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f61e547 | 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 | import OAI.Probability.FreeIsing.Estimates.ExponentialTaylor
namespace OAI
noncomputable section
open Filter
open scoped BigOperators Topology NNReal
namespace Problem367.ExponentialTaylor
theorem norm_gaussian_recurrence_error_le_sum
(z : ℕ → ℂ) (dt e : ℕ → ℝ) (n : ℕ) {c : ℝ}
(hc : 0 ≤ c) (hdt : ∀ k < n, 0 ≤ dt k)
(he : ∀ k < n,
‖z (k + 1) - Complex.exp ((-c * dt k : ℝ) : ℂ) * z k‖ ≤ e k) :
‖z n - Complex.exp (((-c * ∑ k ∈ Finset.range n, dt k : ℝ) : ℂ)) * z 0‖ ≤
∑ k ∈ Finset.range n, e k := by
have h := norm_recurrence_error_le_sum z
(fun k => Complex.exp ((-c * dt k : ℝ) : ℂ)) e n
(fun k hk => ?_) he
· simpa only [← Complex.exp_sum, ← Complex.ofReal_sum, ← Finset.mul_sum] using h
· rw [Complex.norm_exp_ofReal]
exact Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hc) (hdt k hk))
theorem eq_gaussian_mul_of_recurrence_errors_tendsto_zero
{ι : Type*} {l : Filter ι} [NeBot l]
(z : ι → ℕ → ℂ) (dt e : ι → ℕ → ℝ) (N : ι → ℕ)
{c T : ℝ} {z₀ zT : ℂ}
(hc : 0 ≤ c)
(hdt : ∀ i k, k < N i → 0 ≤ dt i k)
(htotal : ∀ i, ∑ k ∈ Finset.range (N i), dt i k = T)
(hstart : ∀ i, z i 0 = z₀) (hend : ∀ i, z i (N i) = zT)
(he : ∀ i k, k < N i →
‖z i (k + 1) - Complex.exp ((-c * dt i k : ℝ) : ℂ) * z i k‖ ≤ e i k)
(hlim : Tendsto (fun i => ∑ k ∈ Finset.range (N i), e i k) l (𝓝 0)) :
zT = Complex.exp ((-c * T : ℝ) : ℂ) * z₀ := by
apply sub_eq_zero.mp
apply norm_eq_zero.mp
apply le_antisymm _ (norm_nonneg _)
apply ge_of_tendsto hlim
exact Eventually.of_forall fun i => by
simpa only [htotal i, hstart i, hend i] using
norm_gaussian_recurrence_error_le_sum (z i) (dt i) (e i) (N i) hc (hdt i) (he i)
theorem sum_mul_sqrt_le_total_mul_sqrt_mesh
(dt : ℕ → ℝ) (n : ℕ) {mesh : ℝ}
(hdt : ∀ k < n, 0 ≤ dt k) (hmesh : ∀ k < n, dt k ≤ mesh) :
(∑ k ∈ Finset.range n, dt k * Real.sqrt (dt k)) ≤
(∑ k ∈ Finset.range n, dt k) * Real.sqrt mesh := by
rw [Finset.sum_mul]
apply Finset.sum_le_sum
intro k hk
exact mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt (hmesh k (Finset.mem_range.mp hk)))
(hdt k (Finset.mem_range.mp hk))
theorem sum_sq_le_total_mul_mesh
(dt : ℕ → ℝ) (n : ℕ) {mesh : ℝ}
(hdt : ∀ k < n, 0 ≤ dt k) (hmesh : ∀ k < n, dt k ≤ mesh) :
(∑ k ∈ Finset.range n, (dt k) ^ 2) ≤
(∑ k ∈ Finset.range n, dt k) * mesh := by
rw [Finset.sum_mul]
apply Finset.sum_le_sum
intro k hk
simpa only [pow_two] using mul_le_mul_of_nonneg_left
(hmesh k (Finset.mem_range.mp hk)) (hdt k (Finset.mem_range.mp hk))
theorem sum_cube_le_total_mul_mesh_sq
