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59d1c80 | 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 | import OAI.Geometry.IsometricImmersion.Energy.DirectedCutoffBalance
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology
namespace SmoothLocal.Weighted
open SmoothLocal.Geometry
def directedGradientEnergy (chi I u : Coord → ℝ) (edge lambda : ℝ) (p : Coord) : ℝ :=
chi p * directedWeight edge lambda I p * ((coordPartial 0 u p)^2 + (coordPartial 1 u p)^2)
def directedForcingEnergy (chi I f : Coord → ℝ) (edge lambda : ℝ) (p : Coord) : ℝ :=
chi p * directedWeight edge lambda I p * (f p)^2
def directedCutoffError (A chi I u : Coord → ℝ) (edge lambda epsilon : ℝ) : Coord → ℝ :=
multiplierCutoffError A chi (directedM edge lambda I) (directedN edge lambda epsilon I) u
def directedCoercivityDefect (A B C chi I u : Coord → ℝ) (edge lambda epsilon c : ℝ) (p : Coord) : ℝ :=
chi p * max 0 (c * directedWeight edge lambda I p *
((coordPartial 0 u p)^2 + (coordPartial 1 u p)^2) -
multiplierQuadratic A B C (directedM edge lambda I) (directedN edge lambda epsilon I) u p)
theorem directedCoercivityDefect_zero
(A B C chi I u : Coord → ℝ) (edge lambda epsilon c : ℝ) (p : Coord)
(hcoercive : c * directedWeight edge lambda I p *
((coordPartial 0 u p)^2 + (coordPartial 1 u p)^2) ≤
multiplierQuadratic A B C (directedM edge lambda I) (directedN edge lambda epsilon I) u p) :
directedCoercivityDefect A B C chi I u edge lambda epsilon c p = 0 := by
simp only [directedCoercivityDefect, max_eq_left (sub_nonpos.mpr hcoercive), mul_zero]
theorem directedCoercivityDefect_domination
(A B C chi I u : Coord → ℝ) (edge lambda epsilon c : ℝ) (p : Coord) (hchi : 0 ≤ chi p) :
c * directedGradientEnergy chi I u edge lambda p ≤
chi p * multiplierQuadratic A B C (directedM edge lambda I) (directedN edge lambda epsilon I) u p +
directedCoercivityDefect A B C chi I u edge lambda epsilon c p := by
have hh := mul_le_mul_of_nonneg_left
(le_max_right (0 : ℝ) (c * directedWeight edge lambda I p *
((coordPartial 0 u p)^2 + (coordPartial 1 u p)^2) -
multiplierQuadratic A B C (directedM edge lambda I) (directedN edge lambda epsilon I) u p)) hchi
dsimp [directedGradientEnergy, directedCoercivityDefect]
nlinarith only [hh]
theorem integrated_directed_energy_with_explicit_costs
{A B C chi I u f : Coord → ℝ} {U : Set Coord} {tl tr sb st : ℝ}
(ht : tl ≤ tr) (hs : sb ≤ st) (hU : IsOpen U)
(hA : ContDiffOn ℝ ∞ A U) (hB : ContDiffOn ℝ ∞ B U) (hC : ContDiffOn ℝ ∞ C U)
(hchi : ContDiffOn ℝ ∞ chi U) (hI : ContDiffOn ℝ ∞ I U) (hu : ContDiffOn ℝ ∞ u U)
(hbox : closedRectangle tl tr sb st ⊆ U)
(hchiSides : ∀ s ∈ Icc sb st, chi (boxPoint tl s) = 0 ∧ chi (boxPoint tr s) = 0)
(hchiBottom : ∀ t ∈ Icc tl tr, chi (boxPoint t sb) = 0)
(hEq : ∀ p ∈ U, multiplierOperator A B C u p = f p)
(hchin : ∀ p ∈ closedRectangle tl tr sb st, 0 ≤ chi p)
(lambda epsilon R c : ℝ) (hc : 0 < c) (heps : 0 ≤ epsilon) (heps1 : epsilon ≤ 1)
(hR : ∀ p ∈ closedRectangle tl tr sb st, |p 0| ≤ R) :
(c / 2) * rectangleIntegral tl tr sb st (directedGradientEnergy chi I u st lambda) ≤
((1 + R^2) / (2 * c)) * rectangleIntegral tl tr sb st (directedForcingEnergy chi I f st lambda) +
rectangleIntegral tl tr sb st (fun p => |directedCutoffError A chi I u st lambda epsilon p|) +
rectangleIntegral tl tr sb st (directedCoercivityDefect A B C chi I u st lambda epsilon c) := by
let m := directedM st lambda I
let n := directedN st lambda epsilon I
let Q := multiplierQuadratic A B C m n u
let E := directedGradientEnergy chi I u st lambda
let F := directedForcingEnergy chi I f st lambda
let Err := directedCutoffError A chi I u st lambda epsilon
let Def := directedCoercivityDefect A B C chi I u st lambda epsilon c
let Source := fun p => f p * multiplierTest (cutoffM chi st lambda I) (cutoffN chi st lambda epsilon I) u p
have hW := directedWeight_contDiffOn st lambda hI
have hm : ContDiffOn ℝ ∞ m U := hW.neg
have hn : ContDiffOn ℝ ∞ n U :=
(contDiffOn_const.mul (contDiffOn_apply ℝ ℝ 0 U)).mul hW
have hcm := hchi.mul hm
have hcn := hchi.mul hn
have hut := partial_contDiffOn hu hU 0
have hus := partial_contDiffOn hu hU 1
have hgrad := (hut.pow 2).add (hus.pow 2)
have hf : ContDiffOn ℝ ∞ f U :=
(ellipticOperator_contDiffOn hU hA hB hC hu).congr (fun p hp => (hEq p hp).symm)
