by that constant times the finer cell's diameter, read off at the depth H + n + 1 the descendant
sum of p.response.transfer actually uses.
Refining a weighted flat cell average by one triadic generation #
The descendant telescoping of p.response.transfer compares the cutoff cell averages at two
consecutive generations. The bookkeeping it needs is that a flat average over the depth-m
subcells of the terminal cell, of a weight c w against the cell average of a fixed integrable
function f, is unchanged when each depth-m cell is refined into its 3 ^ d children: the
weight is then read at the parent index.
avg_{w ∈ box m} c w · ⨍_{V^m_w} f = avg_{W ∈ box (m+1)} c (parent W) · ⨍_{V^{m+1}_W} f.
The proof uses only the landed partition of the centred terminal cell
(avsum_volumeAverage_eq), applied once at depth m and once at depth m + 1.
The parent of a cell of the triadic box lies in the coarser box. If W indexes a cell
of depth m + 1 inside the terminal cell, its parent index of depth m indexes a cell of the
same terminal cell.
A triadic cell sits in its parent. The aligned adapted cell of generation k at index
W is contained in the aligned adapted cell of generation k + 1 at the parent index.
The children of one triadic cell carry its cell average. Summing the cell averages of
f over the depth-(m+1) cells whose parent index is w, normalised by the number of
depth-(m+1) cells, gives the cell average of f over the depth-m cell at w, normalised by
the number of depth-m cells.
Refining a weighted flat cell average by one generation. The flat average over the
depth-m subcells of a weight against the cell average of f equals the flat average over the
depth-(m+1) subcells of the same weight read at the parent index.
The cutoff fluctuation splits into cell means plus a 3^{-n} remainder #
The scale gain of the cutoff estimate of p.response.transfer. Exact partition averaging over the
generation-(t - n) triadic subdivision writes the (φ - 1)-weighted average of a density D as
the normalized average of the per-cell weighted averages of φ - 1 and D. On each descendant
cell V, replacing φ - 1 by its own V-mean costs only the oscillation of φ on V, which is
at most 32 d² responseCutoffProfileConst 3^{-n}. Summing back gives the stated remainder bound;
the remaining cell-mean term is the one cancelled in expectation by stationarity.
Across a descendant adapted cell adaptedCellAtCenter qq (t - n) w, a cutoff φ of the
response
class IsResponseCutoff qq t φ deviates from its cell mean by at most
32 d² responseCutoffProfileConst 3^{-n}. This is the oscillation input to the scale gain of the
cutoff estimate of p.response.transfer.
The exact partition average of the (φ - 1)-weighted density over the adapted parent cell
differs from the normalized sum of the products of the per-cell means of φ - 1 and of D by at
most 32 d² responseCutoffProfileConst 3^{-n} times the parent average of D. This is the scale
gain of the cutoff estimate of p.response.transfer.
The cutoff cell averages of a cell and of a coarser cell containing it #
The descendant telescoping of p.response.transfer needs, at the passage from generation t - m
to generation t - m - 1, the increment of the cutoff cell averages over a subcell W of the
generation-(t - m) aligned cell. Because the cutoff class IsResponseCutoff qq t φ carries
derivatives at the scale 3^t, φ varies over the cell by at most 32 d² Θ 3^{-m}; the two
averages of φ over W and over the cell therefore differ by at most that oscillation. This is
the weight that makes the descendant sum converge with the printed weights 3^{3(k-s)/2}.
The cutoff cell average of a subcell differs from that of the cell by the cutoff
oscillation. If V is the aligned adapted cell of generation t - m at the index v and
W ⊆ V has positive finite volume, the cutoff cell averages of φ - 1 over W and over V
differ by at most 32 d² Θ 3^{-m}. This is the weight of the generation-(t-m-1) term of the
descendant sum of p.response.transfer.
The descendant weight of the cutoff-mean row at the carriers #
The descendant sum of p.response.transfer runs over the generations below the scale-s cells,
i.e. over the depths H + n + 1 of the terminal cell. Its weight at depth H + n + 1 is the
increment of the cutoff cell averages between a cell and the cell of the previous generation
containing it. The cutoff class bounds that increment by 32 d² Θ 3^{-(H+n)}, which is the shape
K · 3^{-n} the descendant row consumes, with K = 32 d² Θ 3^{-H} carrying the printed factor
3^{-H}.
The descendant weight of the cutoff-mean row. At the generation t - (H + n + 1) the
increment of the cutoff cell averages between a cell and its parent is at most
(32 d² Θ 3^{-H}) · 3^{-n}. This is the weight of the depth-n term of the descendant sum of
p.response.transfer.