(dt : ℕ → ℝ) (n : ℕ) {mesh : ℝ}
(hdt : ∀ k < n, 0 ≤ dt k) (hmesh : ∀ k < n, dt k ≤ mesh) :
(∑ k ∈ Finset.range n, (dt k) ^ 3) ≤
(∑ k ∈ Finset.range n, dt k) * mesh ^ 2 := by
rw [Finset.sum_mul]
apply Finset.sum_le_sum
intro k hk
have hk' := Finset.mem_range.mp hk
have hsq : (dt k) ^ 2 ≤ mesh ^ 2 :=
pow_le_pow_left₀ (hdt k hk') (hmesh k hk') 2
convert mul_le_mul_of_nonneg_left hsq (hdt k hk') using 1; ring
theorem tendsto_sum_brownian_drift_errors
{ι : Type*} {l : Filter ι}
(dt : ι → ℕ → ℝ) (N : ι → ℕ) (mesh : ι → ℝ)
{T A B C : ℝ} (hA : 0 ≤ A) (hB : 0 ≤ B) (hC : 0 ≤ C)
(hdt : ∀ i k, k < N i → 0 ≤ dt i k)
(hmesh : ∀ i k, k < N i → dt i k ≤ mesh i)
(htotal : ∀ i, ∑ k ∈ Finset.range (N i), dt i k = T)
(hlim : Tendsto mesh l (𝓝 0)) :
Tendsto (fun i => ∑ k ∈ Finset.range (N i),
(A * (dt i k * Real.sqrt (dt i k)) + B * (dt i k) ^ 2 + C * (dt i k) ^ 3))
l (𝓝 0) := by
have hupper (i) : (∑ k ∈ Finset.range (N i),
(A * (dt i k * Real.sqrt (dt i k)) + B * (dt i k) ^ 2 + C * (dt i k) ^ 3)) ≤
A * (T * Real.sqrt (mesh i)) + B * (T * mesh i) + C * (T * (mesh i) ^ 2) := by
simp only [Finset.sum_add_distrib, ← Finset.mul_sum]
apply add_le_add
· apply add_le_add
· exact mul_le_mul_of_nonneg_left
(by simpa only [htotal i] using (sum_mul_sqrt_le_total_mul_sqrt_mesh
(dt i) (N i) (hdt i) (hmesh i))) hA
· exact mul_le_mul_of_nonneg_left
(by simpa only [htotal i] using (sum_sq_le_total_mul_mesh
(dt i) (N i) (hdt i) (hmesh i))) hB
· exact mul_le_mul_of_nonneg_left
(by simpa only [htotal i] using (sum_cube_le_total_mul_mesh_sq
(dt i) (N i) (hdt i) (hmesh i))) hC
have hright : Tendsto (fun i => A * (T * Real.sqrt (mesh i)) +
B * (T * mesh i) + C * (T * (mesh i) ^ 2)) l (𝓝 0) := by
simpa only [Function.comp_def, Real.sqrt_zero, mul_zero, zero_pow (by norm_num : 2 ≠ 0), add_zero] using
(((Real.continuous_sqrt.tendsto 0).comp hlim).const_mul T |>.const_mul A).add
((hlim.const_mul T).const_mul B) |>.add (((hlim.pow 2).const_mul T).const_mul C)
apply squeeze_zero (fun i => ?_) hupper hright
apply Finset.sum_nonneg
intro k hk
have := hdt i k (Finset.mem_range.mp hk)
positivity
theorem eq_gaussian_mul_of_local_error
(z : ℝ → ℂ) {c T A B C : ℝ}
(hc : 0 ≤ c) (hT : 0 ≤ T)
(hA : 0 ≤ A) (hB : 0 ≤ B) (hC : 0 ≤ C)
(hstep : ∀ s t : ℝ, 0 ≤ s → s ≤ t → t ≤ T →
‖z t - Complex.exp ((-c * (t - s) : ℝ) : ℂ) * z s‖ ≤
A * ((t - s) * Real.sqrt (t - s)) + B * (t - s) ^ 2 + C * (t - s) ^ 3) :
z T = Complex.exp ((-c * T : ℝ) : ℂ) * z 0 := by
let mesh : ℕ → ℝ := fun n => T / ((n : ℝ) + 1)
let grid : ℕ → ℕ → ℝ := fun n k => (k : ℝ) * mesh n
let e : ℕ → ℕ → ℝ := fun n _ =>
