High-contrast homogenization: Provider.Response.CenteredResponseCarriers

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

Centered primal and adjoint response carriers

For a coefficient law and one bounded convex domain, the centered response is the annealed response energy minus one half of the product of the two separately annealed optimizer averages. The adjoint response is evaluated at the transpose of the same coefficient sample; no pushforward of the law is used.

The remaining carriers are the skew and symmetric parts of the Schur coordinate of the annealed block, the corrected response block, its geometric-mean metric, and the two metric-adapted loads.

noncomputable def HCPolySupport.HighContrast.Response.centeredResponseOptimizer {d : ℕ} (U : Book.Ch02.Domain d) (a : CoeffSpace d) (p q : Vec d) : Book.Ch02.Solution U (a.coeffOn U)

The canonical response optimizer v_{p,q} on an arbitrary Chapter 2 domain.

Equations

• HCPolySupport.HighContrast.Response.centeredResponseOptimizer U a p q = (HCPolySupport.Book.Ch02.canonicalMaximizer ⋯ p q).toSolution

theorem HCPolySupport.HighContrast.Response.centeredResponseOptimizer_isMaximizer {d : ℕ} (U : Book.Ch02.Domain d) (a : CoeffSpace d) (p q : Vec d) : Book.Ch02.IsResponseMaximizer U (a.coeffOn U) p q (centeredResponseOptimizer U a p q)

The chosen primal optimizer realizes the response supremum.

noncomputable def HCPolySupport.HighContrast.Response.centeredAdjointOptimizer {d : ℕ} (U : Book.Ch02.Domain d) (a : CoeffSpace d) (p q : Vec d) : Book.Ch02.Solution U (a.transpose.coeffOn U)

The canonical adjoint optimizer v*_{p,q} at the same sample.

Equations

• HCPolySupport.HighContrast.Response.centeredAdjointOptimizer U a p q = HCPolySupport.HighContrast.Response.centeredResponseOptimizer U a.transpose p q

theorem HCPolySupport.HighContrast.Response.centeredAdjointOptimizer_isMaximizer {d : ℕ} (U : Book.Ch02.Domain d) (a : CoeffSpace d) (p q : Vec d) : Book.Ch02.IsResponseMaximizer U (a.transpose.coeffOn U) p q (centeredAdjointOptimizer U a p q)

The chosen adjoint optimizer realizes the adjoint response supremum.

noncomputable def HCPolySupport.HighContrast.Response.annealedOptimizerGradient {d : ℕ} (P : MeasureTheory.Measure (CoeffSpace d)) (U : Book.Ch02.Domain d) (p q : Vec d) : Vec d

The annealed spatial average of the primal optimizer gradient. As for the annealed block, the finite-dimensional expectation is defined entrywise.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def HCPolySupport.HighContrast.Response.annealedOptimizerFlux {d : ℕ} (P : MeasureTheory.Measure (CoeffSpace d)) (U : Book.Ch02.Domain d) (p q : Vec d) : Vec d

The annealed spatial average of the primal optimizer flux.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def HCPolySupport.HighContrast.Response.annealedAdjointOptimizerGradient {d : ℕ} (P : MeasureTheory.Measure (CoeffSpace d)) (U : Book.Ch02.Domain d) (p q : Vec d) : Vec d

The annealed spatial average of the adjoint optimizer gradient.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def HCPolySupport.HighContrast.Response.annealedAdjointOptimizerFlux {d : ℕ} (P : MeasureTheory.Measure (CoeffSpace d)) (U : Book.Ch02.Domain d) (p q : Vec d) : Vec d

The annealed spatial average of the adjoint optimizer flux.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def HCPolySupport.HighContrast.Response.centeredResponse {d : ℕ} (P : MeasureTheory.Measure (CoeffSpace d)) (U : Book.Ch02.Domain d) (p q : Vec d) : ℝ

The deterministic centered primal response J_tilde_a(U;p,q).

Equations

• One or more equations did not get rendered due to their size.

noncomputable def HCPolySupport.HighContrast.Response.centeredAdjointResponse {d : ℕ} (P : MeasureTheory.Measure (CoeffSpace d)) (U : Book.Ch02.Domain d) (p q : Vec d) : ℝ

The deterministic centered adjoint response J_tilde_{aᵗ}(U;p,q).

Equations

• One or more equations did not get rendered due to their size.

noncomputable def HCPolySupport.HighContrast.Response.responseSkew {d : ℕ} (K : Mat d) : Mat d

The skew part h_rsp = 1/2 (K-Kᵗ) of the annealed Schur coordinate.

Equations

• HCPolySupport.HighContrast.Response.responseSkew K = 2⁻¹ • (K - Matrix.conjTranspose K)

noncomputable def HCPolySupport.HighContrast.Response.responseSymmetric {d : ℕ} (K : Mat d) : Mat d

The symmetric part k_rsp = 1/2 (K+Kᵗ) of the annealed Schur coordinate.

Equations

• HCPolySupport.HighContrast.Response.responseSymmetric K = 2⁻¹ • (K + Matrix.conjTranspose K)

noncomputable def HCPolySupport.HighContrast.Response.centeredResponseBlock {d : ℕ} (S SStar K : Mat d) : Mat d

The corrected response block b_rsp = S + k_rspᵗ S_*⁻¹ k_rsp.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def HCPolySupport.HighContrast.Response.centeredResponseMetric {d : ℕ} (S SStar K : Mat d) : Mat d

The response metric m_rsp = b_rsp # S_*.

Equations

• HCPolySupport.HighContrast.Response.centeredResponseMetric S SStar K = HCPolySupport.HighContrast.matGeomMean (HCPolySupport.HighContrast.Response.centeredResponseBlock S SStar K) SStar

noncomputable def HCPolySupport.HighContrast.Response.centeredResponseLoadP {d : ℕ} (S SStar K : Mat d) (e : Vec d) : Vec d

The response-coordinate load p_e = m_rsp^{-1/2} e.

Equations

• HCPolySupport.HighContrast.Response.centeredResponseLoadP S SStar K e = (HCPolySupport.HighContrast.matSqrt (HCPolySupport.HighContrast.Response.centeredResponseMetric S SStar K)⁻¹).mulVec e

noncomputable def HCPolySupport.HighContrast.Response.centeredResponseLoadQ {d : ℕ} (S SStar K : Mat d) (e : Vec d) : Vec d

The response-coordinate load q_e = m_rsp^{1/2} e.

Equations

• HCPolySupport.HighContrast.Response.centeredResponseLoadQ S SStar K e = (HCPolySupport.HighContrast.matSqrt (HCPolySupport.HighContrast.Response.centeredResponseMetric S SStar K)).mulVec e