Metrics of curves in shape optimization and analysis - Andrea Carlo ...

Definition 10.7 (Convolution) A arc-parameterized convolutional kernel Kalong the curve c of length L is a L-periodic function K : lR → lR. Given avector field f : S 1 → lR n and a kernel K, we define the convolution by arcparameter 17 formally as∫(f ⋆ K)(s) := K(s − ŝ)f(ŝ) dŝ . (10.6)By defining the run-length function l : lR → lRl(τ) :=c∫ τ0|c ′ (x)| dxwe can rewrite the above eqn. (10.6) explicitly in θ parameter as(f ⋆ K)(θ) :=∫ 2π0K ( l(θ) − l(τ) ) f(τ)|c ′ (τ)| dτ . (10.7)Recall the definition 3.37 of gradient ∇E by means of the identity〈∇E, h〉 c = DE(c; h) ∀h ∈ T c M .Let f = ∇ H 0E, g = ∇ H 1E be the gradients w.r.to the inner products H 0 andH 1 ; by the definition of gradient, we obtain that〈f, h〉 H0 ,c = 〈g, h〉 H1 ,c∀h ∈ T c Mthat is 18 ∫∫h · f ds = h · g + λL 2 (D s h · D s g) dsccby integrating by parts this becomes∫h · (f − g + λL 2 Dsg) 2 ds = 0 ∀hthen we conclude thatc∀h∇ H 0E = ∇ H 1E − λL 2 D 2 s(∇ H 1E) . (10.8)With similar computation, for ˜H 1 we obtain that∫∇ H 0E = ∇˜H1E ds − λL 2 D s(∇˜H1E) 2 . (10.9)cGiven ∇ H 0E, both equations can be solved for ∇ H 1E and ∇˜H1E, by meansof suitable convolution kernels ˜K λ , K λ , that is, we have the formulas∇ H 1E = ∇ H 0E ⋆ K λ , ∇˜H1E = ∇ H 0E ⋆ ˜K λ ;17 Note this definition is different from the eqn. (13) in [55].18 We use the definition (10.4) of H 0 .64