High-contrast homogenization: Entry.Response.Pairing.CoarseBlockFenchelPairing #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Coarse-Block Toolkit for the Direct Fenchel Pairing #
On a bounded open convex domain carrying a pointwise elliptic coefficient, the canonical coarse
block is the Hessian of an infimum of averages of a nonnegative density, hence positive
semidefinite, so its two diagonal quadratic forms are nonnegative, in particular for a_β = respCoeffβ F a; this file records their sample integrability with that of their crossed
product of square roots and the square of their sum. It records the pathwise mean identity of
AK.HC (2.32), computing the cell average of the doubled optimizer state as x + R π(V) x, and
shows every such average is measurable in the sample, since Chapter-2 gradient uniqueness
identifies each maximizer with the canonical selection. Combining these, it proves the direct
full-dual Fenchel pairing of AK.HC, Lemma A.1, (A.4): the cell average, paired against Y = (P, Q), is controlled by the square root of the cell's own coarse-block energy of Y. This is the
coarse-block estimate consumed by p.response.transfer.
Sample integrability of the quadratic readouts of a coarse block #
The cell half and the oscillation half of the cutoff-mean row of p.response.transfer read the
pathwise coarse block only through four scalars: the two diagonal quadratic forms against the
deterministic dual variable, their crossed product of square roots, and the square of their sum.
All four are integrable as soon as the entries of the block are, because the quadratic form is a
fixed real linear combination of the entries and the crossed product of square roots is dominated
by the arithmetic mean of the two forms.
This module records those four steps, together with the two bookkeeping facts the rows consume about flat averages of integrable families and about domination by an absolute value.
Domination by an integrable envelope. A measurable function whose modulus is bounded by an integrable function is integrable.
Integrability of a flat average of an integrable family. A normalized finite sum of
P-integrable functions is P-integrable.
The crossed product of square roots of two nonnegative integrable functions is integrable. The product is dominated by the arithmetic mean of the two functions, by the nonnegativity of the square of the difference of the square roots.
The crossed product of square roots is integrable without a sign hypothesis. The square root of a negative number is zero, so the product is bounded by the crossed product of the square roots of the moduli, and the previous domination applies to those.
The square of the sum of the two square roots is integrable. Expanding the square gives the two functions plus twice their crossed product of square roots.
The quadratic form of a matrix family against a fixed vector is integrable. The form is
the fixed real linear combination β i β k, Y i * M i k * Y k of the entries.
Positive semidefiniteness of the coarse block of a cell #
On a bounded open convex domain carrying a pointwise elliptic coefficient field, the canonical
coarse block matrix is the Hessian of the minimal block energy Mu with prescribed doubled mean.
That energy is an infimum of averages of a nonnegative density, hence nonnegative, so the block is
positive semidefinite. In particular its two diagonal blocks β the b and the S_*^{-1} appearing
in e.response.cutoff.estimate β are positive semidefinite, which are the side conditions of the
direct full-dual pairing.
The coarse block of a cell is positive semidefinite: for every doubled vector X the
quadratic form X Β· (blockMatVecMul (coarseBlockMatrix V b) X) is nonnegative.
The upper-left diagonal block of the coarse block matrix is positive semidefinite: the block
quadratic form of (p, 0) is p Β· (upperLeft p).
The lower-right diagonal block of the coarse block matrix is positive semidefinite: the block
quadratic form of (0, r) is r Β· (lowerRight r).
Nonnegativity of the two diagonal coarse-block forms of the recentred families #
The source-load head of p.response.transfer is the sum of the square roots of the two diagonal
coarse-block quadratic forms of the recentred coefficient on an aligned cell. Both forms are
nonnegative because the coarse block of an elliptic coefficient is positive semidefinite. The
recentred coefficients a_β = respCoeffβ F a are elliptic only almost everywhere, so each
statement is proved at a pointwise elliptic representative on the aligned cell and transported
back, the coarse block being unchanged by an almost-everywhere replacement of the coefficient.
