High-contrast homogenization: Entry.Response.Core.RecenteredResponseIntegrability #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Law integrability of the recentred and adjoint pathwise response #
The pathwise response J(U_u, p, q; b) of a sample coefficient agrees, on an adapted cell where
b is elliptic, with the quadratic expression ยฝ x ยท ๐(U_u; b) x - p ยท q of the coarse block;
using the deterministic congruence linking the coarse block of a recentred coefficient to that of
the sample, this file shows that J(U_u; a_-), J(U_u; a_+) and their squares are P-integrable
under the coarse ellipticity assumption. It records the square-integrability the weak-norm
estimate needs of the doubled optimizer state on an adapted cell, and the affine invariance of the
scale-average seminorm of a doubled field under a constant block change of variables. It closes
with the singular-grid branch of the cutoff-pairing bound of e.response.cutoff.estimate: the
relevant cell average and its integral both vanish when the selected grid is degenerate, and the
weak response energy respWeakEnergy is nonnegative and realized by any admissible family.
Law integrability of the adjoint recentred pathwise response #
The pathwise response J(U, p, q; b) of a sample coefficient is a supremum, and on an adapted
cell it agrees with the quadratic expression ยฝ x ยท ๐(U; b) x - p ยท q of the coarse block
whenever b is elliptic. This file records that agreement for the adjoint recentred
coefficient aโ = aแต + g, where g is the skew Schur half of the self-dual splitting, and
deduces that the annealed response a โฆ J(U, p, q; aโ) is integrable for the law as soon as
the entries of the coarse block ๐(U; a) are.
The two structural facts are:
- the adjoint of a coefficient reflects its coarse block in the flux slot, and recentring by
the skew half
gis the constant shear congruenceGof the self-dual splitting, so๐(U; aโ) = (GยทD)แต ๐(U; a) (GยทD); - the doubled quadratic form of a constant congruence is a finite linear combination of the
entries of
๐(U; a), hence inherits their integrability.
Coarse block of the adjoint recentred coefficient. For every sample a the coarse
block of the adjoint recentred coefficient aโ = aแต + g is the constant congruence, by the
shear G of the self-dual splitting and the flux reflection D = diag(Id, -Id), of the coarse
block of a. The adjoint transpose reflects the coarse block in the flux slot and the
recentring by the skew Schur half g shears it.
The pathwise response of the adjoint recentred coefficient. On the adapted cell the
pathwise response of aโ = aแต + g is the quadratic expression of the coarse block: the
coefficient is almost everywhere equal to a uniformly elliptic representative, and the response
and the coarse block depend only on the almost everywhere class.
Law integrability of the adjoint recentred pathwise response. For the adjoint recentred
coefficient aโ = aแต + g of a sample a, the pathwise response of the adapted cell is
integrable for the law as soon as the entries of the coarse block ๐(U; a) are: the recentring
is a constant congruence of that coarse block, and the quadratic form of a constant congruence
is a finite combination of its entries.
Law integrability of the pathwise response on an adapted cell #
Let P be a stationary law satisfying the coarse ellipticity assumption, and let U_u be the
adapted cell of generation u on a selected grid. This file proves that for each of the two
recentred coefficients a_- = a - g and a_+ = a^t + g of the self-dual recentring the pathwise
response J(U_u, p, q'; a_\pm) is P-integrable.
The route has three steps. The response of an elliptic field on a bounded open convex domain is
the fixed quadratic \tfrac12 (-p, q') \cdot \mathbf A(U; a)(-p, q') - p \cdot q' of the coarse
block; the coarse block of a recentred field is a deterministic congruence of the coarse block
of the sample, so its entries are fixed linear combinations of the entries of \mathbf A(U; a);
and those entries are P-integrable for every law satisfying the coarse ellipticity assumption.
The only dimensional hypothesis used is d \neq 0.
Paper: p.response.transfer, Step 3 and Step 6; the recentring is the one of e.self.dual.
