approximation of the mumford-shah functional by … · bv-phase field approximation for the...

APPROXIMATION OF THE MUMFORD-SHAH FUNCTIONALBY PHASE FIELDS OF BOUNDED VARIATION

SANDRO BELZ

Department of Mathematics, Technical University of MunichBoltzmannstraße 3, 85748 Garching, Germany

[email protected]

KRISTIAN BREDIES

Institute of Mathematics and Scientific Computing, University of GrazHeinrichstraße 36, 8010 Graz, Austria

[email protected]

Abstract. In this paper we introduce a new phase field approximation ofthe Mumford-Shah functional similar to the well-known one from Ambrosioand Tortorelli. However, in our setting the phase field is allowed to be afunction of bounded variation, instead of an H1-function. In the context ofimage segmentation, we also show how this new approximation can be usedfor numerical computations, which contains a total variation minimization ofthe phase field variable, as it appears in many problems of image processing.A comparison to the classical Ambrosio-Tortorelli approximation, where thephase field is an H1-function, shows that the new model leads to sharper phasefields.

1. Introduction

The Mumford-Shah functional has been introduced by D. Mumford and J. Shahin [37] in the context of image segmentation. For a given image, g ∈ L∞(Ω), whereΩ ⊂ Rn represents the image domain, it is given by

α

2

∫Ω|∇u|2 dx+ β

2

∫Ω|u− g|2 dx+ γH1(Γ) (1.1)

where α, β, γ > 0 are parameters, free to choose. One wants to minimize the func-tional with respect to u ∈ C1(Ω \ Γ), being the segmentally denoised approximationof g, and Γ ⊂ Ω closed, describing the contours of the segments. For β = 0 thisfunctional appeared once more in [30] in the context of fracture mechanics. There,u models the displacement function, and Γ ⊂ Ω being closed represents the fractureset. The minimization is then restricted to some Dirichlet boundary condition.

2010 Mathematics Subject Classification. 49J45, 26A45, 68U10.Key words and phrases. Mumford-Shah, free-discontinuity problem, Γ-convergence, phase field,

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2 S. BELZ AND K. BREDIES

As usual in the theory of free-discontinuity problems (see [8,16]) the Mumford-Shah functional (1.1) is relaxed to the space of special functions of bounded variation(see Section 2.3 for more details on these functions), where the set Γ is replaced bythe discontinuity set Su. Namely, instead of (1.1) one considers

α

2

∫Ω|∇u|2 dx+ β

2

∫Ω|u− g|2 dx+ γH1(Su) (1.2)

for u ∈ SBV(Ω), the set of special functions of bounded variation. In this setting theexistence of the minimizers is well-known and follows from compactness propertiesof SBV(Ω)∩L∞(Ω) and some lower semi-continuity properties (see [5–7]), using thedirect method in the calculus of variations. Furthermore, by the regularity propertyshown in [26] we know that for any minimizer u ∈ SBV(Ω) of (1.2) the pair (u, Su)minimizes (1.1).

As already mentioned, in the case of fracture mechanics, we usually have β = 0and L2-penalization is replaced by a Dirichlet boundary condition. In general, thefunctional must then be defined on GSBV(Ω), the set of generalized special functionsof bounded variation (see Section 2.3), in order to obtain the existence of a minimizer.This is due to the requirement of a uniform bound of the minimizing sequence in thedirect method for applying the above-mentioned compactness properties in SBV(Ω).Only for β > 0, this L∞-bound is automatically achieved, whereas for β = 0 onehas to fall back to GSBV(Ω).

For numerical computations some variational approximations in terms of Γ-convergence (see Section 2.2) turned out to be very useful. It guarantees that aconvergent sequence of minimizers of the approximating functionals converge tominimizers of the Γ-limit. We firstly discuss in Theorem 3.2 an approximation forβ = 0. Hence, we consider the functional

MS(u) := α

2

∫Ω|∇u|2 dx+ γH1(Su) for all u ∈ GSBV(Ω) .

