High-contrast homogenization: Entry.Response.Rows.CellOscillationSplit #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Oscillation and Cell Parts of the Cutoff-Mean Defect #
Weighted averages are not additive in the weight, but they split at a chosen level c: the
average of w * f is the average of the fluctuation (w - c) * f plus c times the unweighted
average of f. Applying this cell by cell, after first replacing the average over the adapted
cell by the flat average over its triadic subcells by exact partition averaging, splits the
normalized average of (φ - 1) * g into a within-cell oscillation part, estimated pathwise, and
a cell part carrying the departure of the subcell cutoff values from a common level; the
state-valued analogue for the doubled optimizer state X = (∇v, b∇v) is recorded
coordinatewise. The file also shows the cell average of the crossed pairing ⟨Y₂, ∇u⟩ + ⟨Y₁, a∇u⟩ is P-integrable on every aligned cell of every generation below the terminal
one, dominating its Fenchel-probe bound by Young's inequality. The splitting of the normalized
cutoff-weighted average into its within-cell oscillation part and its cell part, together with the
exact partition averaging behind it, serves e.response.cutoff.estimate; the P-integrability of
the cell average of the crossed pairing on every aligned cell below the terminal one serves
p.response.transfer.
The cell average of the crossed pairing is integrable in the sample #
The descendant sum of the oscillation half of p.response.transfer reads the cell average of the
crossed pairing ⟨Y₂, ∇u⟩ + ⟨Y₁, a ∇u⟩ of the terminal optimizer on every aligned cell of every
generation below the terminal one. That readout is measurable in the sample by
CarrierIntegrabilityConditions, and the Fenchel probe of DescendantFenchelProbe dominates its
modulus by the
two-term head of the cell times the square root of twice the doubled cell energy. Young's
inequality turns that product into the sum of the squared head and the cell energy, both of which
are P-integrable — the first from the entrywise integrability of the pathwise coarse block on
that cell, the second from the integrability of the terminal response. So the readout is
P-integrable on every cell of every generation.
The annealed cell energy of the terminal optimizer on an aligned cell of any generation,
minus sign. TerminalHalfEnergyIntegrability makes the readout P-integrable on the cells of
one fixed
generation aligned with a scale s under t = s + H; every generation t - m is that statement
at s := t - m and H := m.
The annealed cell energy of the terminal optimizer on an aligned cell of any generation,
plus sign. The adjoint twin of integrable_volumeAverage_energy_depth_respCoeffMinus.
The cell average of the crossed pairing is P-integrable, minus sign. On every aligned
cell of every generation below the terminal one, the cell average of the scalar crossed pairing of
the deterministic dual variable with the terminal optimizer state is P-integrable.
The cell average of the crossed pairing is P-integrable, plus sign. The adjoint twin of
integrable_volumeAverage_cross_optimizerField_respCoeffMinus.
Splitting a weighted cell average over a partition #
Weighted averages are not additive in the weight, but they split at a chosen level c:
the average of w * f is the average of the fluctuation (w - c) * f plus the level c
times the unweighted average of f. Applying this identity cell by cell and recombining
with the equal-volume partition identity gives the cell decomposition behind the two
cutoff-mean rows of e.response.cutoff.estimate.
Splitting a weighted cell average at a level c. For any constant c, the average
of w * f over V is the average of (w - c) * f over V plus c times the average of f
over V. The first summand records the part of the weight that fluctuates about the level c;
the second is that level times the unweighted cell average. The identity is purely pointwise
under the average and needs only the integrability of the two summands on V.
Exact partition averaging on the adapted cell #
The cell decomposition used in the response cutoff estimate first replaces the average over the
adapted cell HighContrast.adaptedCell q t by the flat average of the normalized averages over its
3^{nd} triadic subcells adaptedCellAtCenter q (t - n) w, w ∈ triadicIndexBox d n. Unlike the
recombination of the optimizer energy, this step needs neither positivity of the integrand nor a
bound on it: the parent cell is the disjoint union of its open subcells up to the null grid seams,
the subcells have equal volume, and the identity follows from finite additivity of the integral.
This file records the index box nonemptiness used to normalize the finite sum and the exact
averaging identity itself.
The triadic index box of generation n is nonempty: the constant index 0 has every
coordinate in the interval [-(3^n - 1)/2, (3^n - 1)/2].
