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110f4f3 | 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 | import Mathlib
import OAI.Analysis.BiholderTransport.Regularity.MetricDefinitions
import OAI.Analysis.BiholderTransport.LinearAlgebra.ActualRadial
import OAI.Analysis.BiholderTransport.Coordinates.SplitAction
import OAI.Analysis.BiholderTransport.Regularity.SecondScaled
import OAI.Analysis.BiholderTransport.CostGeometry.BranchCost2
import OAI.Analysis.BiholderTransport.Coordinates.ProperPieces
namespace OAI
section
section
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
namespace WeakMTWTransport
section Crossing
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance (x : M) : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end Crossing
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
namespace WeakMTWTransport
section MiddleHessian
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance (x : M) : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
omit [IsRiemannianManifold 𝓘(ℝ,Model n) M] in
lemma reverseRay_prefix (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ) :
reverseRay (tangentScale h z)=tangentScale (-h) (sprayFlow h z) := by
rw [reverseRay,sprayFlow_scale,mul_one,tangentScale_mul,neg_one_mul]
lemma proper_prefix_reverse_in_injectivityDomain {z : TangentBundle 𝓘(ℝ,Model n) M}
(hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) :
(-h) • (sprayFlow h z).2∈injectivityDomain (sprayFlow h z).1 := by
have H := reverseRay_injectivityDomain (z := tangentScale h z)
(contracted_minimizer_mem_injectivityDomain hz hh hh1)
rw [reverseRay_prefix] at H
exact H
lemma proper_prefix_reverse_endpoint (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ) :
riemannianExp (sprayFlow h z).1 ((-h) • (sprayFlow h z).2)=z.1 := by
have H := reverseRay_endpoint (tangentScale h z)
rw [reverseRay_prefix] at H
exact H
def middleHessian (z : TangentBundle 𝓘(ℝ,Model n) M) (h T : ℝ) :
TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1 →L[ℝ]
TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1 →L[ℝ] ℝ :=
h⁻¹ • normalHessian (sprayFlow h z).1 ((-h) • (sprayFlow h z).2)+
(T-h)⁻¹ • normalHessian (sprayFlow h z).1 ((T-h) • (sprayFlow h z).2)
omit [IsRiemannianManifold 𝓘(ℝ,Model n) M] in
lemma middleHessian_apply (z : TangentBundle 𝓘(ℝ,Model n) M) (h T : ℝ)
(v w : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) :
middleHessian z h T v w=
normalHessian (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) v w/h+
normalHessian (sprayFlow h z).1 ((T-h) • (sprayFlow h z).2) v w/(T-h) := by
simp only [middleHessian,add_apply,smul_apply,
smul_eq_mul,div_eq_mul_inv]
ring
def middleCenteredAction (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ)
(b : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) : ℝ :=
h⁻¹*(normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) b-
normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) 0)-
(-(1-h)⁻¹)*(normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) b-
normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) 0)
lemma middle_normal_prefix (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ)
(b : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) :
normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) b=
cost z.1 (riemannianExp (sprayFlow h z).1 b) := by
unfold normalCost
rw [proper_prefix_reverse_endpoint,cost_symm]
lemma middle_normal_suffix (z : TangentBundle 𝓘(ℝ,Model n) M) (h : ℝ)
(b : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) :
normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) b=
cost (riemannianExp (sprayFlow h z).1 b) (riemannianExp z.1 z.2) := by
unfold normalCost
rw [proper_suffix_endpoint,one_smul]
lemma middle_action_contact {z : TangentBundle 𝓘(ℝ,Model n) M}
(hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) :
normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) 0/h+
normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) 0/(1-h)=
cost z.1 (riemannianExp z.1 z.2) := by
rw [middle_normal_prefix,middle_normal_suffix,riemannianExp_zero]
have H := splitNormalAction_contact hz hh hh1
simp only [splitNormalAction,normalCost,riemannianExp_zero] at H
simpa only [riemannianExp_smul] using H
lemma middleCenteredAction_minimum {z : TangentBundle 𝓘(ℝ,Model n) M}
(hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) :
IsLocalMin (middleCenteredAction z h) 0 := by
filter_upwards [] with b
have H := squared_dist_divided_action_le z.1 (riemannianExp (sprayFlow h z).1 b)
(riemannianExp z.1 z.2) hh hh1
have H' : cost z.1 (riemannianExp z.1 z.2)≤
normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2) b/h+
normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2) b/(1-h) := by
rw [middle_normal_prefix,middle_normal_suffix]
convert H using 1 <;> simp only [cost,div_div,mul_comm]
rw [←middle_action_contact hz hh hh1] at H'
dsimp only [middleCenteredAction]
simp only [sub_self,mul_zero,neg_mul,sub_neg_eq_add,div_eq_mul_inv] at H' ⊢
nlinarith
lemma middleCenteredAction_contDiffAt {z : TangentBundle 𝓘(ℝ,Model n) M}
(hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1) :
ContDiffAt ℝ 2 (middleCenteredAction z h) 0 := by
have hf : ContDiffAt ℝ 2 (normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2)) 0 :=
(normalCost_contDiffAt (proper_prefix_reverse_in_injectivityDomain hz hh hh1)).of_le
(ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)
have hg : ContDiffAt ℝ 2 (normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2)) 0 :=
(normalCost_contDiffAt (proper_suffix_in_injectivityDomain hz hh hh1 le_rfl)).of_le
(ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)
exact (contDiffAt_const.mul (hf.sub contDiffAt_const)).sub
(contDiffAt_const.mul (hg.sub contDiffAt_const))
lemma middleCenteredAction_hessian {z : TangentBundle 𝓘(ℝ,Model n) M}
(hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1)
(v : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) :
fderiv ℝ (fderiv ℝ (middleCenteredAction z h)) 0 v v=middleHessian z h 1 v v := by
have hf : ContDiffAt ℝ 2 (normalCost (sprayFlow h z).1 ((-h) • (sprayFlow h z).2)) 0 :=
(normalCost_contDiffAt (proper_prefix_reverse_in_injectivityDomain hz hh hh1)).of_le
(ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)
have hg : ContDiffAt ℝ 2 (normalCost (sprayFlow h z).1 ((1-h) • (sprayFlow h z).2)) 0 :=
(normalCost_contDiffAt (proper_suffix_in_injectivityDomain hz hh hh1 le_rfl)).of_le
(ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)
unfold middleCenteredAction
rw [second_fderiv_scaled_differences hf hg,middleHessian_apply]
simp only [normalHessian,neg_mul,sub_neg_eq_add,div_eq_mul_inv,mul_comm]
lemma middleHessian_one_nonneg {z : TangentBundle 𝓘(ℝ,Model n) M}
(hz : z.2∈minimizingVectors z.1) {h : ℝ} (hh : 0<h) (hh1 : h<1)
(v : TangentSpace 𝓘(ℝ,Model n) (sprayFlow h z).1) :
0 ≤ middleHessian z h 1 v v := by
rw [←middleCenteredAction_hessian hz hh hh1]
exact localMin_hessian_nonneg (middleCenteredAction_contDiffAt hz hh hh1)
(middleCenteredAction_minimum hz hh hh1) v
end MiddleHessian
end WeakMTWTransport
end
end
end
end OAI
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