Chapter 5 Phase field methods for binary recovery
5.3 Binary recovery application
We now show that (5.7) with both the smooth double well and double obstacle potentials can be fitted into the framework of the previous section.
Smooth double well potential
SetV, W :=H1(Ω), H :=L2(Ω), let S :H →H satisfy the assumptions in Section 5.1.3, and take b(u, η) := (Su, Sη) +σ1ε(∇u,∇η) c(u, η) := σ1 ε (u, η) l(u) := (S∗gd, u) J(u) := σ1 4ε Z Ω u4.
Here and throughout this chapter (·,·) denotes theL2(Ω) inner product. S∗ denotes the adjoint operator of S, which is defined as follows: For real Hilbert spaces U,
V the adjoint operator of a continuous linear operator A : U → V is the unique continuous linear operatorA∗ :V →U such that
(Au, v)V = (u, A∗v)U ∀u∈U, v ∈V.
This definition is a special case of the adjoint for operators involving Banach spaces (see (2.5) and (3.4)). The above objects have the properties required in Section 5.2. Coercivity of b can be shown using a contradiction argument and that S0 = 0. J
is well defined and continuous since H1(Ω) is continuously embedded inL6(Ω) for Ω⊂Rn withn≤3. I satisfies assumption (5.8) since
Z Ω u4 4 − u2 2 ≥ Z Ω u2 2 −1 = 1 2kuk 2 L2(Ω)− |Ω|,
and so using thatkSu−gdk2L2(Ω) ≥0 we get
I(u)≥ σ1ε 2 k∇uk 2 L2(Ω)+ σ1 2εkuk 2 L2(Ω)− σ1 ε |Ω| ≥ σ1 2 min ε,1 ε kuk2V −σ1 ε |Ω|
for allu∈W. MoreoverIequalsFεfrom (5.7) with the smooth double well potential (up to an additive constant), so (5.9) becomes: Givengd∈L2(Ω) find
arg min u∈H1(Ω) F1(u) := 1 2kSu−gdk 2 L2(Ω)+σ1 Z Ω ε 2|∇u| 2+1 εΨ1(u) ! . (5.15)
comes: Givengd∈L2(Ω), findu∈H1(Ω) such that (S∗(Su−gd), η) +σ1ε(∇u,∇η) +
σ1
ε (u
3−u, η) = 0 ∀η∈H1(Ω).
In this example we have an equality instead of a variational inequality because W
is the full spaceV.
(5.13) gives the following iterative method for solving the above variational inequality, and it converges by Theorem 5.6: Givengd∈L2(Ω) andu0∈H1(Ω), for
n= 1,2, ...find u=un∈H1(Ω) such that (S∗(Su−gd), η) +σ1ε(∇u,∇η) +
σ1
ε (u
3−un−1, η) = 0 ∀η ∈H1(Ω). (5.16)
Double obstacle potential
Define K := {u ∈ H1(Ω) : |u| ≤ 1 a.e. in Ω}. Set V := H1(Ω), W := K, H :=
L2(Ω), letS :H→H satisfy the assumptions in Section 5.1.3, and take
b(u, η) := (Su, Sη) +σ2ε(∇u,∇η)
c(u, η) := σ2
ε (u, η) l(u) := (S∗gd, u)
J(u) := 0.
The above objects have the properties required in Section 5.2. As with the smooth double well potential,I satisfies assumption (5.8) since foru∈W we have
− Z Ω u2 2 ≥ Z Ω u2 2 −1 = 1 2kuk 2 L2(Ω)− |Ω|.
