High-contrast homogenization: Provider.Recurrence.SchattenIdeal #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The ideal property of the Schatten size #
The fixed-grid recurrence changes the normalization of the averaged response
from the child mean E_j to the parent mean E_p through the exact congruence
E_p^{-1/2}(G-E_j)E_p^{-1/2} = C^t(E_j^{-1/2}(G-E_j)E_j^{-1/2})C with
C = E_j^{1/2}E_p^{-1/2}, and then transports the Schatten size across it. The
result is the averaged fluctuation after the change of normalization, in the
proof of p.fixed.geometry.parent.child.recurrence, whose constant is allowed to
depend on the dimension and on the exponent.
The transport is the ideal property of the Schatten norm, read through the
scalar Loewner sandwich. A symmetric block H is caught between
∓|H|_{S_Q}I; congruence preserves the Loewner order, so C^tHC is caught
between ∓|H|_{S_Q}C^tC; and C^tC is the relative mean
P = E_p^{-1/2}E_jE_p^{-1/2} of the determinant transport, which lies below
e^{Δ}I. So C^tHC is caught between ∓|H|_{S_Q}e^{Δ}I, and a block caught
between ∓tI has Schatten size at most (2d)^{1/Q}t. The chain costs the
dimensional factor (2d)^{1/Q}, which is the shape of the printed constant
C(d,Q); nothing downstream sees the loss.
The exact ideal inequality |C^tHC|_{S_Q} ≤ |C^tC|·|H|_{S_Q} is stronger than
what is proved here, and is not available by this argument: it asks the spectrum
of C^tHC to be dominated by that of H in the sense of majorization, whereas
the Loewner order only bounds each eigenvalue separately.
The final section carries the pointwise estimate to the mixed norm
‖·‖_{L^Q(S_Q)}, which is where the recurrence uses it: combined with the
averaging estimate of l.fixed.geometry.matrix.averaging at the child
normalization it produces the printed display at the parent normalization.
The normalized block #
The flattening of the F-normalization is the literal congruence by the
inverse square root.
The transported Schatten size #
The ideal property of the Schatten size, in the normalization the
fixed-grid recurrence uses. Changing the normalization of a symmetric block
from the child mean to the parent mean multiplies its Schatten size by at most
(2d)^{1/Q}e^{Δ}, the determinant increment entering through the Loewner bound
P ≤ e^{Δ}I on the relative mean.
The transported average #
The ideal property of the Schatten size in the mixed norm. A pointwise
inequality between Schatten sizes passes to the L^Q norms of a random doubled
block.
The averaged fluctuation after the change of normalization.
The averaged centered response is estimated at the child normalization by
l.fixed.geometry.matrix.averaging; the ideal property carries that estimate
to the parent normalization at the cost of the determinant increment and a
constant depending only on the dimension and the exponent.