File size: 10,708 Bytes
159357f | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 | import Mathlib
import OAI.Analysis.CoulombIonization.Localization.PosteriorAnnularQueryBarrier
namespace OAI
noncomputable section
namespace CoulombBarrier
section
open MeasureTheory Set Metric
open scoped NNReal ContDiff
open CoulombAtom
attribute [local irreducible] masterWidth masterKernel
lemma masterWidth_relative_upper_half {c₁ r₀ s : ℝ} (hc : 0 ≤ c₁)
(hr₀ : 0 < r₀) (hs : 0 < s) (hs1 : s ≤ 1) {y : Space}
(hy : r₀ ≤ 2*‖y‖) : masterWidth c₁ r₀ s y ≤ 2*c₁*‖y‖ := by
have hd : masterBaseDistance r₀ y ≤ 2*‖y‖ :=
max_le (by linarith [norm_nonneg y]) hy
have hp := Real.rpow_le_one hs.le hs1 masterExponent_nonneg
calc
_ ≤ c₁*masterBaseDistance r₀ y*s^masterExponent := masterWidth_upper hc hr₀ hs y
_ ≤ c₁*masterBaseDistance r₀ y*1 :=
mul_le_mul_of_nonneg_left hp (mul_nonneg hc (le_max_of_le_left (norm_nonneg y)))
_ ≤ _ := by nlinarith [mul_le_mul_of_nonneg_left hd hc]
lemma masterKernel_ne_zero_expanded_annular {g : Space → ℝ}
(hgs : tsupport g ⊆ ball 0 1) {B c₁ r₀ s r : ℝ} (hB : 1 ≤ B) (hc : 0 < c₁)
(hcL : c₁ < (10*(100000:ℝ))⁻¹) (hr₀ : 0 < r₀) (hs : 0 < s)
(hs1 : s ≤ 1) (hrr : r₀ ≤ 2*r) {x y : Space} (hy : r ≤ ‖y‖)
(hyB : ‖y‖ ≤ B*r) (hxy : masterKernel c₁ r₀ s g x y ≠ 0) :
r/2 ≤ ‖x‖ ∧ ‖x‖ ≤ 2*B*r := by
have hr : 0 < r := by linarith
rw [masterKernel_eq_widthKernel hc hr₀ hs g] at hxy
have hm := widthKernel_support_local (masterWidth_small_lipschitz hc hcL hr₀ hs hs1)
(masterWidth_pos hc hr₀ hs) (by norm_num) (masterProfile_support hgs) y hxy
simp only [mem_closedBall,dist_eq_norm] at hm
have ht := masterWidth_relative_upper_half hc.le hr₀ hs hs1 (by linarith : r₀ ≤ 2*‖y‖)
have hc2 : c₁ < 1/8 := by norm_num at hcL ⊢; linarith
have hd : ‖x-y‖ ≤ ‖y‖/2 := by nlinarith [norm_nonneg y]
have hl := norm_sub_norm_le y x
rw [norm_sub_rev y x] at hl
have hu := norm_sub_norm_le x y
constructor
· nlinarith only [hl,hd,hy]
· calc
‖x‖ ≤ (3/2:ℝ)*‖y‖ := by linarith only [hu,hd]
_ ≤ (3/2:ℝ)*(B*r) := mul_le_mul_of_nonneg_left hyB (by norm_num)
_ ≤ 2*(B*r) := mul_le_mul_of_nonneg_right (by norm_num)
(mul_nonneg (zero_le_one.trans hB) hr.le)
_ = 2*B*r := by ring
theorem exists_master_expanded_annular_query_constant {g : Space → ℝ}
(hg : ContDiff ℝ ∞ g) (hgs : tsupport g ⊆ ball 0 1) {B c₁ : ℝ} (hB : 1 ≤ B)
(hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹) :
∃ C : ℝ, 0 ≤ C ∧ ∀ {r₀ s r : ℝ}, 0 < r₀ → 0 < s → s ≤ 1 →
r₀ ≤ 2*r → r ≤ s → ∀ x y z : Space,
r ≤ ‖y‖ → ‖y‖ ≤ B*r → r ≤ ‖z‖ → ‖z‖ ≤ B*r →
‖masterKernel c₁ r₀ s g x y-masterKernel c₁ r₀ s g x z‖ ≤
C*r^(-4-4*masterExponent)*‖y-z‖*
(if r/2 ≤ ‖x‖ ∧ ‖x‖ ≤ 2*B*r then 1 else 0) := by
have hcomp : HasCompactSupport (fun q => (g q)^2) :=
HasCompactSupport.of_support_subset_isCompact (isCompact_closedBall (0 : Space) 1)
(masterProfile_support hgs)
obtain ⟨H,hH⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hcomp (hg.pow 2) (by simp)
refine ⟨(H:ℝ)*16*c₁⁻¹^4,by positivity,?_⟩
intro r₀ s r hr₀ hs hs1 hrr hrs x y z hy hy2 hz hz2
have hr : 0 < r := by linarith
have hzero : ‖masterKernel c₁ r₀ s g x y-masterKernel c₁ r₀ s g x z‖ = 0 →
