High-contrast homogenization: Provider.Quenched.SmallContrastMetricFactor

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

The weak-norm metric factor of the small-contrast estimate

The weak value of the small-contrast one-step estimate carries the metric factor

K_{M,F} = |M_0^{-1/2} F M_0^{-1/2}|^{1/2}, M_0 = diag(m_0, m_0^{-1}),

at F the recentered reference G_{h_0}^t (c 𝐄) G_{h_0} and at F the recentered terminal adapted mean. This file bounds the first one by data of the reference alone — the printed M_0 is built from the reference, so the factor must not grow with the generation.

The route is the canonical factorization M(𝐄) = G_{-g(𝐄)}^t M_0 G_{-g(𝐄)} at m_0 = m(𝐄) (e.scale.selection.canonical.metric) together with the two-sided bound 𝐄 ≤ 𝔡(𝐄)^{1/2} M(𝐄) between a positive block and its canonical metric: the two shears compose, so the whole factor is the imbalance times the distortion of the single shear h_0 - g(𝐄), and that distortion is quadratic in the size τ of m^{-1/2}(h_0 - g(𝐄))m^{-1/2}.

The g(𝐄) half of the shear is removed once and for all by the recentering of SmallContrastCanonRecenter (canonicalShear_recentered_eq_zero), which is the normalization of Section 2.5 of HC; the remaining h_0 half is the response skew of the terminal mean, controlled in SmallContrastMetricFactorTau.

Two elementary quadratic-form facts

Shears compose

theorem HCPolySupport.HighContrast.Quenched.skewBlockCongr_skewBlockCongr {d : ℕ} (a b : Mat d) (A : BlockMat d) : Response.skewBlockCongr a (Response.skewBlockCongr b A) = Response.skewBlockCongr (b + a) A

Shear congruences compose additively.

The shear distortion of the split metric

theorem HCPolySupport.HighContrast.Quenched.blockMatLoewnerLE_skewBlockCongr_diagonalMetric {d : ℕ} {m h : Mat d} (hm : Matrix.PosDef m) {tau : ℝ} (htau : ∀ (x : Vec d), vecDot (matVecMul h x) (matVecMul m⁻¹ (matVecMul h x)) ≤ tau ^ 2 * vecDot x (matVecMul m x)) : BlockMatLoewnerLE (Response.skewBlockCongr h (Book.Ch02.blockDiag m m⁻¹)) (blockScale (2 + 2 * tau ^ 2) (Book.Ch02.blockDiag m m⁻¹))

The distortion of the split metric under a shear. If the shear h has m-normalized size at most τ, the shear congruence of M_0 = diag(m,m^{-1}) is at most (2 + 2τ²) M_0. This is the only place the shear enters the metric factor.

The metric factor at the reference

theorem HCPolySupport.HighContrast.Quenched.blockMatLoewnerLE_skewBlockCongr_reference {d : ℕ} {E : BlockMat d} (hE : IsSymmetricBlockMat E) (hEpd : Book.Ch02.BlockPosDef E) (hsharp : BlockMatLoewnerLE (blockSharp E) E) (h : Mat d) {c : ℝ} (hc : 0 ≤ c) {tau : ℝ} (htau : ∀ (x : Vec d), vecDot (matVecMul (h - canonicalShear E) x) (matVecMul (canonicalMetric E)⁻¹ (matVecMul (h - canonicalShear E) x)) ≤ tau ^ 2 * vecDot x (matVecMul (canonicalMetric E) x)) : BlockMatLoewnerLE (Response.skewBlockCongr h (blockScale c E)) (blockScale (c * √(kappaRef E) * (2 + 2 * tau ^ 2)) (Book.Ch02.blockDiag (canonicalMetric E) (canonicalMetric E)⁻¹))

The reference metric factor, squared. With the adapted-cube metric m_0 = m(𝐄) — the canonical metric of the reference — the M_0-size of the recentered dilated reference is at most c · κ_𝐄^{1/2} · (2 + 2τ²), where τ bounds the m_0-normalized size of the total shear h - g(𝐄).

The generation never appears: the bound is reference data times the shear distortion.

theorem HCPolySupport.HighContrast.Quenched.blockSize_skewBlockCongr_reference_le {d : ℕ} {E : BlockMat d} (hE : IsSymmetricBlockMat E) (hEpd : Book.Ch02.BlockPosDef E) (hsharp : BlockMatLoewnerLE (blockSharp E) E) (h : Mat d) {c : ℝ} (hc : 0 ≤ c) {tau : ℝ} (htau : ∀ (x : Vec d), vecDot (matVecMul (h - canonicalShear E) x) (matVecMul (canonicalMetric E)⁻¹ (matVecMul (h - canonicalShear E) x)) ≤ tau ^ 2 * vecDot x (matVecMul (canonicalMetric E) x)) : blockSize (Response.skewBlockCongr h (blockScale c E)) (Book.Ch02.blockDiag (canonicalMetric E) (canonicalMetric E)⁻¹) ≤ c * √(kappaRef E) * (2 + 2 * tau ^ 2)

The reference metric factor, squared. With the adapted-cube metric m_0 = m(𝐄) the M_0-size of the recentered dilated reference is at most c · κ_𝐄^{1/2} · (2 + 2τ²). The generation never appears.