have hEc : ContinuousOn E (closedRectangle tl tr sb st) :=
(((hchi.mul hW).mul hgrad).continuousOn.mono hbox)
have hFc : ContinuousOn F (closedRectangle tl tr sb st) :=
(((hchi.mul hW).mul (hf.pow 2)).continuousOn.mono hbox)
have hQ := multiplierQuadratic_contDiffOn hU hA hB hC hm hn hu
have hchiQ : ContinuousOn (fun p => chi p * Q p) (closedRectangle tl tr sb st) :=
(hchi.mul hQ).continuousOn.mono hbox
have hErr := multiplierCutoffError_contDiffOn hU hA hchi hm hn hu
have hErrc : ContinuousOn Err (closedRectangle tl tr sb st) := hErr.continuousOn.mono hbox
have hDef : ContinuousOn Def (closedRectangle tl tr sb st) :=
(hchi.continuousOn.mul ((continuousOn_const (c := (0 : ℝ))).sup
((((contDiffOn_const (c := c)).mul hW).mul hgrad).sub hQ).continuousOn)).mono hbox
have htest : ContDiffOn ℝ ∞ (multiplierTest (cutoffM chi st lambda I) (cutoffN chi st lambda epsilon I) u) U :=
(hcm.mul hus).add (hcn.mul hut)
have hSource : ContinuousOn Source (closedRectangle tl tr sb st) := (hf.mul htest).continuousOn.mono hbox
have hsplit : rectangleIntegral tl tr sb st Source =
rectangleIntegral tl tr sb st (fun p => chi p * Q p) - rectangleIntegral tl tr sb st Err := by
calc
_ = rectangleIntegral tl tr sb st (fun p => multiplierOperator A B C u p *
multiplierTest (cutoffM chi st lambda I) (cutoffN chi st lambda epsilon I) u p) := by
apply rectangleIntegral_congr ht hs
intro p hp
change f p * multiplierTest (cutoffM chi st lambda I) (cutoffN chi st lambda epsilon I) u p =
multiplierOperator A B C u p *
multiplierTest (cutoffM chi st lambda I) (cutoffN chi st lambda epsilon I) u p
rw [hEq p (hbox hp)]
_ = rectangleIntegral tl tr sb st
(multiplierQuadratic A B C (cutoffM chi st lambda I) (cutoffN chi st lambda epsilon I) u) :=
integrated_cutoff_directed_identity ht hs hU hA hB hC hchi hI hu hbox hchiSides hchiBottom lambda epsilon
_ = rectangleIntegral tl tr sb st (fun p => chi p * Q p - Err p) := by
apply rectangleIntegral_congr ht hs
intro p hp
exact multiplierQuadratic_cutoff_split hU hA hchi hm hn hu (hbox hp)
_ = _ := rectangleIntegral_sub ht hs hchiQ hErrc
have hdom := rectangleIntegral_mono ht hs (continuousOn_const.mul hEc) (hchiQ.add hDef)
(fun p hp => directedCoercivityDefect_domination A B C chi I u st lambda epsilon c p (hchin p hp))
change rectangleIntegral tl tr sb st (fun p => c * E p) ≤
rectangleIntegral tl tr sb st (fun p => chi p * Q p + Def p) at hdom
rw [rectangleIntegral_const_mul, rectangleIntegral_add ht hs hchiQ hDef] at hdom
have hsourceBound (p : Coord) (hp : p ∈ closedRectangle tl tr sb st) :
Source p ≤ (c / 2) * E p + ((1 + R^2) / (2 * c)) * F p := by
have hWn : 0 ≤ directedWeight st lambda I p :=
mul_nonneg (pow_nonneg (sub_nonneg.mpr hp.2.2) 8) (Real.exp_pos _).le
have hpoint := directed_source_young_scalar (mul_nonneg (hchin p hp) hWn) hc heps heps1 (hR p hp)
(f := f p) (ut := coordPartial 0 u p) (us := coordPartial 1 u p)
have hfac := cutoff_test_factor chi I u st lambda epsilon p
dsimp [Source]
rw [hfac]
exact (le_abs_self _).trans (by
convert hpoint using 1
dsimp [E, F, directedGradientEnergy, directedForcingEnergy]
ring)
have hsourceIntegral := rectangleIntegral_mono ht hs hSource
((continuousOn_const.mul hEc).add (continuousOn_const.mul hFc)) hsourceBound
change rectangleIntegral tl tr sb st Source ≤ rectangleIntegral tl tr sb st
(fun p => (c / 2) * E p + ((1 + R^2) / (2 * c)) * F p) at hsourceIntegral
have hcE : ContinuousOn (fun p => (c / 2) * E p) (closedRectangle tl tr sb st) :=
continuousOn_const.mul hEc
have hcF : ContinuousOn (fun p => ((1 + R^2) / (2 * c)) * F p) (closedRectangle tl tr sb st) :=
continuousOn_const.mul hFc
rw [rectangleIntegral_add ht hs hcE hcF,
rectangleIntegral_const_mul, rectangleIntegral_const_mul] at hsourceIntegral
have herrorIntegral := rectangleIntegral_mono ht hs hErrc hErrc.abs (fun p hp => le_abs_self (Err p))
change (c / 2) * rectangleIntegral tl tr sb st E ≤
((1 + R^2) / (2 * c)) * rectangleIntegral tl tr sb st F +
rectangleIntegral tl tr sb st (fun p => |Err p|) + rectangleIntegral tl tr sb st Def
nlinarith only [hdom, hsplit, hsourceIntegral, herrorIntegral]
end SmoothLocal.Weighted
end
end OAI
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