High-contrast homogenization: Entry.Response.Direct.TerminalEnergyDeficitBound #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The Terminal-Optimizer Replacement Engine #
At a response maximizer the pathwise response is half the unweighted variation energy, so
replacing the terminal-cell optimizer's energy by the pathwise response leaves exactly the (φ - 1)-weighted optimizer energy, the cost of the terminal-optimizer replacement in the centred
cutoff decomposition of AK.HC (3.45)-(3.54). The competitor-splitting identity ∇v · b ∇v − ∇w · b ∇w = (∇v − ∇w) · symmPart(b) (∇v + ∇w) together with Cauchy-Schwarz
for the positive semidefinite averaged form symmPart(a) bounds the energy defect between two
competitors by the square roots of their averaged difference and sum energies. Restricting the
terminal optimizer to a subcell gives an admissible competitor there, and combining these facts
bounds the discrepancy between the terminal optimizer's half energy and a subcell's response by
that subcell's deficit, |½ ⨍_V ⟨∇u, symmPart(a) ∇u⟩ - J(V)| ≤ D + 2√(J(V) · D).
Paper: the terminal-optimizer replacement of p.response.transfer.
The congruence toolbox of the almost-everywhere elliptic representative #
A point of the coefficient carrier is elliptic only almost everywhere, while the variational
identities of the response are stated for a pointwise elliptic coefficient. The elliptic
representative supplies, for each sample, a pointwise elliptic field agreeing almost everywhere
with the carrier coefficient on the parent domain, and Response.aHarmonicOfAEEq carries a
harmonic function along that replacement. This module records the invariances the first error row
of p.response.transfer still needs: restriction of ellipticity to a measurable subset,
invariance of ResponseJ on a subset, the averaged response and the ψ-weighted variation energy
of a carried-along harmonic function, and the transport of their IntegrableOn facts.
Ellipticity restricts to a measurable subset.
The response ResponseJ on a subset is unchanged by an almost-everywhere replacement of the
coefficient on the parent.
The averaged response integrand of a carried-along harmonic function, on a subset.
The averaged ψ-weighted variation energy of a carried-along harmonic function, on a subset:
weighting by an arbitrary scalar function changes nothing in the transport.
An IntegrableOn fact transports across the carried-along harmonic function, for the
response integrand of the parent, restricted to a subset.
The cutoff energy defect as a (φ − 1)-weighted energy #
At a response maximizer the pathwise response J(U; p, r; b) of AK.HC (2.9) is half the
unweighted variation energy. Consequently, in the centred cutoff decomposition of
AK.HC (3.45)-(3.54), replacing the terminal optimizer energy by the pathwise response leaves
exactly the (φ − 1)-weighted optimizer energy: the cost of the terminal-optimizer replacement
is the fluctuation of the optimizer energy against the cutoff, and φ − 1 has mean zero on the
cell where φ has mean one.
cutoffHalfEnergy_sub_respJ_eq— the first error rowe1: the terminal-optimizer replacement costs exactly the(φ − 1)-weighted energy.volumeAverage_sub_one_eq_zero— a cutoff of mean one has mean-zero fluctuation.
Paper: AK.HC (3.45)-(3.54).
The cutoff energy defect e1. For a response maximizer v of the loads (p, r) in the
adapted cell ⋄_t^q, the cutoff-weighted half optimizer energy differs from the pathwise
response J(U; p, r; b) by half the (φ − 1)-weighted optimizer energy. This is the exact
algebraic form of the first error row of the centred cutoff decomposition AK.HC (3.45)-(3.54):
the terminal-optimizer replacement costs the (φ − 1)-weighted energy, and φ − 1 has mean
zero on the cell. The side conditions are the integrability hypotheses of the CoarseGraining
energy–response identity together with the integrability of the weighted energy.
Mean-zero fluctuation of a mean-one cutoff. If φ averages to 1 over a cell of finite
nonzero volume, then φ − 1 averages to 0.
The competitor split of the weighted optimizer energy #
The first error row of the cutoff estimate of p.response.transfer replaces the terminal
optimizer energy of a cell by the scale-s cell optimizers. The pathwise algebra behind that
replacement is the identity
∇v · b ∇v − ∇w · b ∇w = (∇v − ∇w) · symmPart b (∇v + ∇w),
which holds because the quadratic form only sees the symmetric part of b. Combined with the
Cauchy–Schwarz inequality for the positive semidefinite symmetric part, it bounds the
(φ − 1)-weighted energy defect of two competitors by the square roots of the averaged
difference and sum energies.
abs_vecDot_matVecMul_le_sqrt_mul_sqrt— Cauchy–Schwarz for a symmetric positive semidefinite matrix.- the pathwise competitor identity for the optimizer field.
- the weighted energy defect bound.
Paper: the first error row of p.response.transfer.