One of the first and most popular results in this direction was given by L.Ambrosio and V. M. Tortorelli in [10]. They introduced the functionals

AT ε(u, v) =∫

Ω(v2 + ηε)|∇u|2 dx+

∫Ω

14ε (1− v)2 + ε|∇v|2 dx (1.3)

for u ∈ H1(Ω) and v ∈ H1(Ω; [0, 1]) and showed via a Γ-convergence argument thatany limit point (u, 1) of a sequence of minimizers (uε, vε) of AT ε is a minimizer ofMS, provided that ηε

ε → 0. Many other approximations based on this result havebeen proven. Just recently, we proved that the Euclidean norms of the gradientscan be replaced by Riemannian norms (see [2]). This result finds application infracture mechanics applied to surfaces. Another approach considering higher orderterms of the phase field has been studied e.g. in [14] and [20]. What happens withthe approximation AT ε when ηε

ε does not converge to zero is investigated in [25]and [34]. A totally different idea of approximating MS by finite differences wasproposed by E. De Giorgi and proven by M. Gobbino in [32]. In [18] A. Braides andG. Dal Maso used non-local functionals depending on the average of the gradientof u on small balls. From the work presented in [16] one gets an approximation ofMS for the following functional with small ε > 0:∫

Ω(v2 + ηε)|∇u|2 dx+ 1

2p′ε

∫Ω

(1− v)p′dx+ εp−1

∫Ω|∇v|p dx (1.4)

BV-PHASE FIELD APPROXIMATION FOR THE MUMFORD-SHAH FUNCTIONAL 3

for u ∈ H1(Ω) and v ∈W 1,p(Ω) with p > 1 and p′ being the Hölder conjugate of p.In the approximations (1.3), (1.4) and in the functionals we are going to study

in this paper the additional function v works as a phase field variable describing thediscontinuity set of u. To be more precise, for small ε > 0 the minimizing functionv is close to 0 where u is “steep” or jumps, which means in the context of fracturemechanics the presence of a crack and in the context of image segmentation thepresence of a contour. Elsewhere, the phase field variable is close to 1 and u isexpected to be “flat” in this area. In practice the weights of the different integralterms declare what is meant to be “steep” or “flat”.

In this paper we present a new approximation of the Mumford-Shah functional,allowing the phase field variable v to be in BV(Ω), the set of functions of boundedvariation. Namely, as a special case of our main result we consider the functionals

α

2

∫Ω

(v2 + ηε)|∇u|2 dx+ γ

2ε

∫Ω

(1− v) dx+ γ

2 |Dv|(Ω)

for u ∈ H1(Ω) and v ∈ BV(Ω), which Γ-converge in some sense toMS and representthe case with p = 1 in (1.4). From there we derive the required setting for β > 0.

In this way the phase field variable v can have jumps, which is exploited in theproof of the lim sup-inequality (see Proposition 2.2), when constructing the recoverysequence for the Γ-convergence result. Moreover, we expect from this fact thatthe phase fields become somewhat sharper than the ones obtained from (1.3). Weapprove this expectation with some numerical computations in the context of imagesegmentation. The application of this theory to fracture mechanics remains for futurework, which includes the studying the convergence behaviour of a time-discreteevolution as it was done – and is still ongoing – for the classical approach of Ambrosioand Tortorelli. For more details on this topic we refer to [1, 3, 4, 31,36,38,39].

The paper is structured as follows: In Section 2 we start with some preliminariesrecalling the necessary technical issues. For the versed reader this section mightbe skipped or only used as a reference text. In Section 3 we formulate our mainresult, Theorem 3.2, from which we directly infer all other necessary theoremsand corollaries. Section 4 is then dedicated to the proof of Theorem 3.2 and inSection 5 we provide some numerical comparison of our new model and the classicalAmbrosio-Tortorelli approximation.

2. Preliminaries and Notation

In this section, we collect the notation and the well-known results from theliterature which are used in this paper.

With Bρ(x) we denote the Euclidean ball with radius ρ > 0 and center x ∈ Rn.For some S ⊂ Rn the set Bρ(S) refers to the ρ-neighborhood of S. The set Sn−1 isthe n− 1-dimensional sphere in Rn. At some places it is convenient to use the shortnotation a ∨ b and a ∧ b for maxa, b and mina, b, respectively.

The essential supremum and the essential infimum of some measurable func-tion u is written as ess supu and ess inf u, respectively. The essential support of ameasurable function u is denoted by suppu.

