High-contrast homogenization: Entry.Response.Rows.StationaryAnnealedErrorScalars #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Stationarity of the Annealed Block and the Cutoff Rows' Error Scalars #
For a stationary law and a scale above the threshold j_*, every aligned cell of a generation is
an integer translate of the centred adapted cell; since the law is translation invariant, the two
diagonal sub-blocks of the annealed coarse block of a_- = a - g or a_+ = a^T + g are the same
matrix at every cell of the generation, and the annealed response of that family is likewise
independent of the aligned cell. Consequently the error scalars respEJMinus/respTauMinus,
each a Bochner integral of the pathwise response, and respLsMinus, a Bochner integral of a
coarse-block entry, are algebraic functions of the annealed recentred block under its entrywise
integrability. This file also identifies the annealed flat average of one descendant generation's
squared pathwise head, built from the two diagonal quadratic forms of that generation's own
coarse block, with the corresponding expression built from the single annealed block. The
stationarity and these closed block forms are the ingredients consumed by the transfer step
p.response.transfer.
The annealed head of one descendant generation #
The source load L_s of p.response.transfer is the sum over the descendant generations of
3^{-3n/2} times the flat average over that generation's cells of (√α + √β)^2, where α and β
are the two diagonal quadratic forms of the annealed coarse block of the cell. The pairing of
the cutoff row produces instead the flat average of the same expression built from the pathwise
coarse blocks, followed by the expectation. Moving the expectation inside a single generation is
the content here: for each cell the expectation of (√α + √β)^2 is at most (√E[α] + √E[β])^2
by the sample Cauchy--Schwarz inequality, and the expectation of each pathwise coarse-block
quadratic form is the quadratic form of the annealed block. Averaging over the cells of the
generation therefore recovers that generation's own summand of L_s, with the expectation on the
inside and no inversion and no Jensen step in the wrong direction.
The annealed head of one descendant generation. The flat cell average over the generation
of 3^{-3n/2} subcells of (√α + √β)^2 integrates to at most the corresponding flat average of
(√E[α] + √E[β])^2, where α and β are the upper-left and lower-right pathwise coarse-block
quadratic forms and the square roots on the right are taken of their annealed counterparts. This is
the step that moves the expectation inside one generation's summand of the source load L_s of
p.response.transfer.
The annealed-block form of the two error scalars of the cutoff rows #
The cutoff rows of the response estimate bound the centred response by expressions in
respEJMinus, respTauMinus and respLsMinus. The first two are Bochner integrals of the
pathwise response; under the entrywise integrability hypotheses the cutoff rows carry, both are
algebraic functions of the annealed recentred blocks. This file records the two identities that
replace those integrals by the quadratic form of the annealed block.
The second identity is the underlying entrywise statement, e.annealed.schur: the quadratic form
of a doubled block commutes with the Bochner integral entrywise. The first identity is then the
source form of the centred response energy e.response.cutoff.estimate: the pathwise response
J(U_u, p, q'; b a) is the block energy x. A x / 2 - p. q', so its P-expectation is the
same block energy of the annealed block E[A].
The quadratic form commutes with the annealed expectation. For any doubled vector X,
the P-integral of the pathwise quadratic form X. (coarseBlockMatrix V (b a)) X equals the
quadratic form of the entrywise annealed block annealedBlockOf P V b, whenever every entry of
the pathwise block is P-integrable. This is the entrywise expectation identity E[A] of
e.annealed.schur.
The pathwise response integrates to the annealed block energy. If the pathwise response
is the doubled block energy x. A x / 2 - p. q' with x = (-p, q') and A the coarse block
matrix of the adapted cell, and if the entries of A are P-integrable, then the expectation of
the response is the same block energy evaluated at the annealed block Aann:
∫ J = x. Aann x / 2 - p. q'. This is the first step of the cutoff rows of
e.response.cutoff.estimate.
The annealed energies of the cutoff rows in closed block form #
The cutoff rows of the response estimate bound the centred response with the two error scalars
respEJMinus and respTauMinus. The first is the P-expectation of the recentred pathwise
response at one adapted cell; the second is the defect of that expectation between two scales.
Under the entrywise integrability of the coarse block carried by the cutoff rows, both are
algebraic functions of the recentred annealed block respEhatMinus: the pathwise response is the
block energy x · A x / 2 - p · q', so its expectation is the same block energy of the annealed
block E[A], and the pairing p · q' cancels in the defect. These are the closed block forms
against which the cutoff rows are stated (e.response.cutoff.estimate).
The annealed energy of the cutoff minus rows as a block quadratic. Under entrywise
integrability of the coarse block, the P-expectation of the recentred pathwise response
J(U_t, p, q⁻; a₋) is the quadratic form ½ x · E[A] x of the recentred annealed block
respEhatMinus at the load x = (-p, q⁻), minus the pairing p · q⁻. This is the closed block
form of the centred response energy e.response.cutoff.estimate.
The annealed response is the same on every aligned subcell #
The mean cancellation behind p.response.transfer — the constant cell means cancel after
expectation because the scale-s translations are integral — rests on the statement that,
for a stationary law, the annealed response of a recentred coefficient family on an aligned
cell of a fixed scale is independent of the particular aligned cell. The aligned cells of a
scale are integer translates of the centred cell, the annealed block of a translate is the
annealed block of the scale, and the pathwise response on any aligned cell is the block energy
of that cell's coarse block; hence the expectation is a function of the scale alone.
This module records that identity for both recentred families a_- and a_+.
Stationary aligned cells have a scale-determined annealed response, minus sign. For a
stationary law P and the recentred coefficient a_- = a - g of p.response.transfer, the
P-expectation of the pathwise response on the aligned cell adaptedCellAtCenter (respGrid jStar F) j w
is the block energy of the annealed block Gᵀ E_j G, independently of the cell index w.
Stationary aligned cells have a scale-determined annealed response, plus sign. For a
stationary law P and the recentred coefficient a_+ = aᵀ + g of p.response.transfer, the
P-expectation of the pathwise response on the aligned cell adaptedCellAtCenter (respGrid jStar F) j w
is the block energy of the annealed block G_+ᵀ E_j G_+, independently of the cell index w.
The annealed block of an aligned cell is independent of the cell #
For a stationary law P and a scale j above j_*, the annealed block of a recentred
coefficient family a_- = a - g or a_+ = aᵀ + g on an aligned cell of the adapted grid is
the same matrix at every cell of the scale. Every scale-j aligned cell is an integer
translate of the centred cell, the law is invariant under integer translations, and the
recentring subtracts (respectively adds) a constant matrix field, which commutes with
translation. This is the w-independence that makes the flat average over the 3^{H d}
subcells of the direct full-dual pairing collapse to the head term of respSourceLoad in
p.response.transfer.
The annealed block of the recentred field is the same on every aligned cell, minus
sign. For a stationary law P and the recentred coefficient a_- = a - g of
p.response.transfer, the P-entrywise annealed block of the pathwise coarse block on an
aligned cell of scale j ≥ j_* is the annealed block of the centred cell, independently of the
cell index w.
The annealed block of the recentred field is the same on every aligned cell, plus
sign. For a stationary law P and the recentred coefficient a_+ = aᵀ + g of
p.response.transfer, the P-entrywise annealed block of the pathwise coarse block on an
aligned cell of scale j ≥ j_* is the annealed block of the centred cell, independently of the
cell index w.