High-contrast homogenization: Provider.Quenched.SmallContrastCenterSchur #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The calibrated profile centers in closed Schur form #
At the recentered samples, the two independently annealed profile centers of
the one-step hub are the closed Schur pairs of the terminal adapted mean: the
recentered mean has Schur data (S, σ*, k_sym), and the reflected-center
identities evaluate to the gradient component -p + σ*⁻¹(r + k_sym p) and
the flux component (1 - k_sym σ*⁻¹) r - (S + k_sym σ*⁻¹ k_sym) p, with the
sign of k_sym flipped for the adjoint.
theorem
HCPolySupport.HighContrast.Quenched.profilePrimalCenter_calibrated_eq
{d : ℕ}
{P : MeasureTheory.Measure (CoeffSpace d)}
[MeasureTheory.IsProbabilityMeasure P]
{q : Mat d}
(hq : Matrix.PosDef q)
(t : ℤ)
(hint : HasFiniteAdaptedMean P q t)
{S SStar K : Mat d}
(hform : toFullBlockMat (adaptedMean P q t) = schurBlock S SStar K)
(p r : Vec d)
:
Response.profilePrimalCenter P hq t (fun (a : CoeffSpace d) => a.subSkew (Response.responseSkew K) ⋯) p r = (-p + SStar⁻¹.mulVec (r + Matrix.mulVec (Response.responseSymmetric K) p), (1 - Response.responseSymmetric K * SStar⁻¹).mulVec r - Matrix.mulVec (S + Response.responseSymmetric K * SStar⁻¹ * Response.responseSymmetric K) p)
The primal center in closed Schur form.
theorem
HCPolySupport.HighContrast.Quenched.profileAdjointCenter_calibrated_eq
{d : ℕ}
{P : MeasureTheory.Measure (CoeffSpace d)}
[MeasureTheory.IsProbabilityMeasure P]
{q : Mat d}
(hq : Matrix.PosDef q)
(t : ℤ)
(hint : HasFiniteAdaptedMean P q t)
{S SStar K : Mat d}
(hform : toFullBlockMat (adaptedMean P q t) = schurBlock S SStar K)
(p r : Vec d)
:
Response.profileAdjointCenter P hq t (fun (a : CoeffSpace d) => a.subSkew (Response.responseSkew K) ⋯) p r = (-p + SStar⁻¹.mulVec (r - Matrix.mulVec (Response.responseSymmetric K) p), (1 + Response.responseSymmetric K * SStar⁻¹).mulVec r - Matrix.mulVec (S + Response.responseSymmetric K * SStar⁻¹ * Response.responseSymmetric K) p)
The adjoint center in closed Schur form.