| import OAI.Analysis.AncientFlat.Kernel.KernelChangingPair |
| import OAI.Analysis.AncientFlat.Kernel.SchwartzIBP |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ContDiff SchwartzMap Laplacian |
| namespace FlatDuality |
| open BiharmonicKernel InnerProductSpace |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] |
| def adjointTime (c ε : ℝ) (s : ℝ) : ℝ := ε+c^2/16*(Real.exp (-s)-Real.exp (-1)) |
| def adjointAmplitude (s : ℝ) : ℝ := Real.exp ((s-1)/2) |
| lemma adjointTime_bounds (c ε : ℝ) {s : ℝ} (hs : s ∈ Icc (0:ℝ) 1) : |
| adjointTime c ε s ∈ Icc ε (ε+c^2/16) := by |
| have h0 : 0≤c^2/16 := by positivity |
| have h1 : Real.exp (-1)≤Real.exp (-s) := Real.exp_le_exp.mpr (by linarith [hs.2]) |
| have h2 : Real.exp (-s)≤1 := Real.exp_le_one_iff.mpr (by linarith [hs.1]) |
| dsimp [adjointTime] |
| constructor |
| · exact le_add_of_nonneg_right (mul_nonneg h0 (sub_nonneg.mpr h1)) |
| · nlinarith [Real.exp_pos (-1)] |
| lemma adjointAmplitude_bounds {s : ℝ} (hs : s ∈ Icc (0:ℝ) 1) : |
| 0<adjointAmplitude s ∧ adjointAmplitude s≤1 := by |
| exact ⟨Real.exp_pos _,Real.exp_le_one_iff.mpr (by linarith [hs.2])⟩ |
| lemma hasDerivAt_adjointTime (c ε s : ℝ) : |
| HasDerivAt (adjointTime c ε) (-c^2/16*Real.exp (-s)) s := by |
| convert! ((((hasDerivAt_id s).neg.exp).sub_const (Real.exp (-1))).const_mul (c^2/16)).const_add ε using 1 |
| dsimp only [adjointTime,id_eq,Pi.neg_apply] |
| ring |
| lemma hasDerivAt_adjointAmplitude (s : ℝ) : |
| HasDerivAt adjointAmplitude (adjointAmplitude s/2) s := by |
| convert! ((((hasDerivAt_id s).sub_const 1).div_const 2).exp) using 1 |
| dsimp only [adjointAmplitude,id_eq] |
| ring |
| lemma contDiff_adjointTime (c ε : ℝ) : ContDiff ℝ 1 (adjointTime c ε) := by unfold adjointTime; fun_prop |
| lemma contDiff_adjointAmplitude : ContDiff ℝ 1 adjointAmplitude := by unfold adjointAmplitude; fun_prop |
| lemma laplacian_translate (f : V → ℂ) (z : V) (x : V) : |
| Δ (fun y => f (y-z)) x=Δ f (x-z) := by |
| simp only [laplacian_eq_iteratedFDeriv_stdOrthonormalBasis,iteratedFDeriv_comp_sub] |
| lemma bilaplacian_translate (f : V → ℂ) (z : V) (x : V) : |
| Δ (Δ (fun y => f (y-z))) x=Δ (Δ f) (x-z) := by |
| have he : Δ (fun y => f (y-z))=fun y => Δ f (y-z) := funext (laplacian_translate f z) |
| rw [he,laplacian_translate] |
| variable [MeasurableSpace V] [BorelSpace V] |
| def adjointSchwartz (c ε s : ℝ) (z : V) (h : 0<adjointTime c ε s) : SchwartzMap V ℂ := |
| adjointAmplitude s • SchwartzMap.compSubConstCLM ℝ z (heatKernel _ h) |
| lemma adjointSchwartz_apply (c ε s : ℝ) (z : V) (h : 0<adjointTime c ε s) (x : V) : |
| adjointSchwartz c ε s z h x=kernelTest (adjointTime c ε) adjointAmplitude z s x := by |
| simp only [adjointSchwartz,smul_apply,SchwartzMap.compSubConstCLM_apply, |
| heatKernel_eq_kernel,Complex.real_smul,kernelTest] |
| lemma adjointSchwartz_bilap (c ε s : ℝ) (z : V) (h : 0<adjointTime c ε s) (x : V) : |
| (Δ (Δ (adjointSchwartz c ε s z h) : SchwartzMap V ℂ) : SchwartzMap V ℂ) x= |
| (adjointAmplitude s : ℂ)*bilapKernel (adjointTime c ε s) (x-z) := by |
| let φ := SchwartzMap.compSubConstCLM ℝ z (heatKernel (adjointTime c ε s) h) |
| change (Δ (Δ ((adjointAmplitude s) • φ) : SchwartzMap V ℂ) : SchwartzMap V ℂ) x=_ |
| simp only [←SchwartzMap.laplacianCLM_eq',map_smul,smul_apply] |
| rw [Complex.real_smul] |
| congr 1 |
| have he : (φ : V → ℂ)=fun y => kernel (adjointTime c ε s) (y-z) := by |
| funext y |
| exact congrFun (heatKernel_eq_kernel _ h) (y-z) |
| change (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) x=_ |
| simp only [SchwartzMap.laplacian_apply] |
| have he' : ((Δ φ : SchwartzMap V ℂ) : V → ℂ)=Δ (φ : V → ℂ) := |
| funext (SchwartzMap.laplacian_apply φ) |
| rw [he',he,bilaplacian_translate] |
| rfl |
| lemma adjointSchwartz_equation (c ε s : ℝ) (z : V) (h : 0<adjointTime c ε s) (x : V) : |
| kernelTestDeriv (adjointTime c ε) adjointAmplitude z s x= |
| ((c^2*Real.exp (-s)/16 : ℝ) : ℂ)*(Δ (Δ (adjointSchwartz c ε s z h) : SchwartzMap V ℂ) : SchwartzMap V ℂ) x+ |
| (1/2:ℂ)*adjointSchwartz c ε s z h x := by |
| rw [adjointSchwartz_apply,adjointSchwartz_bilap] |
| simp only [kernelTestDeriv,kernelTest,(hasDerivAt_adjointTime c ε s).deriv, |
| (hasDerivAt_adjointAmplitude s).deriv,Complex.ofReal_div,Complex.ofReal_mul, |
| Complex.ofReal_neg,Complex.ofReal_pow,Complex.ofReal_ofNat] |
| ring |
| end FlatDuality |
| |
| end |
| |
| end OAI |
| |