MoreoverIequalsFεfrom (5.7) with the double obstacle potential (up to an additive constant), so (5.9) becomes: Given gd∈L2(Ω) find
arg min u∈K F2(u) := 1 2kSu−gdk 2 L2(Ω)+σ2 Z Ω ε 2|∇u| 2+ 1 2ε(1−u 2) ! . (5.17)
Solutions to (5.17) satisfy (5.11), which becomes: Given gd ∈ L2(Ω), find
u∈K such that
(S∗(Su−gd), η−u) +σ2ε(∇u,∇η− ∇u)−
σ2
ε (u, η−u)≥0 ∀η∈K.
inequality, which converges by Theorem 5.6: Given gd ∈ L2(Ω) and u0 ∈ K, for
n= 1,2, ...find u=un∈K such that
(S∗(Su−gd), η−u) +σ2ε(∇u,∇η− ∇u)−
σ2
ε (u
n−1, η−u)≥0 ∀η∈K. (5.18)
5.3.1 Alternative iterative methods
In (5.16) and (5.18) theS∗S term is taken implicitly (i.e. we solve for it rather than evaluate it), so we need to be able to invert the operator S∗S−σiε∆ efficiently, otherwise these iterative methods will be too computationally expensive. In some cases this may be possible, for example ifS is the identity, but in general this is not the case.
As we remarked earlier, the definitions ofband cthat makeI correspond to (5.15) and (5.17) are not unique. For example we can setb(u, η) =B(u, η) +ρ(u, η) and c(u, η) =C(u, η) +ρ(u, η) for some ρ ≥ 0. The ρ(u, η) terms cancel out in I, so definingB and C the same wayb and cwere defined earlier in this section gives the same optimisation problems (5.15) and (5.17). But the corresponding iterative methods are different. The point of this is that the ρ(u, η) term is convex (when
η=u), so it gives us more flexibility in how we define B and C while still having b
andc satisfy the coercivity and positivity assumptions.
In particular, for suitably largeρwe can take theS∗S term explicitly (which in our framework corresponds to moving it fromb toc), and also take the σi
ε(u, η) term implicitly (i.e. move it fromc to b). So for our examples this corresponds to taking
b(u, η) :=ρ(u, η) +σiε(∇u,∇η)−
σi
ε(u, η), c(u, η) :=ρ(u, η)−(S∗Su, η).
A restriction such asρ >max(σi
ε, Cs2), whereCs is the stability constant from (5.6), is then sufficient for both b to be coercive and c to be nonnegative. So we have the following iterative methods, which are in general easier to solve computationally than (5.16) and (5.18).
Example 5.7 (Smooth double well). Given gd ∈ L2(Ω) and u0 ∈ H1(Ω), for
n= 1,2, ... findu=un∈H1(Ω) such that
ρ(u−un−1, η) + (S∗(Sun−1−gd), η) +σ1ε(∇u,∇η) +
σ1
ε (u
3−u, η) = 0 (5.19)
Example 5.8 (Double obstacle). Given gd ∈ L2(Ω) and u0 ∈ K, for n = 1,2, ... findu=un∈K such that
ρ(u−un−1, η−u) + (S∗(Sun−1−gd), η−u) +σ2ε(∇u,∇η− ∇u)−
σ2
ε (u, η−u)≥0
(5.20) for allη∈K.
When solving Example 5.7 in practice, it is more convenient for us to solve a linear equation. Therefore we linearise theJ0(u) term in (5.19) and consider the following iterative method.
Example 5.9 (Smooth double well). Given gd ∈ L2(Ω) and u0 ∈ H1(Ω), for
n= 1,2, ... findu=un∈H1(Ω) such that
ρ(u−un−1, η) + (S∗(Sun−1−gd), η) +σ1ε(∇u,∇η) +
σ1
ε((u
n−1)2u−u, η) = 0 (5.21)
for allη∈H1(Ω).
This iterative method lies outside of our framework, so the convergence the- ory does not necessarily hold. However it works well in practice.
To finish this section we show how we can reformulate the iterative methods to removeS∗(Sun−1−gd) whenSis defined as in (5.5). For example, (5.20) becomes the following.
Example 5.10 (Double obstacle). Given gd ∈L2(Ω) and u0 ∈ K, for n= 1,2, ... findu=un∈K such that
ρ(u−un−1, η−u) + (pn−1, η−u) +σ2ε(∇u,∇η− ∇u)−
σ2
ε (u, η−u)≥0
for allη∈K, where pn−1∈H1(Ω)solves
α(∇pn−1,∇η) + (pn−1, η) = (yn−1−gd, η) ∀η ∈H1(Ω), andyn−1 solves the weak form of (5.5) withu=un−1.