High-contrast homogenization: Entry.Response.Core.BlockAdjointCalculus #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Schur calculus for the centred block response #
This file develops the block algebra behind the centred-response identity
e.response.by.centered.energies, for a single symmetric positive doubled block. It proves the
Schur formulas expressing the block energy, the block mean and the centred response of a doubled
block in terms of its constituent pieces, together with the calculus of the block adjoint
K ↦ -K — its positivity, symmetry and skewness behaviour under conjugation — and the
cancellation of the basis trace the identity relies on. Every statement concerns a single
symmetric positive doubled block in isolation; none of it refers to a raw coefficient sample.
The block algebra of the centred responses #
The Schur formulas for the block energy, the block means and the centred response, the
block-adjoint calculus (K -> -K), and the basis-trace cancellation, all used by the
centred-response identity e.response.by.centered.energies
(p.response.transfer). Nothing here mentions RawOutput; every statement is
about a single symmetric positive doubled block.
blockAdjoint A negates the off-diagonal blocks of A and keeps its diagonal blocks.
The skew Schur coefficient flips sign under the block adjoint.
The Schur complement schurSigma is invariant under the block adjoint.
The block adjoint preserves symmetry.
The block adjoint preserves positive definiteness.
The block response mean blockResponseMean A (-p, q'), expanded through the Schur
cancellation identities schur_cancel into the skew and Schur-complement pieces of A.
The block response energy blockResponseEnergy A p q', expanded through the Schur
cancellation identities into the skew and Schur-complement pieces of A.
The centred response blockCenteredResponse A p q', expanded via
response_by_centered_energies_aux_energy and response_by_centered_energies_aux_mean.
The sum of the centred responses at (p, q - h_t p) and at the adjoint (p, q + h_t p),
expanded into the Schur-complement and respSym cross terms: the key algebraic identity behind
the centred-response identity.
The dual Schur complement schurSigmaStar A is Loewner-below the primal one
schurSigma A, given the swap-conjugate Loewner hypothesis ho.
A uniform bound |Jminus e| + |Jplus e| ≤ M at every unit basis vector sums, over the d
standard basis vectors, to at most d * M.
The basis sum of the Schur-complement and respSym cross terms is bounded by d * M,
combining response_by_centered_energies_aux_centered_sum with the uniform response bound
hM.
The basis sum ∑ᵢ (m^{-1/2} eᵢ). C (m^{1/2} eᵢ) equals trace C, for m positive
definite, by cyclicity of the trace and cancellation of the two square roots.
The metric dichotomy of the selected grid #
respGrid jStar F is 𝒬(m(F)), the rounded grid of the canonical metric of F. The metric
is positive semidefinite for every F -- no hypothesis -- because the junk values of
matSqrt and of Matrix.inv are positive semidefinite
(Geometry.explicitCanonicalMetric_posSemidef). So exactly one of two things happens: the metric
is
positive definite, which is the branch on which the pullback estimates apply, or it is
singular, and then the selected grid is the zero matrix, which is the branch on which the
adapted cell is Lebesgue-null. This is Step 0' of the response cutoff kernel; it is why that
kernel needs no positivity hypothesis on F.