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import OAI.Analysis.BiholderTransport.Contact.HopfSupports
namespace OAI
noncomputable section
open Set Filter Manifold Bundle Metric
open scoped Topology ContDiff NNReal
namespace WeakMTWTransport
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty 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]
omit [CompactSpace M] [Nonempty M] in
lemma cost_divided_triangle_two_times (x y z : M) {s t : ℝ} (hs : 0<s) (ht : 0<t) :
cost x z/(s+t) ≤ cost x y/s+cost y z/t := by
have hst : 0<s+t := add_pos hs ht
have ha : 0<s/(s+t) := div_pos hs hst
have ha1 : s/(s+t)<1 := (div_lt_one hst).mpr (lt_add_of_pos_right s ht)
have H := div_le_div_of_nonneg_right (divided_cost_triangle x y z ha ha1) hst.le
calc
_ ≤ _ := H
_ = _ := by
have he : 1-s/(s+t)=t/(s+t) := by field_simp; ring
rw [he]
field_simp
omit [CompactSpace M] [Nonempty 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] in
lemma mdifferentiableAt_of_chart_differentiable {f : M → ℝ} {x : M}
(hd : DifferentiableAt ℝ (fun y => f ((extChartAt 𝓘(ℝ,Model n) x).symm y))
(extChartAt 𝓘(ℝ,Model n) x x)) :
MDifferentiableAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) f x := by
have H := hd.hasFDerivAt.hasMFDerivAt.comp x
(mdifferentiableAt_extChartAt (I := 𝓘(ℝ,Model n))
(show x∈(chartAt (Model n) x).source from mem_chart_source _ _)).hasMFDerivAt
apply (H.congr_of_eventuallyEq ?_).mdifferentiableAt
filter_upwards [extChartAt_source_mem_nhds (I := 𝓘(ℝ,Model n)) x] with y hy
simp only [Function.comp_apply,(extChartAt 𝓘(ℝ,Model n) x).left_inv hy]
lemma WeakMTW.mdifferentiable_hopfLax (hmtw : WeakMTW (n := n) (M := M))
{u v : M → ℝ} (hu : Continuous u) (hv : Continuous v) (hdual : IsCostDualPair u v)
{t : ℝ} (ht : 0<t) (ht1 : t<1) :
MDifferentiable 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) (hopfLax t u) := by
intro z
obtain ⟨B,hB,r,hr,_,H⟩ := hmtw.uniform_hopfLax_C11 z
apply mdifferentiableAt_of_chart_differentiable
exact (H t ht ht1 u v hu hv hdual).1 _ (mem_ball_self hr)
lemma hopfLax_derivative_of_minimizer {u : M → ℝ} (hu : Continuous u)
{x : M} {p : TangentSpace 𝓘(ℝ,Model n) x} (hp : p∈minimizingVectors x)
{t : ℝ} (ht : 0<t) (ht1 : t≤1)
(heq : hopfLax t u (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1 =
u x+cost x (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1/t)
(hd : MDifferentiableAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) (hopfLax t u)
(sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1) :
HasMFDerivAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) (hopfLax t u)
(sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1
(innerSL ℝ (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2) := by
let w : TangentBundle 𝓘(ℝ,Model n) M := ⟨x,p⟩
let z := sprayFlow t w
obtain ⟨ε,hε,hεt,_,_,Hε⟩ := exists_two_short_legs x p ht
(show t<t+1 by linarith) zero_lt_one
have hshort := (Filter.Eventually.self_of_nhds Hε).2.2.2
let a := (sprayFlow (t-ε) w).1
let G : M → ℝ := fun y => u x+cost x a/(t-ε)+cost a y/ε
have hG : HasMFDerivAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) G z.1 (innerSL ℝ z.2) := by
have H := (hasMFDerivAt_const (I := 𝓘(ℝ,Model n)) (I' := 𝓘(ℝ,ℝ))
(u x+cost x a/(t-ε)) z.1).add (hshort.const_smul ε⁻¹)
