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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | import OAI.Geometry.Perimeter.Regularity.EllipticContraction
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
theorem linear_iteration_scalar {K H a B s Q E E' d : ℝ}
(hK : 1 ≤ K) (hH : 0 ≤ H) (ha : 0 ≤ a) (hB : 0 ≤ B) (hs : 0 ≤ s)
(hsmall : 32*K^2*a ≤ 1) (hforce : 16*K*B ≤ H)
(hQ0 : 0 ≤ Q) (hQ : Q ≤ 2*K*H) (hE : E ≤ H^2*s^2)
(hd0 : 0 ≤ d) (hd : d^2 ≤ K*E)
(hnext : E' ≤ E/16+K*(a*s*Q+B*s)^2) :
d ≤ K*H*s ∧ E' ≤ H^2*(s/2)^2 := by
have hK0 : 0 ≤ K := by linarith
have hKH : 0 ≤ K*H*s := by positivity
have hdE := mul_le_mul_of_nonneg_left hE hK0
have hKK : K ≤ K^2 := by nlinarith
have hdE' := mul_le_mul_of_nonneg_right hKK (mul_nonneg (sq_nonneg H) (sq_nonneg s))
have hd' : d ≤ K*H*s := (sq_le_sq₀ hd0 hKH).mp (by nlinarith)
have hsmallH := mul_le_mul_of_nonneg_right hsmall hH
have hAQ := mul_le_mul_of_nonneg_left hQ (mul_nonneg hK0 ha)
have hbase : K*(a*Q+B) ≤ H/8 := by nlinarith
have hb : K*(a*s*Q+B*s) ≤ H*s/8 := by
have hh := mul_le_mul_of_nonneg_right hbase hs
nlinarith
have hb0 : 0 ≤ K*(a*s*Q+B*s) := by positivity
have hb2 := pow_le_pow_left₀ hb0 hb 2
have hb3 := mul_le_mul_of_nonneg_right hKK (sq_nonneg (a*s*Q+B*s))
refine ⟨hd',?_⟩
nlinarith [sq_nonneg (H*s)]
theorem linear_energy_iteration {θ δ K a B H R : ℝ}
(hθ : 0 < θ) (hθ1 : θ < 1) (hK : 1 ≤ K)
(hc : LinearContractionProperty n θ δ K)
(ha : 0 ≤ a) (hB : 0 ≤ B) (hH : 0 ≤ H) (hR : 0 < R)
(hsmall : 32*K^2*a ≤ 1) (hforce : 16*K*B ≤ H)
{A : Euc n → Euc n →L[ℝ] Euc n} (hA : Continuous A)
{f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) {F : Euc n → Euc n} (hF : Continuous F)
(p : Euc n) (h : WeakLinearEquation A f F (Metric.ball p R))
(hnear : ∀ x∈Metric.ball p R, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖ ≤ δ)
(hosc : ∀ j x, x∈Metric.ball p (θ^j*R) →
‖A p-A x‖ ≤ a*(1/2)^j ∧ ‖F x-F p‖ ≤ B*(1/2)^j)
(hinit : scaledGradientEnergy f p R 0 ≤ H^2) :
∃ q : ℕ → Euc n, q 0=0 ∧
(∀ j, ‖q j‖ ≤ 2*K*H*(1-(1/2)^j)) ∧
(∀ j, scaledGradientEnergy f p (θ^j*R) (q j) ≤ H^2*((1/2)^j)^2) ∧
(∀ j, ‖q (j+1)-q j‖ ≤ K*H*(1/2)^j) := by
have hball (j : ℕ) : Metric.ball p (θ^j*R) ⊆ Metric.ball p R :=
Metric.ball_subset_ball (by nlinarith [pow_le_one₀ (n := j) hθ.le hθ1.le])
have hstep (j : ℕ) (q : Euc n) : ∃ q' : Euc n,
‖q'-q‖^2 ≤ K*scaledGradientEnergy f p (θ^j*R) q ∧
scaledGradientEnergy f p (θ^(j+1)*R) q' ≤ scaledGradientEnergy f p (θ^j*R) q/16+