Cauchy–Schwarz for a symmetric positive semidefinite matrix. For a real matrix S with
matTranspose S = S whose quadratic form is nonnegative, the pairing u · S w is bounded by the
square roots of the two diagonal energies u · S u and w · S w, i.e.
|u · S w| ≤ √(u · S u) · √(w · S w).
Averaged Cauchy--Schwarz in the coefficient metric #
The terminal-optimizer comparison of p.response.transfer pairs one gradient field against
another in the metric symmPart a, and controls the pairing by the two diagonal energies through
Cauchy--Schwarz for the averaged bilinear form. This module records that averaged inequality on a
set carrying a pointwise elliptic coefficient, in the normalized-average form the estimate uses.
abs_volumeAverage_vecDot_symmPart_le— averaged Cauchy--Schwarz forsymmPart a.
Paper: p.response.transfer.
Averaged Cauchy–Schwarz in the coefficient metric. On a set U carrying a coefficient
a elliptic pointwise, the volume average of the symmPart a-pairing of two vector fields is
bounded by the geometric mean of the averages of their two diagonal energies:
|⨍_U f · symmPart a g| ≤ √(⨍_U f · symmPart a f) · √(⨍_U g · symmPart a g). No hypothesis on
the measure of U is needed: the normalizing constant (volume U).toReal⁻¹ is nonnegative and
factors through both sides, so the degenerate cases are covered as well.
The restricted terminal optimizer and its subcell responses #
The terminal-optimizer replacement of p.response.transfer restricts the optimizer of the terminal
cell to each aligned subcell of the coarse scale and compares the restricted response value there
with the subcell's own response. The response integrand and the variation-energy integrand are
pointwise expressions in the coefficient and the gradient of the harmonic function, so restricting
the function to a subdomain leaves both integrands unchanged as functions. Exact partition
averaging then expresses the terminal response value as the flat average of the restricted response
values over the aligned subcells; the entire scale defect therefore sits in the gap between each
subcell's own response and the restricted one.
Restricting a harmonic function to a subdomain leaves the scalar response integrand unchanged.
Both integrands are pointwise expressions in the coefficient and the gradient, and the restricted
harmonic function has the same gradient as the parent, so the two functions agree. This is the
integrand identity used in the terminal-optimizer replacement of p.response.transfer.
Restricting a harmonic function to a subdomain leaves the scalar variation-energy integrand
unchanged. The integrand is a pointwise expression in the coefficient and the gradient, and the
restricted harmonic function has the same gradient as the parent. This is the energy identity used
in the terminal-optimizer replacement of p.response.transfer.
Exact partition averaging at a response maximizer: the flat average over the depth-n aligned
subcells of the parent response integrand is the parent response value, and at a maximizer that
value is the terminal response J. Thus no response is lost in the subdivision; the entire scale
defect sits in the gap between each subcell's own response and the restricted one. This is the
bookkeeping identity of the terminal-optimizer replacement of p.response.transfer.
The subcell response deficit controls the terminal optimizer's energy #
The terminal-optimizer replacement of p.response.transfer compares the energy of the terminal
optimizer with the subcell responses. Let V ⊆ U, let u optimize the response problem on the
large cell U, and let v optimize the same load on the small cell V. Restricted to V, u
is an admissible competitor there, so its response value on V falls short of J(V) by the
nonnegative deficit
D = J(V) - ⨍_V g_V(u).
The quadratic-response identity makes the symmPart(a)-energy of ∇u - ∇v on V exactly 2 D.
Splitting the two energies and bounding the cross term by Cauchy--Schwarz against
⨍_V ⟨∇v, symmPart(a) ∇v⟩ = 2 J(V) bounds the terminal optimizer's energy against the subcell
response:
|½ ⨍_V ⟨∇u, symmPart(a) ∇u⟩ - J(V)| ≤ D + 2 √(J(V) · D).
This is the pathwise engine of the terminal-optimizer replacement of p.response.transfer.
Paper: p.response.transfer.
The subcell deficit controls the terminal optimizer's energy. Let V ⊆ U be open sets,
a a coefficient elliptic on V, v a response maximizer on V for the load (p, r), and u
any a-harmonic function on U. Writing J = J(V; p, r; a) for the subcell response and
D = J - ⨍_V g_V(u) for the deficit of the restricted parent u against it, the terminal
optimizer's energy on V is controlled by the subcell response through
|½ ⨍_V ⟨∇u, symmPart(a) ∇u⟩ - J| ≤ D + 2 √(J · D).
The difference energy of ∇u - ∇v is 2 D by the quadratic-response identity, and the cross
term ⨍_V ⟨∇v, symmPart(a)(∇u - ∇v)⟩ is bounded by averaged Cauchy--Schwarz against the two
diagonal energies 2 J and 2 D. The side conditions are the integrability hypotheses of the
difference-energy identity, the energy--response identity at v, the cross integrability, and the
parent's energy integrability.