A * (mesh n * Real.sqrt (mesh n)) + B * (mesh n) ^ 2 + C * (mesh n) ^ 3
have hm (n : ℕ) : 0 ≤ mesh n := by dsimp [mesh]; positivity
have htotal (n : ℕ) : (∑ _k ∈ Finset.range (n + 1), mesh n) = T := by
simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_add, Nat.cast_one]
dsimp [mesh]
field_simp
have hmeshlim : Tendsto mesh atTop (𝓝 0) := by
simpa only [mesh, div_eq_mul_inv, one_div, one_mul, mul_zero] using
(tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul T
have herrlim : Tendsto (fun n => ∑ k ∈ Finset.range (n + 1), e n k) atTop (𝓝 0) :=
tendsto_sum_brownian_drift_errors (fun n _ => mesh n) (fun n => n + 1) mesh
hA hB hC (fun n _ _ => hm n) (fun _ _ _ => le_rfl) htotal hmeshlim
apply eq_gaussian_mul_of_recurrence_errors_tendsto_zero
(fun n k => z (grid n k)) (fun n _ => mesh n) e (fun n => n + 1) hc
(fun n _ _ => hm n) htotal (fun n => by simp [grid])
(fun n => ?_) (fun n k hk => ?_) herrlim
· congr 1
dsimp [grid, mesh]
rw [Nat.cast_add, Nat.cast_one]
field_simp
· have hkn : (k : ℝ) + 1 ≤ (n : ℝ) + 1 := by exact_mod_cast (Nat.succ_le_of_lt hk)
have hstart : 0 ≤ grid n k := mul_nonneg (Nat.cast_nonneg k) (hm n)
have hdiff : grid n (k + 1) - grid n k = mesh n := by
simp only [grid, Nat.cast_add, Nat.cast_one]
ring
have horder : grid n k ≤ grid n (k + 1) := by linarith [hm n]
have hend : grid n (k + 1) ≤ T := by
calc
grid n (k + 1) = ((k : ℝ) + 1) * mesh n := by simp [grid]
_ ≤ ((n : ℝ) + 1) * mesh n := mul_le_mul_of_nonneg_right hkn (hm n)
_ = T := by dsimp [mesh]; field_simp
simpa only [hdiff, e] using hstep (grid n k) (grid n (k + 1)) hstart horder hend
theorem eq_gaussian_mul_of_local_error_nnreal
(z : ℝ≥0 → ℂ) (T : ℝ≥0) {c A B C : ℝ}
(hc : 0 ≤ c) (hA : 0 ≤ A) (hB : 0 ≤ B) (hC : 0 ≤ C)
(hstep : ∀ s t : ℝ≥0, s ≤ t → t ≤ T →
‖z t - Complex.exp ((-c * ((t : ℝ) - (s : ℝ)) : ℝ) : ℂ) * z s‖ ≤
A * (((t : ℝ) - (s : ℝ)) * Real.sqrt ((t : ℝ) - (s : ℝ))) +
B * ((t : ℝ) - (s : ℝ)) ^ 2 + C * ((t : ℝ) - (s : ℝ)) ^ 3) :
z T = Complex.exp ((-c * (T : ℝ) : ℝ) : ℂ) * z 0 := by
have h := eq_gaussian_mul_of_local_error (fun r => z r.toNNReal) hc T.coe_nonneg
hA hB hC (fun s t hs hst ht => ?_)
· simpa only [Real.toNNReal_coe, Real.toNNReal_zero] using h
· have ht0 : 0 ≤ t := hs.trans hst
have hst' : s.toNNReal ≤ t.toNNReal := Real.toNNReal_mono hst
have ht' : t.toNNReal ≤ T := by
simpa only [Real.toNNReal_coe] using Real.toNNReal_mono ht
simpa only [Real.coe_toNNReal _ hs, Real.coe_toNNReal _ ht0] using
hstep s.toNNReal t.toNNReal hst' ht'
end Problem367.ExponentialTaylor
end
end OAI
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