‖masterKernel c₁ r₀ s g x y-masterKernel c₁ r₀ s g x z‖ ≤
((H:ℝ)*16*c₁⁻¹^4)*r^(-4-4*masterExponent)*‖y-z‖*
(if r/2 ≤ ‖x‖ ∧ ‖x‖ ≤ 2*B*r then 1 else 0) := by
intro he
rw [he]
split_ifs <;> positivity
by_cases ha : masterKernel c₁ r₀ s g x y = 0 ∧ masterKernel c₁ r₀ s g x z = 0
· exact hzero (by rw [ha.1,ha.2,sub_self,norm_zero])
have hex : masterKernel c₁ r₀ s g x y ≠ 0 ∨ masterKernel c₁ r₀ s g x z ≠ 0 := by tauto
have hax : r/2 ≤ ‖x‖ ∧ ‖x‖ ≤ 2*B*r := hex.elim
(fun h => masterKernel_ne_zero_expanded_annular hgs hB hc hcL hr₀ hs hs1 hrr hy hy2 h)
(fun h => masterKernel_ne_zero_expanded_annular hgs hB hc hcL hr₀ hs hs1 hrr hz hz2 h)
rw [ite_eq_left hax,mul_one]
have hlow : c₁*r^(1+masterExponent)/2 ≤ masterWidth c₁ r₀ s x := by
have haux (q : Space) (hq : r ≤ ‖q‖) (hxq : masterKernel c₁ r₀ s g x q ≠ 0) :
c₁*r^(1+masterExponent)/2 ≤ masterWidth c₁ r₀ s x := by
rw [masterKernel_eq_widthKernel hc hr₀ hs g] at hxq
exact (div_le_div_of_nonneg_right (masterWidth_radius_lower hc.le hr hrs hq) (by norm_num)).trans
(widthKernel_ne_zero_width_lower (masterWidth_small_lipschitz hc hcL hr₀ hs hs1)
(masterWidth_pos hc hr₀ hs) (masterProfile_support hgs) hxq)
exact hex.elim (haux y hy) (haux z hz)
have hinv := pow_le_pow_left₀ (inv_nonneg.mpr (masterWidth_pos hc hr₀ hs x).le)
(inv_anti₀ (by positivity : 0 < c₁*r^(1+masterExponent)/2) hlow) 4
rw [inverse_fourth_radius_scale hc hr] at hinv
rw [masterKernel_eq_widthKernel hc hr₀ hs g]
exact (widthKernel_query_difference (masterWidth_pos hc hr₀ hs) hH x y z).trans
((mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hinv H.coe_nonneg)
(norm_nonneg _)).trans_eq (by ring))
end
open MeasureTheory ProbabilityTheory Filter Set Metric
open scoped BigOperators Topology ContDiff
open CoulombAtom CoulombObservation
attribute [local irreducible] masterWidth masterKernel originalRawKernel physicalObservationLaw
originalDatum jointMasterPosterior
lemma posterior_density_le_raw_annular_count {N K : ℕ} (μ : Measure (Configuration N))
[IsFiniteMeasure μ] (ell : Fin K → ℝ) (j : ℕ) {c₁ r₀ s : ℝ}
(hc : 0 < c₁) (hr : 0 < r₀) (hs : 0 < s) {g : Space → ℝ}
(hg : Continuous g) (y : Space) {a b A : ℝ} (hA : 0 ≤ A)
(hbound : ∀ x, masterKernel c₁ r₀ s g x y ≤ A)
(hsupport : ∀ x, masterKernel c₁ r₀ s g x y ≠ 0 → a ≤ ‖x‖ ∧ ‖x‖ ≤ b)
(d : OriginalDatum N K ell j) :
jointMasterPosterior μ ell j c₁ r₀ s g d y ≤
A * ∫ x, rawAnnularCount a b x ∂originalRawKernel μ ell j d := by
let f : Configuration N → ℝ := fun x => ∑ i, masterKernel c₁ r₀ s g (x i) y
have hm : Measurable f := Finset.measurable_sum _ (fun i _ =>
((masterKernel_joint_continuous hc hr hs hg).measurable.comp
(f := fun x : Configuration N => (x i,y))
((measurable_pi_apply i).prodMk measurable_const)))
have hn (x) : 0 ≤ f x := Finset.sum_nonneg fun i _ => masterKernel_nonneg _ _ _ _ _ _
have hb (x) : f x ≤ A*rawAnnularCount a b x := by
rw [rawAnnularCount,Finset.mul_sum]
apply Finset.sum_le_sum
intro i _
split_ifs with h
· simpa only [mul_one] using hbound (x i)
· have hz : masterKernel c₁ r₀ s g (x i) y = 0 := by
by_contra hh
exact h (hsupport (x i) hh)
simp only [hz,mul_zero,le_refl]
