4.4 Numerical experiments
4.4.3 Experimental results
The algorithms were implemented in MATLAB○c; parameters reported in this section refer to the rescaled cost (4.11). We first present simulation results for a single phase field and no diffuse mass flux, N = 1 and α0 = ∞. Figure 4.2 and 4.3 show solutions for a source and a number of equal sinks arranged as a regular polygon. If α1 is small as in Figure 4.2, the solution looks similar to the Steiner tree (which would correspond
Algorithm 6 Minimization for N > 1, α0 <∞
function MPFSD(εstart, εend, Niter, α1, . . . , αN, β1, . . . , βN, µ+, µ−, ρεend) set fε = (µ+− µ−)∗ ρεend
set (σ0,·, λ0) = SP F S(εstart, εend, Niter, α1, β1, µ+, µ−, ρεend) compute regions R0ε, ˜Rε1, . . . , ˜RεN via (4.12)
for j = 1, . . . , Niter do
set εj = εstart− (j − 1)εstartNiter−ε−1end
set ϕji to the minimizer of (4.13) for given fixed σ = σj−1, i = 1, . . . , N update regions Rε0, ˜Rε1, . . . , ˜RεN via (4.8) and (4.12)
set γεj = mini=1,...,N (ϕji)2+ α2iε2/βi
set (ˆσj, ˆλj) = (ˆσj−1, ˆλj−1)− DR(ˆσj−1, ˆλj−1)−1R(ˆσj−1, ˆλj−1) for γε= γεj−1 end for
end function
return σNiter, ϕN1iter, . . . , ϕNNiter
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Figure 4.2: Optimal transportation networks for branched transportation from a single source to a number of identical sinks at the corners of regular polygons. The top row shows the ground truth, computed by finite-dimensional optimization of the vertex locations in a network with straight edges, the bottom rows show the computation results from the phase field model, the mass flux σ (middle, only support shown) and phase field ϕ (bottom). Parameters were α1 = 0.05, β1 = 1, ε = 0.005.
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Figure 4.3: Truly optimal network (top), computed mass flux σ (middle), and phase field ϕ (bottom) for same branched transportation problems as in Figure 4.2, only with α1 = 1, β1 = 1, ε = 0.005.
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Figure 4.4: Computed mass flux σ and phase field ϕ for same parameters as in Fig-ure 4.2.
to α1 = 0), while the solutions become much more asymmetric for larger α1 as in Figure 4.3. More complex examples are displayed in Figure 4.4.
Figure 4.5 shows simulation results for the same source and sink configuration as in Figure 4.4, but this time with N = 3 different linear segments in h and corresponding phase fields. It is clear that different phase fields become active on the different network branches according to the mass flux through each branch. This can be interpreted as having streets of three different qualities: the street ϕ3 allows faster (cheaper) transport, but requires more maintenance than the others, while street ϕ1 requires the least maintenance and only allows expensive transport.
The case α0 <∞ finally can be interpreted as the situation in which mass can also be transported off-road, that is, part of the transport may happen without a street network, thus having maintenance cost β0 = 0, but at the price of large transport expenses α0 per unit mass. Corresponding results for again the same source and sink configuration are shown in Figure 4.6. In contrast to the case α0 = ∞ it is now also possible to have sources and sinks that are not concentrated in a finite number of points. A corresponding example is shown in Figure 4.7.
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Figure 4.5: Computed mass flux σ and phase fields ϕ1, ϕ2, ϕ3 for the same source and sink as in Figure 4.4 and for the cost function shown on the right, ε = 0.005. The color in σ indicates which phase field is active.
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Figure 4.6: Computed mass flux σ and phase fields ϕ1, ϕ2 for the same source and sink as in Figure4.4 and for the cost function shown on the right, ε = 0.005.
Figure 4.7: Computed mass flux σ and phase field ϕ1 for a central point source and a spatially uniform sink outside a circle of radius and for the cost function shown on the right, ε = 0.005.
Generalized cost functions
5.1 Introduction
In the previous chapters we have considered the phase field function (1− ϕ)2 to model the characteristic function of the support of the vector valued measure σ. The main idea behind this approach was, for a weakly differentiable function ϕ, to consider the measure
µϕ(A) = Z
Ω∩A
|∇ϕ| |1 − ϕ| dx = Z
Ω∩A
|∇W (ϕ)| dx.
where A is any Borel set and W (t) = t2/2. As a matter of fact, given a sequence of functions ϕε for which the quantity
Z
Ω
ε|∇ϕε|2+ (1− ϕε)2 ε
dx
is bounded independently of ε then the sequence of Radon measures µϕε weakly-∗ converges up to subsequence to the measure Hn−1
x
{ϕ 6= 1}. As already stated in the introduction of this thesis, this fact is essential when approximating the Mumford -Shah functional as in the limit energy one aims at recovering the length of the jump set of some BV function which is contained in the set{ϕ 6= 1}. Later on some more general functionals have been introduced with different penalization of the jump set [BBB95]to model fractures. Recently [ABS99, DMOT16] other phase-field methods have dealt with the problem of efficiently approaching these energies. The main idea is that the limit ϕ rather then acquiring only the values{0, 1} should range over [0, +∞). In this chapter we follow this method to approximate any functionalEh where h is a concave cost function.
A way to approximate an energyEh is to substitute for h a sequence hkof piecewise affine functions pointwise converging to h and apply the method described in Chapter 4.
The quality of such approximation depends on the number of phase fields used and for an accurate approximation, the numerical complexity of the method may become prohibitive. Here we propose a different model with a single phase function ϕ its main drawback is that the term in the energy that penalizes σ is linear in σ, it was quadratic or at least strictly convex in the preceding models of the thesis. Let us be more formal.
Let us consider a convex open set Ω ⊂ R2. We let µ−, µ+ ∈ P(Ω) and denote by Xε
the subset of M(Ω, R2)× L2(Ω) of those pairs (σ, ϕ) such that
∇ · σ = (µ+− µ−)∗ ρε
for a standard symmetric convolution kernel ρε. For a couple (σ, ϕ)∈ M(Ω) × L2(Ω) we set
Fε(σ, ϕ) :=
Z
Ω
f (ϕ)|σ| + 1 2
ε|∇ϕ|2+ ϕ2 ε
dx if (σ, ϕ)∈ Xε
+∞ otherwise.
where the function f : R→ R is defined as
f (t) := (−h∗)−1(t2). (5.1) In the above formula h∗ is the concave Legendre transform of h (see Section 5.2 for its precise definition). The limit energy E is defined in equation (4.2) of Chapter 4. We prove:
Theorem 5.1. Let
h : R→ [0, +∞) be an even, continuous function
such that h(0) = 0 and h is concave on [0, +∞). (5.2) Let f be defined as above then
Fε
−Γ
→ E (5.3)
as ε→ 0.
In Section 5.2 we recall the definition of Legendre transform for a concave function and obtain some properties of the function f which are essential to the Γ-convergence result. The convergence result is obtained again by slicing and we will take advantage of the results in Appendix C following a strategy similar to the one in Chapter 4. We introduce the reduced dimension problem and study the upper and lower bound for the Γ-convergence result in Section 5.3.