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

65λ=0.01λ=0.04λ=0.08108λ=0.01λ=0.04λ=0.0846432201−20−0.5 0 0.5−4−0.5 0 0.5Figure 11: Plots of K λ (left) and ˜K λ (right) for various λ with L = 1. The plotsshow the kernels over one period.The kernels ˜K λ , K λ are known in closed form:˜K λ (s) = 1 L( ) s− L√ 2λLcoshK λ (s) =2L √ (λ sinh(12 √ λ1 + (s/L)2 − (s/L) + 1/62λ), for s ∈ [0, L], (10.10)and K λ , ˜K λ are periodically extended to all of lR. See Fig. 11.), s ∈ [0, L]. (10.11)The above idea extends to any j: it is possible to obtain ∇ H j E and ∇˜HjEfrom ∇ H 0E, by convolution. But there is also a way to compute ∇˜HjE withoutresolving to convolutions, see Prop. 10.10.The bad news is that, when shape optimization is implemented using a levelset method (as is usually the case), the curve must be traced out to computegradients. 19 This in particular leads to some problems when a curve undergoesa topological change, since the Sobolev gradient depends on the global shape ofthe curve; but those problems are easily bypassed when the optimization energyis continuous across topological changes of the curves: see Sec. 5.1 in [55].10.3.1 Smoothing of gradients, coarse-to-fine flowingIn Sundaramoorthi et al. [57] it was noted that the regularizing properties maybe explained in the Fourier domain: indeed, if we calculate Sobolev gradients∇ H j E of an arbitrary energy E in the frequency domain, then̂∇ H j E(l) =̂∇ H 0E(l)1 + λ(2πl) 2j for l ∈ Z (10.12)19 Though, an alternate method that does not need the tracing of the curve is described inSec. 5.3 in [55] – but it was not successively used, since it depends on some difficult-to-tuneparameters.65