2.1. Measure theory. For any set Ω ⊂ Rn we denote by Ln(Ω) the n-dimensionalLebesgue measure and by Hk(Ω) the k-dimensional Hausdorff measure. Instead ofH0 we also use the symbol # for the counting measure. For a (signed, vector-valued)measure µ we write |µ| for its total variation.


2.2. Γ-convergence. For some sequence of functionals (Fj) and a functional Fdefined on some metric space X we say that Fj Γ-converges to F as j → ∞ andwrite Γ-limj→∞ Fj = F if there holds thelim inf-inequality: for all u ∈ X and all sequences (uj) in X with uj → u there

holdsF (u) ≤ lim inf

j→∞Fj(uj) . (2.1)

lim sup-inequality: for all u ∈ X there exists a sequence (uj) in X such thatuj → u and

lim supj→∞

Fj(uj) ≤ F (u) . (2.2)

One often defines

Γ-lim infj→∞

Fj(u) := inflim infj→∞

Fj(uj) : uj ∈ X for all j > 0, uj → u as j →∞ ,

Γ-lim supj→∞

Fj(u) := inflim supj→∞

Fj(uj) : uj ∈ X for all j > 0, uj → u as j →∞ .

Then the lim inf-inequality is equivalent to F ≤ Γ-lim infj→∞ Fj and the lim sup-inequality is equivalent to Γ-lim supj→0 Fj ≤ F . Note that Γ-lim infj→∞ Fj as wellas Γ-lim supj→∞ Fj are lower semi-continuous.

If one has a family of functionals (Fε) for ε ∈ I ⊂ R the definition is adapted inthe usual way, i.e. Fε Γ-converges to F as ε→ a (for some a ∈ I) if Fεj Γ-convergesto F for all sequences (εj) in I with εj → a.

The most important property of Γ-convergent sequences is the convergence ofminimizers to a minimizer of the limit functional, which is stated in the followingproposition.

Proposition 2.1. Let (Fε), where Fε : X → R ∪ ∞, be a sequence of functionalsΓ-converging to F : X → R ∪ ∞ with respect to the metric space X. Assume thatinfX Fε = infK Fε for some compact set K ⊂ X. Then, there holds limε→0 infX Fε =infX F . Furthermore, for any sequence (uε) in X converging to u ∈ X with Fε(uε) =infX Fε we have F (u) = infX F .

If F = Γ-limj→∞ Fj and u ∈ X, a sequence (uj), for which (2.2) holds, is called arecovery sequence for u, and there clearly holds limj→∞ Fj(uj) = F (u). It is actuallythe case that a convergent sequence of minimizers is a recovery sequence for theminimizer of the Γ-limit. For this reason knowing the recovery sequences provideslots of information about the structure of the limit behaviour of the functionalsequence.

For more details on the concept of Γ-convergence we refer to [17] and [24].

2.3. Functions of bounded variation. In the following we describe the conceptand some essential results of functions of bounded variation. For an extensivemonograph on this topic we refer to [8]. A more basic introduction can be foundin [28].

Let Ω ⊂ Rn be non-empty and open for the rest of this section. The set offunctions of bounded variation, in short BV(Ω), contains all functions u ∈ L1(Ω)whose distributional derivative is a Radon measure, denoted by Du, i.e. there holds∫

Ωudivw dx = −

∫Ωw dDu for all w ∈ C1

c (Ω;Rn) . (2.3)


Defining the total variation

V (u,Ω) = sup∫

Ωudivw dx : w ∈ C1

c (Ω;Rn), ‖w‖∞ ≤ 1

(2.4)

we obtain from the Riesz representation theorem that (2.3) is equivalent toV (u,Ω) <∞. Furthermore, there holds |Du|(Ω) = V (u,Ω) for all u ∈ BV(Ω).

For any measurable function u : Ω → R we define for all x ∈ Ω the upper andlower approximate limit, respectively, by

u+(x) = inft ∈ R : lim

ρ→0

Ln(u > t ∩Bρ(x)

)ρn

= 0,

u−(x) = supt ∈ R : lim

ρ→0

Ln(u < t ∩Bρ(x)

)ρn

= 0.

