High-contrast homogenization: Provider.Recurrence.StationarityTransport #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Translation covariance of the coarse block and the stationarity transport #
This is the translation covariance of the variational coarse block, among the
properties taken from HC in s.introduction: the pair
๐(U + z; a) = ๐(U; T_z a), ๐_*(U + z; a) = ๐_*(U; T_z a), z โ โค^d,
and the consequence drawn from it, that under stationarity of the coefficient law the translated blocks have the same law and hence the same expectation whenever that expectation exists.
The pathwise identity is a change of variables in the variational definition of
the coarse block (s.introduction): translating a competitor is a bijection
between the admissible classes over U + z and over U that preserves the
energy average, so the two infima agree. The coefficient field of the
coefficient space is an almost everywhere class, and the translation action T_z
of the coefficient-law setup in s.introduction is the class of a(ยท + z); the
coarse response sees only the class, so the identity holds at every sample, with
no null set left
over. The sharp half of the display is the image of the first half under the
primal-adjoint involution H โฆ R Hโปยน R of s.scale.selection, which is how
๐_* is written here.
The annealed half is the change of variables E[g โ T_z] = E[g] under the law,
whose only input beyond stationarity of the coefficient law is that the entries
of the coarse response are measurable for the law, another of the properties
taken from HC. The printed hedge "whenever they are finite" is the integrability
guard: it transports too, so an aligned cell inherits the finiteness of the mean
of the cell at the origin.
At the aligned adapted cells of a rounded geometry (s.scale.selection) the
translation vectors are integral, so the transport applies to them directly. The resulting identity
E[A_j^q(z)] = E_j^q at every aligned center is what the averaging step of
p.fixed.geometry.parent.child.recurrence consumes when it replaces the annealed block of
each child by the mean at the parent scale, together with its averaged form
E[G] = E_j^q.
Translates of a set #
The translated sample #
The field of a translated sample is the translate of its field. The
translation action T_z on the coefficient space acts on the almost everywhere
class; this is the identity of representatives it is built from.
The pathwise covariance #
๐(U + z; a) = ๐(U; T_z a). The first half of the translation
covariance of the coarse block, at every sample of the coefficient space.
The annealed transport #
The integer translations preserve the law. This is stationarity of the coefficient law under integer translations, restated as the measure-preserving property of the translation action, the form in which a change of variables applies to an arbitrary statistic of the coefficient field.
The translated blocks have the same law. This is the first conclusion drawn from the pathwise covariance, read at each entry of the doubled block.
The expectation of an entry of the coarse response is unchanged by an integer
translation of the cell. This is the change of variables E[g โ T_z] = E[g]
under stationarity of the coefficient law.
๐_bar(U + z) = ๐_bar(U). The consequence drawn from the pathwise
covariance: under stationarity of the coefficient law the annealed blocks of a
cell and of its integer translate agree.
The finiteness hedge of the translation clause transports: if the coarse response over a cell is integrable for the law, so is the coarse response over every integer translate of it.
The aligned adapted cells #
An aligned adapted cell with integral center is an integer translate of the cell at the origin.
A_r^q(z; a) = A_r^q(0; T_z a). The translation covariance of the
coarse block at an aligned adapted cell: the adapted response at an aligned
center is the response at the origin of the translated sample.
The same identity with the integral translation vector supplied by the
lattice alignment of the rounded grid in s.scale.selection.
E[A_j^q(z)] = E_j^q. The adapted means at aligned translated cells
coincide: this is the identity the averaging step of
p.fixed.geometry.parent.child.recurrence consumes when it replaces the annealed block of
each child by the mean at the parent scale.
The finiteness of the adapted mean transports to every aligned cell at the
same scale, so the guard HasFiniteAdaptedMean at the origin is the only one the
averaging step needs.