Exact partition averaging on the adapted cell (e.response.cutoff.estimate). For an
invertible grid q, generation t and depth n, the normalized average of an integrand g
over the adapted cell HighContrast.adaptedCell q t is the flat average of the normalized averages
of g over the 3^{nd} triadic subcells adaptedCellAtCenter q (t - n) w. Only the
integrability of
g on the parent cell is needed; the subcells cover the parent up to the null grid seams, and
their equal volumes make the unweighted mean the correct recombination.
The cutoff-fluctuation average splits on the adapted cell #
The cutoff estimate e.response.cutoff.estimate bounds the normalized average over the terminal
cell HighContrast.adaptedCell q t of the pairing (φ - 1) * g. The two terms of that bound have
different natures: the oscillation of the cutoff inside a scale-s subcell is estimated pathwise,
whereas the departure of the subcell values from a common level is only controlled after taking
expectations. This file records the exact algebraic identity that separates them.
The terminal average is first written as the flat average of the normalized averages over the
3^{nd} triadic subcells adaptedCellAtCenter q (t - n) w by
volumeAverage_adaptedCell_eq_flat_average.
Each subcell term is then split at the level ⨍_{cell} φ - 1 of its own cutoff average by
volumeAverage_weight_mul_split: the fluctuation part (φ - ⨍_{cell} φ) * g carries the
within-cell oscillation, and the cell part (⨍_{cell} φ - 1) * ⨍_{cell} g carries the subcell
cutoff values. Distributing the flat mean over the two parts gives the stated decomposition.
Splitting the cutoff-fluctuation average over the adapted cell
(e.response.cutoff.estimate). Let q be an invertible grid, t a generation, n a depth and
φ, g functions on the terminal cell HighContrast.adaptedCell q t, with (φ - 1) * g
integrable
there. Assume that on every depth-n triadic subcell adaptedCellAtCenter q (t - n) w the two
summands
of the split of (φ - 1) * g at the level ⨍_{cell} φ - 1 are integrable, namely the fluctuation
(φ - ⨍_{cell} φ) * g and the level term (⨍_{cell} φ - 1) * g. Then the terminal average of
(φ - 1) * g is the flat average over the subcells of the within-cell oscillation
⨍_{cell} (φ - ⨍_{cell} φ) * g plus the flat average over the subcells of the cell part
(⨍_{cell} φ - 1) * ⨍_{cell} g. The identity is purely algebraic and needs no positivity of
either function.
The cutoff-mean defect splits coordinatewise on the adapted cell #
The cutoff-mean defect of e.response.cutoff.estimate is, in each coordinate of each slot, the
(φ - 1)-weighted average of the optimizer state X = (∇v, b ∇v)
(cutoffStateMeanAux_fst_sub, cutoffStateMeanAux_snd_sub). The cell decomposition of
volumeAverage_fluct_eq_osc_add_cell then splits that average over the depth-n triadic
subcells into a within-cell oscillation part of size 3^{-n} and a cell part that cancels
after expectation. This is the state-valued analogue of the split already landed for the
energy.
Coordinatewise split of the first cutoff-mean defect (e.response.cutoff.estimate).
For an invertible grid q, a generation t, a depth n and a cutoff φ, the first-coordinate
cutoff-mean defect at the load coordinate i, namely (cutoffStateMeanAux U φ b v).1 i minus
the plain cell average (cellAverage U (optimizerField b v)).1 i, equals the flat average over
the depth-n triadic subcells of the within-cell oscillation
⨍_{cell} (φ - ⨍_{cell} φ) * (X).1 i plus the flat average of the cell part
(⨍_{cell} φ - 1) * ⨍_{cell} (X).1 i. The identity is purely algebraic: it needs only the
integrability of the cutoff-weighted coordinate, of the coordinate itself, and of the two
summands of the split on every subcell.
Coordinatewise split of the second cutoff-mean defect (e.response.cutoff.estimate).
For an invertible grid q, a generation t, a depth n and a cutoff φ, the second-coordinate
cutoff-mean defect at the flux coordinate i, namely (cutoffStateMeanAux U φ b v).2 i minus
the plain cell average (cellAverage U (optimizerField b v)).2 i, equals the flat average over
the depth-n triadic subcells of the within-cell oscillation
⨍_{cell} (φ - ⨍_{cell} φ) * (X).2 i plus the flat average of the cell part
(⨍_{cell} φ - 1) * ⨍_{cell} (X).2 i. The identity is purely algebraic: it needs only the
integrability of the cutoff-weighted coordinate, of the coordinate itself, and of the two
summands of the split on every subcell.