Step 1: the coarse block of an adapted cell is law integrable in every dimension #
Every finite moment of the Schatten norm of the coarse block of an adapted cell is finite,
for any law satisfying the coarse ellipticity assumption. This is the statement of the
bounded-window source estimate with the dimensional hypothesis relaxed to d \neq 0, which is
all its proof uses.
Every entry of the coarse block of an adapted cell is P-integrable, in every dimension
d \neq 0.
The same at the selected grid of the response argument.
Square integrability of the recentred pathwise response #
On an adapted cell the pathwise response J(U_u; a_-(a)) of the recentred coefficient is the
fixed quadratic form AK.HC (2.15) of the coarse block ๐(U_u; a_-(a)), and that coarse block is
a deterministic congruence Gแต ๐(U_u; a) G of the coarse block of the sample. Hence the response
is an affine function of the entries of ๐(U_u; a), and the square of the response is a finite
combination of products of pairs of those entries.
This file records the two consequences used by the cutoff rows:
- each entry of
๐(U_u; ยท)is square integrable as soon as the coarse block lies in every Schatten momentL^N(S_N), because the operator norm โ and so each entry โ is dominated by the Schatten norm; - consequently the square of the pathwise response is
P-integrable for both recentringsa_- = a - ganda_+ = aแต + g, with the Schatten-moment hypothesis discharged from the stationary law and the coarse ellipticity assumption.
Paper: AK.HC (2.15), and the bounded-window source estimate for the Schatten moments.
Entrywise square integrability of the coarse block #
Square integrability of the pathwise response #
Square integrability of the recentred pathwise response. For the recentred coefficient
a_- = a - g of the self-dual splitting, whose coarse block is the fixed shear congruence
Gแต ๐(U_u; a) G, the square of the pathwise response is P-integrable as soon as the coarse
block of the sample lies in every Schatten moment. The response is the block quadratic
AK.HC (2.15) of that congruence, hence an affine function of the entries of ๐(U_u; ยท); each
entry is square integrable because it is dominated by the operator norm, which the Schatten norm
dominates.
Square integrability of the adjoint recentred pathwise response. For the adjoint recentred
coefficient a_+ = aแต + g, whose coarse block is the fixed congruence (GยทD)แต ๐(U_u; a) (GยทD),
the square of the pathwise response is P-integrable as soon as the coarse block of the sample
lies in every Schatten moment. The proof is the adjoint twin of
integrable_respJ_sq_respCoeffMinus.
Discharging the Schatten-moment hypothesis #
Square integrability of the recentred pathwise response from the law hypotheses. For a
stationary law satisfying the coarse ellipticity assumption, the coarse block of the response cell
lies in every Schatten moment L^N(S_N) by the bounded-window source estimate, so
integrable_respJ_sq_respCoeffMinus applies and the square of the recentred pathwise response is
P-integrable.
Square integrability of the adjoint recentred pathwise response from the law hypotheses.
The adjoint twin of integrable_respJ_sq_respCoeffMinus_of_dagger: the coarse block of the response
cell lies in every Schatten moment, and integrable_respJ_sq_respCoeffPlus applies.
Square integrability of the doubled optimizer state on an adapted cell #
The scale-average seminorm bound of e.response.weak.estimate is applied to the doubled
optimizer state M_0^{1/2}\bigl((\nabla v, a\nabla v) - Y\bigr) on an adapted cell. Its
hypothesis is that the squared Euclidean length of that state is integrable on the cell, and
this file supplies that hypothesis.
The gradient slot is square integrable because the optimizer is an H^1 function; the flux slot
is square integrable because the coefficient is uniformly elliptic on the cell after modification
on a null set, which is the form in which a qualitatively locally uniformly elliptic sample
provides ellipticity; the constant Y is square integrable because the cell has finite measure;
and a constant block matrix carries square integrable coordinates to square integrable
coordinates.
Paper: e.response.weak.estimate, and the normalization of the doubled state that precedes it.
Coordinatewise square integrability #
If every coordinate of a doubled field is square integrable, its squared Euclidean length is integrable.
A constant block matrix carries a doubled field with square integrable coordinates to a doubled field with square integrable coordinates.
