High-contrast homogenization: Provider.Response.LoadCalibrationLoads #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The exact energy sum of the two signed loads #
The boxed display of the response-load calibration:
(L_t^-)² + (L_t^+)² = 4 eᵗ(m^{1/2}S_*^{-1}m^{1/2}) e ≤ 4√θ,
with L_t^∓ = |Ê_t^{1/2}x^∓|, Ê_t = 𝐆₀ᵗE_t𝐆₀, and the two signed profile
vectors x^- = (-p, q^-), x^+ = (p, q^+), q^∓ = q ± (g₀ - h)p,
p = m^{-1/2}e, q = m^{1/2}e, m = B # S_*.
The proof is the printed one, through the skew-recentered blocks and the
coefficient transpose. Recentering by 𝐆₀ replaces
q^∓ by q ∓ hp, so the Schur factorization gives
(L_t^-)² = ⟨Sp, p⟩ + ⟨S_*⁻¹(q + rp), q + rp⟩,
(L_t^+)² = ⟨Sp, p⟩ + ⟨S_*⁻¹(q - rp), q - rp⟩,
whose sum is 2⟨Bp, p⟩ + 2⟨S_*⁻¹q, q⟩ because the two cross terms cancel and
⟨S_*⁻¹rp, rp⟩ = ⟨rᵗS_*⁻¹r p, p⟩. The Riccati characterization of the metric
geometric mean gives mS_*⁻¹m = B, so both summands equal
eᵗ(m^{1/2}S_*^{-1}m^{1/2})e and the sum is four times it.
For the bound, the printed proof passes through the eigenvalues of
S_*^{-1/2}mS_*^{-1/2}. The route here is the equivalent Loewner one, which
needs no spectral theory: joint monotonicity and homogeneity of the geometric
mean turn B ≤ θS_* into m = B # S_* ≤ (θS_*) # S_* = √θ S_*, and inverting
and conjugating by m^{1/2} turns that into
m^{1/2}S_*^{-1}m^{1/2} ≤ √θ I.
There are no definitions in this file.
Square roots against the matrix #
m^{-1/2}m = m^{1/2}.
m m^{-1/2} = m^{1/2}.
The two signed loads #
The negative load in Schur coordinates. Recentering by
𝐆₀ replaces q^- by q - hp, and the Schur factorization then produces
q + rp in the second slot.
The positive load in Schur coordinates, through the Riccati characterization of the matrix geometric mean.
The exact sum #
The exact energy sum:
(L_t^-)² + (L_t^+)² = 4 eᵗ(m^{1/2}S_*^{-1}m^{1/2})e.
The bound by the response ratio #
m ≤ √θ S_* from B ≤ θ S_*, by joint monotonicity and homogeneity of
the metric geometric mean.
The exact sum and its bound, the boxed display: for a
unit vector e the two signed loads have energy sum
4 eᵗ(m^{1/2}S_*^{-1}m^{1/2})e, and that is at most 4√θ.