High-contrast homogenization: Entry.Response.Rows.SourceLoadHeadBound #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Source-Load Head Bound for the Cell-Pairing Row #
The cutoff-mean rows average the cutoff fluctuation over a partition of the terminal cell against
a pathwise pairing; discrete Cauchy-Schwarz for weighted flat averages bounds this by the
geometric mean of the pairing's flat-averaged head and twice the flat-averaged scalar deficit,
and annealed Cauchy-Schwarz then moves the expectation inside each factor. The head factor is the
depth-zero summand of the source load L_s, whose series sums over triadic generations the
squared (b^{1/2}, S_*^{-1/2})-norm of the annealed mean; the expectation of its squared
two-term head is bounded, again by annealed Cauchy-Schwarz, by the square of the sum of the
square roots of the expectations of its summands. Since every summand of L_s is nonnegative,
the convergent series dominates each of them, and the expectation of a pathwise coarse-block
quadratic form equals the same form of the annealed block. The head bound at the terminal scale
is what the cell half of p.response.transfer consumes.
Cauchy--Schwarz for the square roots of two integrable functions #
For a measure P and integrable nonnegative functions f and g, the integral of the product of
the square roots is bounded by the product of the square roots of the integrals:
∫ √f √g ≤ √(∫ f) · √(∫ g).
This is the annealed Cauchy--Schwarz inequality behind the cell part of the first error row of
e.response.cutoff.estimate: it compares the difference energy f with the cell energy g of
the sample. The proof integrates the pointwise Young inequality
√f √g ≤ (lam f + g / lam) / 2 against the measure; the optimal choice of lam reproduces the
geometric mean √(∫ f) · √(∫ g).
Cauchy--Schwarz for the square roots of two integrable functions: if f and g are integrable
and nonnegative, then ∫ √f √g ≤ √(∫ f) · √(∫ g). The pointwise Young inequality
2 √(f a) √(g a) ≤ lam f a + g a / lam is integrated against P, and the parameter lam is then
chosen as √(∫ g) / √(∫ f); when one of the integrals vanishes, the corresponding function vanishes
almost everywhere and so does the product. This is the annealed Cauchy--Schwarz inequality behind
the cell part of the first error row of e.response.cutoff.estimate.
Weighted flat averages over the cells of a partition #
The first error row of e.response.cutoff.estimate averages over the scale-s subcells of a
terminal cell. The cutoff fluctuations about one are bounded by one in absolute value, so the
weighted flat average of the fluctuation times an integrand is controlled by the flat average of
the absolute integrand; the cross terms are then averaged by the discrete Cauchy--Schwarz
inequality for flat averages. This file records those two finite-sum steps, together with the
additivity of the flat average.
Weighted flat average against a bounded weight. If every weight fl w is bounded in
absolute value by one on the finite index set Z, then the flat average of fl w * v w is
bounded by the flat average of |v w|. The weight bound enters pointwise through
|fl w * v w| ≤ |v w|, and the nonnegative factor (Z.card : ℝ)⁻¹ passes through the resulting
sum inequality. This is the weighted cell step of e.response.cutoff.estimate.
Discrete Cauchy--Schwarz for flat averages. If x and y are nonnegative on Z, the
flat average of √(x w) * √(y w) is bounded by the product of the square roots of the flat
averages of x and of y. The Cauchy--Schwarz inequality for finite sums gives the unnormalized
bound, and multiplying by the nonnegative factor (Z.card : ℝ)⁻¹ turns the product of the two
square roots into the product of the normalized square roots. This controls the cross terms of
the optimizer replacement in e.response.cutoff.estimate.
Cauchy--Schwarz with the expectation outside the cell average #
The first error row of e.response.cutoff.estimate pairs the cutoff fluctuation cell averages
c w, which are bounded by one in absolute value, against the pathwise product √(f w a) √(g w a)
of the difference and cell energies. The weighted flat average over the scale-s subcells is
controlled pathwise by the flat Cauchy--Schwarz product of the two energies, and the sample
expectation is then moved inside each factor by the annealed square-root Cauchy--Schwarz
inequality. The result is the direct full-dual pairing estimate of e.response.cutoff.estimate:
the weights stay inside the cell average, while the expectation acts linearly on each energy.
Expectation outside the weighted cell average. Let c be the cutoff fluctuation cell
averages, bounded by one on the finite index set Z, and let f and g be nonnegative families of
sample functions. Then the modulus of the sample expectation of the weighted flat average of
c w * (√(f w a) * √(g w a)) is bounded by the product of the square roots of the sample
expectations of the flat averages of f and of g.
Pathwise, the bounded weights are peeled off by the flat weighted-average bound, and the remaining
flat average of √(f w a) √(g w a) is estimated by the discrete Cauchy--Schwarz inequality for
flat averages. Integrating the resulting pointwise bound and applying the annealed square-root
Cauchy--Schwarz inequality moves the expectation inside each factor. This is the weighted cell
step of e.response.cutoff.estimate.
The cell pairing row of the cutoff-mean estimate, abstractly #
The cell half of the two cutoff-mean rows of e.response.cutoff.estimate pairs the cutoff
fluctuation cell averages c w, bounded by one in absolute value, against a full-dual pairing
pairing w a. Each subcell contributes a pairing controlled pathwise by the cell's own
source-load head G w a times the square root of twice its scalar deficit D w a. Averaging the
bounded weight against that pathwise control and then applying the discrete and the annealed
Cauchy--Schwarz inequalities bounds the modulus of the sample expectation by the geometric mean of
the annealed head and the annealed flat deficit, the latter being exactly 2 tau.