The doubled optimizer state #
Both slots of the centred doubled optimizer field are square integrable on an adapted cell, for a coefficient that agrees almost everywhere on the cell with a uniformly elliptic field.
The scale-average seminorm of a recentred doubled field #
The scale averages that enter the weak quantity of e.response.weak.estimate are the cell
averages of a doubled field after an affine change Z \mapsto S(Z - Y) with a constant block
matrix S and a constant doubled vector Y. Because the cell average is linear, those averages
are the affine images of the cell averages of the field itself, so the Jensen/partition bound
applies verbatim to the recentred family.
The affine identity needs the two side conditions of an average over a cell: the cell has finite
nonzero volume, and each of the 2d coordinates of the field is integrable on it. Both are
supplied here from square integrability on the ambient cell.
Paper: e.response.weak.estimate.
Linearity of the cell average under a constant affine map #
The average of c (g - k) over a cell of finite nonzero volume is c times the average of
g minus k.
Linearity of the doubled cell average under the constant affine map Z \mapsto S(Z - Y).
The squared seminorm bound for a recentred family #
The two recentred response coefficients #
The singular-grid branch of the cutoff pairing bound #
The selected grid is respGrid jStar F = Geometry.explicitRoundedGrid jStar (explicitCanonicalMetric F), and
every entry of Geometry.explicitRoundedGrid carries a factor built from the inverse of its metric
argument. Mathlib's matrix inverse of a singular matrix is 0
(Matrix.nonsing_inv_apply_not_isUnit), so when the canonical metric is singular the selected
grid degenerates to the zero matrix and the adapted cell carries no Lebesgue volume. This file
records those degenerations: every volume average over a degenerate adapted cell is the junk value
0, and the determinant of the degenerate selected grid vanishes.
On a singular grid q the adapted cell HighContrast.adaptedCell q t is Lebesgue null, so
every
volume average over it is the junk value 0. Indeed
Geometry.volume_adaptedCell_toReal gives
(volume (HighContrast.adaptedCell q t)).toReal = |q.det| * ((3 : โ) ^ t) ^ d, which vanishes when
q.det = 0, and volumeAverage multiplies by the inverse of that volume.
Elementary facts about the weak response energy #
The weak response energy respWeakEnergy is the supremum, over all families of cell
maximizers of the response functional, of the weak energies
3^{-t} โซ [M_0^{1/2}(X_t - Y)]^2 dP. This file records the two facts used whenever it is
consumed as an upper bound: the supremum is nonnegative, and any single admissible family of
maximizers realizes a member of the defining set, so its weak energy lies below the supremum
once that set is bounded above. These are the elementary parts of the weak estimate
e.response.weak.estimate.
Every admissible family of cell maximizers realizes a member of the defining set of the
weak response energy W; consequently its weak energy is at most W whenever that set is
bounded above. This is the direction in which W is consumed as an upper bound.
The singular-grid branch of the annealed cutoff pairing bound #
The cutoff pairing of the response estimate is an expectation of absolute volume averages of an
adapted cell. When the selected grid is singular the adapted cell is Lebesgue null, so every such
average is the junk value 0 and the expectation vanishes. Together with the metric dichotomy
for the canonical metric, which always places an arbitrary doubled block on either the positive
definite branch or the singular-grid branch, this disposes of the branch on which the change of
variables underlying the cutoff argument is unavailable.
Paper: e.response.cutoff.estimate, e.scale.selection.canonical.metric.
On a singular grid q every adapted cell is Lebesgue null, so the absolute volume average of
any function over it is the junk value 0; consequently the integral of those absolute averages
against any measure vanishes. This is the degenerate branch of the cutoff pairing of
e.response.cutoff.estimate.
The canonical metric of an arbitrary doubled block is either positive definite or so
degenerate that the selected grid it produces is singular. No symmetry or positivity hypothesis is
needed, because the junk values of the square root and of the matrix inverse are themselves
positive semidefinite. This is the dichotomy of e.scale.selection.canonical.metric that selects
the branch of e.response.cutoff.estimate.