High-contrast homogenization: Provider.Recurrence.AdaptedCellPositivity #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The pathwise symmetry and positivity of the response over adapted cells #
The first of the coarse-block properties taken from HC asserts that the coarse
block response 𝐀(U; a) of a Lipschitz cell is, pathwise in the coefficient
field, a symmetric positive definite doubled matrix. For the adapted cubes
⋄_j^q = q□_j of a rounded geometry and their aligned translates
3^j q w + ⋄_j^q this file proves both halves.
Symmetry needs no hypothesis at all: the coarse block response of any subset
of ℝ^d is symmetric, the off-diagonal entries being the polarization of the
variational quantity.
Positivity is the deterministic coarse-graining theory of a bounded open convex domain, transported to the adapted cell. A rounded adapted grid is invertible, so the adapted cell is a nonempty bounded open convex domain, hence a chapter 2 domain; the coarse response of a field of the coefficient space on it is then the chapter 2 coarse block matrix of that field read as a coefficient object, which is positive definite.
Two consequences are recorded for the fixed-grid recurrence. The annealed block
of an adapted cell inherits positivity as soon as the response is integrable, so
the positivity binders on the adapted means E_j^q follow from the finiteness
binders. And the Schur ordering σ_* ≤ σ becomes available on every adapted
cell, as it already was on every centered cube.
The pathwise symmetry clause #
The coarse response on an adapted cell is symmetric, pathwise in the
coefficient field. This is the symmetry half of the first of the coarse-block
properties taken from HC; it holds on every subset of ℝ^d.
The coarse response on an aligned adapted cell is symmetric, pathwise in the coefficient field.
The adapted response A_r^q(z) is symmetric, in the carrier the
fixed-grid estimates are written in.
The pathwise positivity clause #
The coarse response of a field of the coefficient space is positive definite on every bounded open convex nonempty domain: it is the chapter 2 coarse block matrix of the field read as a coefficient object on that domain.
The coarse response on an adapted cell is positive definite, pathwise in the coefficient field. This is the positivity half of the first of the coarse-block properties taken from HC, at the cells the fixed-grid recurrence is written on.
The coarse response on an aligned adapted cell is positive definite, pathwise in the coefficient field.
The adapted response is positive definite, in the carrier the fixed-grid estimates are written in.
The pathwise positivity clause at a rounded adapted grid: no positivity hypothesis on the grid is needed, since a rounded grid is positive definite.
The annealed block of an adapted cell #
The adapted mean E_r^q is positive definite as soon as it is an
expectation at all. The pathwise positivity of the response integrates, so the
positivity binders of p.fixed.geometry.parent.child.recurrence follow from its finiteness
binders on a rounded adapted grid.
The adapted mean of a rounded adapted grid is positive definite as soon as
it is finite: the grid's own positivity is part of IsRoundedGrid.
The annealed block of an aligned adapted cell is positive definite as soon as the coarse response is integrable over the law.