High-contrast homogenization: Provider.PolynomialHomogenization.NormalizedAffineResponse

Imported from the Apache-2.0 HighContrastHomogenization development at commit 7a13dbcd8d6609264a713373f5c69ceeac870472.

Affine covariance of the doubled response

An exact positive-definite affine map transports a cube coefficient family to an adapted cell. The doubled response is covariant only after transforming the primal and dual loads in their distinct contragredient ways.

theorem HCPolySupport.HighContrast.doubledResponseJ_affineResponseCell {d : ℕ} {q : Mat d} (hq : Matrix.PosDef q) (t k : ℤ) (w : Fin d → ℤ) (a : CoeffSpace d) (aRef : Book.Ch03.CoeffFamily d) (haRef : ∀ (Q : TriadicCube d), (aRef.coeffOn Q).toCoeffField = affineCoefficient q ⋯ (a.coeffOn (Response.adaptedDomain hq t)).toCoeffField) (P Q : BlockVec d) : Book.Ch02.doubledResponseJ (Book.Ch02.cubeDomain (translateCube w (originCube d k))) (aRef.coeffOn (translateCube w (originCube d k))) (affineReferencePrimalLoad q P) (affineReferenceDualLoad q Q) = Book.Ch02.doubledResponseJ (Response.adaptedDomainAt hq k w) (a.coeffOn (Response.adaptedDomainAt hq k w)) P Q

Doubled affine covariance on an aligned cell, with the distinct primal and dual transformed loads.