K*(a*(1/2)^j*‖q‖+B*(1/2)^j)^2 := by
have hh := hc A hA f hf F hF p (θ^j*R) (by positivity) (h.mono (hball j))
(A p) (F p) q (a*(1/2)^j) (B*(1/2)^j) (by positivity) (by positivity)
(fun x hx ↦ ⟨hnear x (hball j hx),(hosc j x hx).1,(hosc j x hx).2⟩)
simpa only [pow_succ, mul_assoc,mul_left_comm,mul_comm] using hh
choose next hnext using hstep
let q : ℕ → Euc n := fun j ↦ Nat.rec 0 (fun j z ↦ next j z) j
have hq0 : q 0=0 := rfl
have hqnext (j : ℕ) : q (j+1)=next j (q j) := rfl
have hbounds : ∀ j, ‖q j‖ ≤ 2*K*H*(1-(1/2)^j) ∧
scaledGradientEnergy f p (θ^j*R) (q j) ≤ H^2*((1/2)^j)^2 ∧
‖q (j+1)-q j‖ ≤ K*H*(1/2)^j := by
intro j
induction j with
| zero =>
have hinit' : scaledGradientEnergy f p (θ^0*R) (q 0) ≤ H^2*((1/2)^0)^2 := by
simpa only [hq0,pow_zero,one_pow,one_mul,mul_one] using hinit
obtain ⟨hd,_⟩ := linear_iteration_scalar hK hH ha hB (by positivity : 0 ≤ (1/2:ℝ)^0)
hsmall hforce (norm_nonneg (q 0)) (by simp only [hq0,norm_zero]; positivity) hinit'
(norm_nonneg (next 0 (q 0)-q 0)) (hnext 0 (q 0)).1 (hnext 0 (q 0)).2
exact ⟨by simp only [hq0,norm_zero,pow_zero,sub_self,mul_zero,le_refl],hinit',by simpa only [hqnext] using hd⟩
| succ j ih =>
have hqnorm : ‖q j‖ ≤ 2*K*H := ih.1.trans (by
have hh : 0 ≤ 2*K*H*(1/2)^j := by positivity
nlinarith)
obtain ⟨_,hE⟩ := linear_iteration_scalar hK hH ha hB (by positivity : 0 ≤ (1/2:ℝ)^j)
hsmall hforce (norm_nonneg (q j)) hqnorm ih.2.1
(norm_nonneg (next j (q j)-q j)) (hnext j (q j)).1 (hnext j (q j)).2
have hE' : scaledGradientEnergy f p (θ^(j+1)*R) (q (j+1)) ≤ H^2*((1/2)^(j+1))^2 := by
simpa only [hqnext,pow_succ,div_eq_mul_inv,one_div,one_mul] using hE
have hn : ‖q (j+1)‖ ≤ 2*K*H*(1-(1/2)^(j+1)) := by
have htri := norm_add_le (q j) (q (j+1)-q j)
have he : q j + (q (j+1)-q j) = q (j+1) := by abel
rw [he] at htri
rw [pow_succ]
nlinarith [ih.1,ih.2.2]
have hn' : ‖q (j+1)‖ ≤ 2*K*H := hn.trans (by
have hh : 0 ≤ 2*K*H*(1/2)^(j+1) := by positivity
nlinarith)
obtain ⟨hd,_⟩ := linear_iteration_scalar hK hH ha hB (by positivity : 0 ≤ (1/2:ℝ)^(j+1))
hsmall hforce (norm_nonneg (q (j+1))) hn' hE'
(norm_nonneg (next (j+1) (q (j+1))-q (j+1))) (hnext (j+1) (q (j+1))).1 (hnext (j+1) (q (j+1))).2
exact ⟨hn,hE',by simpa only [hqnext] using hd⟩
exact ⟨q,hq0,fun j ↦ (hbounds j).1,fun j ↦ (hbounds j).2.1,fun j ↦ (hbounds j).2.2⟩
end RPISingularHarnack
end
end OAI
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