have hfb (x) : ‖f x‖ ≤ A*N := by
rw [Real.norm_of_nonneg (hn x)]
exact (hb x).trans (mul_le_mul_of_nonneg_left (rawAnnularCount_le a b x) hA)
have hci : Integrable (rawAnnularCount a b) (originalRawKernel μ ell j d) := by
apply Integrable.of_bound (rawAnnularCount_measurable a b).aestronglyMeasurable (N:ℝ)
exact ae_of_all _ (fun x => by
rw [Real.norm_of_nonneg (rawAnnularCount_nonneg a b x)]; exact rawAnnularCount_le a b x)
unfold jointMasterPosterior
calc
_ ≤ ∫ x, A*rawAnnularCount a b x ∂originalRawKernel μ ell j d :=
integral_mono (Integrable.of_bound hm.aestronglyMeasurable _ (ae_of_all _ hfb))
(hci.const_mul _) hb
_ = _ := integral_const_mul _ _
theorem exists_posterior_expanded_annular_density_constant {g : Space → ℝ} (hg : Continuous g)
(hgs : tsupport g ⊆ ball 0 1) {B c₁ : ℝ} (hB : 1 ≤ B) (hc : 0 < c₁)
(hcL : c₁ < (10*(100000:ℝ))⁻¹) :
∃ D : ℝ, 0 ≤ D ∧ ∀ {N K : ℕ} (μ : Measure (Configuration N)),
∀ [IsFiniteMeasure μ] (ell : Fin K → ℝ) (j : ℕ) {r₀ s r : ℝ},
0 < r₀ → 0 < s → s ≤ 1 → r₀ ≤ 2*r → r ≤ s →
∀ (d : OriginalDatum N K ell j) (y : Space), r ≤ ‖y‖ → ‖y‖ ≤ B*r →
jointMasterPosterior μ ell j c₁ r₀ s g d y ≤
D*r^(-3-3*masterExponent) *
∫ x, rawAnnularCount (r/2) (2*B*r) x ∂originalRawKernel μ ell j d := by
obtain ⟨A,hAn,hA⟩ := masterKernel_amplitude hg hgs
refine ⟨A*c₁⁻¹^3,by positivity,?_⟩
intro N K μ hμ ell j r₀ s r hr₀ hs hs1 hrr hrs d y hy hyB
have hr : 0 < r := by linarith
let D := A*c₁⁻¹^3*r^(-3-3*masterExponent)
have hDn : 0 ≤ D := by dsimp only [D]; positivity
have hb (x) : masterKernel c₁ r₀ s g x y ≤ D :=
(hA hc hcL hr₀ hs hs1 x y).trans (by
simpa only [D,mul_assoc] using mul_le_mul_of_nonneg_left
(masterWidth_inverse_cube_radius (r₀ := r₀) hc hr hrs hy) hAn)
exact posterior_density_le_raw_annular_count μ ell j hc hr₀ hs hg y hDn hb
(fun x hx => masterKernel_ne_zero_expanded_annular hgs hB hc hcL hr₀ hs hs1 hrr hy hyB hx) d
theorem exists_posterior_expanded_annular_query_constant {g : Space → ℝ}
(hg : ContDiff ℝ ∞ g) (hgs : tsupport g ⊆ ball 0 1) {B c₁ : ℝ} (hB : 1 ≤ B)
(hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹) :
∃ C : ℝ, 0 ≤ C ∧ ∀ {N K : ℕ} (μ : Measure (Configuration N)),
∀ [IsFiniteMeasure μ] (ell : Fin K → ℝ) (j : ℕ) {r₀ s r : ℝ},
0 < r₀ → 0 < s → s ≤ 1 → r₀ ≤ s → r₀ ≤ 2*r → r ≤ s →
∀ (d : OriginalDatum N K ell j) (y z : Space),
r ≤ ‖y‖ → ‖y‖ ≤ B*r → r ≤ ‖z‖ → ‖z‖ ≤ B*r →
‖jointMasterPosterior μ ell j c₁ r₀ s g d y-
jointMasterPosterior μ ell j c₁ r₀ s g d z‖ ≤
C*r^(-4-4*masterExponent)*‖y-z‖*
∫ x, rawAnnularCount (r/2) (2*B*r) x ∂originalRawKernel μ ell j d := by
obtain ⟨C,hC,hbound⟩ := exists_master_expanded_annular_query_constant hg hgs hB hc hcL
refine ⟨C,hC,?_⟩
intro N K μ hμ ell j r₀ s r hr₀ hs hs1 hr₀s hrr hrs d y z hy hy2 hz hz2
have hcg : HasCompactSupport g :=
HasCompactSupport.of_support_subset_isCompact (isCompact_closedBall (0 : Space) 1)
((subset_tsupport g).trans (hgs.trans ball_subset_closedBall))
exact posterior_query_difference_le_count μ ell j hc hr₀ hs hr₀s hg.continuous hcg
y z (fun x => hbound hr₀ hs hs1 hrr hrs x y z hy hy2 hz hz2) d
end CoulombBarrier
end
end OAI
|