| import OAI.Geometry.Perimeter.Regularity.EllipticContraction |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal ContDiff InnerProductSpace |
| namespace RPISingularHarnack |
| variable {n : ℕ} |
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| 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)] |
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| 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⟩ |
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| end RPISingularHarnack |
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| end |
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| end OAI |
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