The cell pairing row of the cutoff-mean estimate. Let c be the cutoff fluctuation cell
averages, bounded by one on the finite index set Z, and let pairing be a full-dual pairing whose
subcell values are controlled pathwise by the source-load head G and the scalar deficit D
through |pairing w a| ≤ G w a * √(2 * D w a). If the annealed flat average of G ^ 2 is at most
Lhead and the annealed flat average of 2 * D equals 2 * tau, then the modulus of the sample
expectation of the weighted flat average of c w * pairing w a is at most √Lhead * √(2 * tau).
This is the cell half of the two cutoff-mean rows of e.response.cutoff.estimate.
The source load dominates its own scale-m term #
The source load L_s of e.response.cutoff.estimate is the sum over generations of the
weighted flat averages of the squared (b_{s-n,z}, S_{*,s-n,z})-norms of the annealed mean.
Every one of its summands is nonnegative, so whenever the series converges it dominates each of
its terms -- in particular the terminal term m = 0, which is the flat average over the
terminal partition that the cutoff-mean rows pair against.
Every summand of the source load L_s of e.response.cutoff.estimate is nonnegative: the
geometric weight 3^{-3n/2} is positive, the reciprocal cell count is nonnegative, and the
inner average is a sum of squares.
The head term of the source load #
The source load L_s of e.response.cutoff.estimate is the weighted sum over the triadic
generations of the squared (b^{1/2}, S_*^{-1/2})-norm of the annealed mean. At generation
n = 0 the weight is 1, the index box is the single scale-s cell itself, and the printed
summand (|b^{1/2} P| + |S_*^{-1/2} Q|)^2 dominates the flat head energy
|b^{1/2} P|^2 + |S_*^{-1/2} Q|^2. This module identifies that head term and bounds the flat
head energy by the source load whenever the defining series converges. It is the only term of
L_s that the cutoff-mean row of the estimate consumes.
At generation n = 0 the triadic index box is the singleton box {0}: the only cell of the
partition at the terminal scale is the scale-s cell itself.
The depth-zero head of the source load #
The cell half of the cutoff-mean row of p.response.transfer pairs the mean defect against the
dual variable Y and produces the square of the two-term head
(|b_s^{1/2}P| + |(S_{*,s})^{-1/2}Q|) of the source load at the terminal generation's own
scale. The source load L_s is the sum over the descendant generations of 3^{-3n/2} times the
flat average of the same expression, so the head is the n = 0 summand of that sum and is
therefore at most L_s as soon as the family is summable. This module exposes that terminal
summand and bounds the head by the load.
The terminal (n = 0) summand of the source load L_s of p.response.transfer: at
generation 0 the weight 3^{-3n/2} is 1, the triadic index box is the singleton box {0},
and the flat average collapses to the value at the scale-s cell itself, so the summand is
exactly the squared two-term head (|b_s^{1/2}P| + |(S_{*,s})^{-1/2}Q|)².
The squared two-term head (|b_s^{1/2}P| + |(S_{*,s})^{-1/2}Q|)² at the terminal scale is
the n = 0 summand of the source load L_s of p.response.transfer, so whenever the defining
series converges the head is at most the load itself.
Expectation of a pathwise coarse-block quadratic form #
The quadratic form of a coarse block 𝐀(V; b a) in a fixed vector is a finite linear combination
of the entries of the block. The Bochner integral therefore commutes with the quadratic form, and
the expectation of the pathwise form is the form of the annealed block E[𝐀(V; b)]. These are the
identities at which the expectation lands on the diagonal blocks used by the source load of
p.response.transfer, so that the load is read from E[𝐀(V; b)] rather than from a pointwise
Cauchy--Schwarz bound.
The expectation of the upper-left pathwise quadratic form is the annealed quadratic form.
For an integrable family of coarse blocks, the expectation of Y · (𝐀(V; b a).upperLeft) Y equals
Y · (E[𝐀(V; b)].upperLeft) Y, the quadratic form of the upper-left diagonal block of the
annealed block that defines the source load of p.response.transfer.
The expectation of the lower-right pathwise quadratic form is the annealed quadratic form.
For an integrable family of coarse blocks, the expectation of Y · (𝐀(V; b a).lowerRight) Y equals
Y · (E[𝐀(V; b)].lowerRight) Y, the quadratic form of the lower-right diagonal block of the
annealed block that defines the source load of p.response.transfer.
The expectation of the squared two-term head #
For a measure P and integrable nonnegative functions f and g, the expectation of the square
of the sum of the two square roots is bounded by the square of the sum of the square roots of the
expectations:
∫ (√f + √g)^2 ≤ (√(∫ f) + √(∫ g))^2.
Expanding the square, the two pure terms integrate to the annealed quantities ∫ f and ∫ g,
while the cross term is controlled in the sample by the annealed Cauchy--Schwarz inequality
integral_sqrt_mul_sqrt_le_sqrt_mul_sqrt; the pieces reassemble into the square of the sum of the
square roots of the annealed quantities. This is how the pathwise head of the direct full-dual
pairing is dominated by the head of respSourceLoad.
The expectation of the squared two-term head: if f and g are integrable and nonnegative
and the square (√f + √g)^2 is integrable, then
∫ (√f + √g)^2 ≤ (√(∫ f) + √(∫ g))^2. The square is expanded pointwise into
f + g + 2 √f √g, the two pure terms integrate to ∫ f and ∫ g, and the cross term is bounded
by the annealed Cauchy--Schwarz inequality integral_sqrt_mul_sqrt_le_sqrt_mul_sqrt; the result
recombines into the square of the sum of the square roots of the annealed quantities. This is the
estimate behind the head of respSourceLoad in the first error row of e.response.cutoff.estimate.