High-contrast homogenization:

Provider.PolynomialHomogenization.Root.RowSupply.EpsilonAffineDoubledResponseRealization

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

Exact doubled-response realization on a translated observation cube

The observation coefficient is identified locally with the microscopic affine pullback. Affine, scalar, skew, and translation covariance then give an exact physical response on the corresponding translated adapted cell.

theorem HCPolySupport.HighContrast.RowSupply.translateSet_adaptedCellAt_eq_translate {d : ℕ} (z : Vec d) (q : Mat d) (k : ℤ) (w : Fin d → ℤ) : translateSet z (adaptedCellAt q k w) = adaptedCellTranslate q k (z + adaptedCellCenter q k w)

theorem HCPolySupport.HighContrast.RowSupply.doubledResponseJ_observation_epsilonAffine_eq {d : ℕ} [NeZero d] {epsilon : ℝ} (hepsilon : 0 < epsilon) (center : Vec d) (a : CoeffSpace d) (abar : Mat d) (hS : Matrix.PosDef (symmPart abar)) (k : ℤ) (w : Fin d → ℤ) (aObs : Book.Ch02.CoeffOn (Book.Ch02.cubeDomain (translateCube w (originCube d k)))) (hObs : aObs.toCoeffField =ᵐ[volumeMeasureOn (openCubeSet (translateCube w (originCube d k)))] fun (x : Vec d) => affineCoefficient (matSqrt (symmPart abar)) ⋯ (fun (y : Vec d) => scaledCoeff epsilon a y - skewPart abar) (x + center)) (P Q : BlockVec d) : Book.Ch02.doubledResponseJ (Book.Ch02.cubeDomain (translateCube w (originCube d k))) aObs (normalizedReferencePrimalLoad abar P) (normalizedReferenceDualLoad abar Q) = Transport.coeffSpaceDoubledResponse (adaptedCellTranslate (epsilonAffineGrid epsilon abar) k (epsilon⁻¹ • matVecMul (matSqrt (symmPart abar)) center + adaptedCellCenter (epsilonAffineGrid epsilon abar) k w)) a P Q

Exact realization of a locally identified observation response as the physical response on the microscopic affine image of the cube.