have heD : 0+ε⁻¹ • innerSL ℝ (ε • (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2) =
innerSL ℝ z.2 := by
ext v
simp only [zero_add,_root_.smul_apply,smul_eq_mul,innerSL_apply_apply,
real_inner_smul_left]
dsimp [z,w]
field_simp
replace H := H.congr_mfderiv heD
have heG : G = ((fun _ => u x+cost x a/(t-ε)) +
ε⁻¹ • fun y => dist (sprayFlow (t-ε) (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1 y^2/2) := by
funext y
dsimp [G,cost,a,w]
ring
rw [heG]
exact H
have hmin : dist x z.1=t*‖p‖ := by
dsimp [z,w]
rw [←riemannianExp_smul,minimizingVectors_smul hp ht.le ht1,norm_smul,
Real.norm_eq_abs,abs_of_pos ht]
have hleft : dist x a=(t-ε)*‖p‖ :=
sprayFlow_minimizing_prefix w (sub_pos.mpr hεt).le (by linarith) hmin
have hright : dist a z.1=ε*‖p‖ := by
have H := sprayFlow_minimizing_subinterval w (sub_pos.mpr hεt).le (by linarith) hmin
simpa only [sub_sub_cancel] using H
have hcontact : hopfLax t u z.1=G z.1 := by
rw [heq]
dsimp [G,cost]
rw [hmin,hleft,hright]
field_simp
ring
have hle : ∀ y, hopfLax t u y≤G y := by
intro y
have H := cost_divided_triangle_two_times x a y (sub_pos.mpr hεt) hε
rw [sub_add_cancel] at H
exact (hopfLax_le hu t y x).trans (by dsimp [G]; linarith)
exact hd.hasMFDerivAt.congr_mfderiv
((upper_contact_mfderiv hd hG.mdifferentiableAt (Filter.Eventually.of_forall hle) hcontact).trans hG.mfderiv)
def riemannianGradient (f : M → ℝ) (x : M) : TangentSpace 𝓘(ℝ,Model n) x := by
letI : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
exact (InnerProductSpace.toDual ℝ (TangentSpace 𝓘(ℝ,Model n) x)).symm
(mfderiv 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) f x)
omit [CompactSpace M] [Nonempty M]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M] in
lemma riemannianGradient_eq_of_derivative {f : M → ℝ} {x : M}
{v : TangentSpace 𝓘(ℝ,Model n) x}
(h : HasMFDerivAt 𝓘(ℝ,Model n) 𝓘(ℝ,ℝ) f x (innerSL ℝ v)) :
riemannianGradient (n := n) f x=v := by
let : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
rw [riemannianGradient,h.mfderiv]
exact (InnerProductSpace.toDual ℝ (TangentSpace 𝓘(ℝ,Model n) x)).symm_apply_apply v
lemma WeakMTW.hopfLax_backward_flow (hmtw : WeakMTW (n := n) (M := M))
{v : M → ℝ} (hv : Continuous v) {t : ℝ} (ht : 0<t) (ht1 : t<1)
(q : subgradientGraph (n := n) (cTransform v)) :
sprayFlow (-t)
(⟨graphProjection (cTransform v) t q,
riemannianGradient (hopfLax t (cTransform v)) (graphProjection (cTransform v) t q)⟩ :
TangentBundle 𝓘(ℝ,Model n) M) = q.1 := by
have hdual : IsCostDualPair (cTransform v) (cTransform (cTransform v)) :=
⟨(cTransform_triple hv).symm,rfl⟩
have hu := continuous_cTransform hv
have hp := (hmtw.global_support hv q.2).1
have hbase : graphProjection (cTransform v) t q=(sprayFlow t q.1).1 :=
riemannianExp_smul _ _ _
have hd := hmtw.mdifferentiable_hopfLax hu (continuous_cTransform hu) hdual ht ht1
have hval := hmtw.hopfLax_graph_value hv ht ht1 q
rw [hbase] at hval
have hder := hopfLax_derivative_of_minimizer hu hp ht ht1.le hval (hd _)
rw [hbase,riemannianGradient_eq_of_derivative hder]
change sprayFlow (-t) (sprayFlow t q.1)=q.1
rw [←sprayFlow_add,neg_add_cancel,sprayFlow_zero]
end WeakMTWTransport
end
end OAI
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