High-contrast homogenization: Provider.Response.CutoffGradientScale

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

The gradient scale of the adapted cutoff

The adapted cutoff is the canonical smooth product cutoff between two concentric reference cubes of relative radii 1 - 1/(2d) and 1 - 1/(4d), normalized to have reference mean one and pulled forward by the grid matrix.

Two quantitative facts about that construction are recorded here.

• Its normalizing average is at most one, because the unnormalized product cutoff is bounded by one. Consequently the normalized cutoff takes a value at least one on the inner cube and vanishes off the outer cube, and a one-dimensional mean value argument along a coordinate segment produces a point at which the adapted-coordinate gradient is at least the reciprocal collar width 8d·3^{-t}. Since 4(d+1) < 8d whenever d ≥ 2, no bound of the form ‖q∇φ‖ ≤ 4(d+1)·3^{-t} can hold for this cutoff.
• In the opposite direction the same construction obeys the sharper first derivative bound with coefficient 32 d² θ', where θ' is the derivative bound of the one-dimensional transition profile. This is the smallest coefficient the construction supports; the coefficient recorded downstream is the common second-order enlargement of it.

theorem HCPolySupport.HighContrast.Response.adaptedPreYoungCutoff_pullback_fderiv_bound_sharp {d : ℕ} [NeZero d] {q : Mat d} (hq : Matrix.PosDef q) (t : ℤ) (x : Vec d) : ‖fderiv ℝ (fun (y : Vec d) => adaptedPreYoungCutoff q hq t (matVecMul q y)) x‖ ≤ 32 * ↑d ^ 2 * smoothTransitionProfile.derivBound * 3 ^ (-t)

The smallest first-derivative coefficient the adapted construction supports: the reference gradient bound of the transition profile, doubled by normalization and multiplied by the reciprocal collar width.