The two diagonal coarse-block forms of a_- = a - g are nonnegative on every aligned
cell. Both entries of the source-load head of p.response.transfer are well defined on every
aligned cell of the response grid.
The two diagonal coarse-block forms of a_+ = aα΅ + g are nonnegative on every aligned
cell. The adjoint twin of
zero_le_vecDot_coarseBlockMatrix_respCoeffMinus_adaptedCellAtCenter.
The optimizer mean identity on an aligned subcell at an almost-everywhere elliptic coefficient #
The pathwise mean identity AK.HC (2.32) computes the cell mean of the doubled optimizer state of
a response maximizer as x + R π(V) x with x = (-p, r) and π(V) the coarse block of the cell.
The response coefficients of p.response.transfer are elliptic only almost everywhere, while the
identity is available at a pointwise elliptic coefficient. Carrying the harmonic field along an
almost everywhere elliptic representative preserves the maximizer property and the optimizer field
up to a null set, so the identity transfers verbatim to the aligned subcell.
The optimizer mean identity AK.HC (2.32) on an aligned adapted subcell adaptedCellAtCenter q j w,
at a coefficient b that agrees almost everywhere there with a pointwise elliptic representative
f: the cell average of the doubled optimizer field of any response maximizer is
(-p, r) + R π(adaptedCellAtCenter q j w; b) (-p, r).
Cell averages of the doubled optimizer field are measurable in the sample #
The doubled optimizer field of a response maximizer for the recentred coefficients a_- and
a_+ is determined, up to a null set of the response cell, by the coefficient sample alone:
Chapter-2 almost-everywhere gradient uniqueness identifies every response maximizer with the
canonical selection. Consequently every coordinate of the cell average of the doubled optimizer
field over a measurable subcell is a measurable function of the sample, for an arbitrary family of
maximizers. These are the subcell averages entering the weak quantity of
e.response.weak.estimate.
Each coordinate of the cell average of the doubled optimizer field of an arbitrary family of
response maximizers for the recentred coefficient a_- is measurable in the coefficient sample.
The field need not be measurable in the sample, but its cell average is determined by the
measurable canonical selection through Chapter-2 a.e. gradient uniqueness, so the subcell average
entering the weak quantity of e.response.weak.estimate is measurable.
Each coordinate of the cell average of the doubled optimizer field of an arbitrary family of
response maximizers for the recentred coefficient a_+ is measurable in the coefficient sample.
This is the plus twin of measurable_cellAverage_optimizerField_respCoeffMinus.
The direct full-dual pairing of a cell average against a state #
The direct Fenchel pairing of AK.HC, Lemma A.1, display (A.4): the cell average of the doubled
optimizer field, paired across its two slots with a state Y = (P, Q), is controlled by the square
root of the cell's own coarse-block energy of Y β read through the two diagonal blocks of the
coarse block, b in the gradient slot and S_*^{-1} in the flux slot, exactly the two blocks from
which the source load L_s^{Β±} is built β times the square root of the pathwise symmetric energy of
the field. The pairing is Y.2 against the cell-average gradient plus Y.1 against the
cell-average flux, the full-dual form consumed by the cutoff-mean rows.
Algebraic core of the full-dual pairing AK.HC (A.4): a real number X satisfying the Fenchel
inequality lam * X β€ lam^2 A / 2 + E / 2 for every real lam, with A β₯ 0, obeys
X^2 β€ A E.
Gradient half of the direct full-dual pairing AK.HC (A.4): the cell average of the gradient of
a b-harmonic field, tested against r, is bounded by the square root of the lower-right coarse
block form of r times the square root of the pathwise symmetric energy of the field.
Flux half of the direct full-dual pairing AK.HC (A.4): the cell average of the flux of a
b-harmonic field, tested against p, is bounded by the square root of the upper-left coarse
block form of p times the square root of the pathwise symmetric energy of the field.
The direct full-dual pairing AK.HC (A.4): the cell average of the doubled optimizer field,
paired across the two slots with a state Y = (P, Q), is bounded by the sum of the square roots
of the two diagonal coarse-block forms of Y times the square root of the pathwise symmetric
energy of the field.