For all x ∈ Ω there obviously holds u−(x) ≤ u+(x). If u−(x) = u+(x) we write fortheir common value u∗(x). The set Su is the discontinuity set containing all thosepoints x ∈ Ω for which there holds u−(x) < u+(x).

In what follows let u ∈ BV(Ω). Then, Su has Lebesgue measure 0 and forHn−1-almost all points x ∈ Su one can find a unit normal vector νu(x) such thatu+(x) =

(u|H+(x)

)∗(x) and u−(x) =(u|H−(x)

)∗(x) with

H+(x) =y ∈ Ω:

⟨y − x, νu(x)

⟩> 0

H−(x) =y ∈ Ω:

⟨y − x, νu(x)

⟩< 0.

If this is the case one says that x is a jump point. We call u a precise representativeof u if u(x) = u∗(x) for all x ∈ Ω \ Su and u(x) = 1

2 (u+(x) + u−(x)) for all jumppoints x ∈ Su.

For functions of bounded variation on the real line we actually have that everypoint in Su is a jump point. Furthermore, on an open interval the pointwisevariation of u and the variation as defined in (2.4) coincide. Precisely, for a < b andu ∈ BV(a, b) there holds

V(u, (a, b)

)= sup

N∑i=1

∣∣u(ti)− u(ti−1)∣∣ : N ∈ N, a < t0 < · · · < tN < b

. (2.5)

For any u ∈ BV(Ω) one can split the measure Du in the following way

Du = Dau+ Dju+ Dc ,

where Dau = ∇uLn denotes the absolutely continuous part of Du with respect tothe Lebesgue measure. Therefore, with ∇u we denote its density function, whichwe also call the approximate gradient of u. With Dju we denote the jump part of u,which can be written as Dju = (u+ − u−) · νuHn−1xSu, and Dcu is the Cantor part.

The set of special functions of bounded variation, denoted by SBV(Ω), con-tains those functions of bounded variation whose Cantor part is zero, i.e. wehave SBV(Ω) = u ∈ BV(Ω) : Dcu = 0. The singular part of such functions istherefore only concentrated on the set of jump points.

A measurable function u : Ω → R is a generalized special function of boundedvariation, where we write u ∈ GSBV(Ω), if any truncation of u is locally a specialfunction of bounded variation, i.e. uM ∈ SBVloc(Ω) for all M > 0, with uM =(−M)∨ u∧M . Note that for u ∈ GSBV(Ω) we cannot define ∇u as above, becausethe distributional derivative does not need to be a measure on that space. However,


∇uM is well defined for all M > 0 and converges pointwise a.e. as M →∞. Thus,we simply define ∇u(x) = limM→∞∇uM (x) for a.e. x ∈ Ω. Furthermore, onecan show that Su =

⋃M>0 SuM . These results and more details can be found

in [8, Section 4.5] and the references therein.Moreover, we will use the following two subspaces of GSBV(Ω) and SBV(Ω)

defined for every p > 0 bySBVp(Ω) =

u ∈ SBV(Ω): ∇u ∈ Lp(Ω),Hn−1(Su) <∞

GSBVp(Ω) =

u ∈ GSBV(Ω): ∇u ∈ Lp(Ω),Hn−1(Su) <∞

.

A density result, which plays an important role in the proof of the lim sup-inequality for our main assertion, is stated in the next theorem. It follows directlyfrom [23, Theorem 3.1] and the following remarks therein.

Theorem 2.2. Let Ω ⊂ Rn be non-empty, open and bounded with Lipschitz bound-ary, and take u ∈ SBV2(Ω) ∩ L∞(Ω). Then, there exists a sequence (wj) inSBV2(Ω) ∩ L∞(Ω) such that

1. Swj is a polyhedral set,2. Hn−1(Swj \ Swj) = 0 ,3. wj ∈W 1,∞(Ω \ Swj ) for all j ∈ N ,4. wj → u in L1(Ω) as j →∞ ,

5. ∇wj → ∇u in L2(Ω) as j →∞ ,

6. Hn−1(Swj )→ Hn−1(Sw) as j →∞ .

We now shortly introduce the concept of slicing, which is essential for the proofof the lim inf-inequality. For that purpose, let Ω ⊂ Rn be open and bounded, andlet ξ ∈ Sn−1 be a unique normal vector. Then, we write Ωξ for the projection of Ωonto ξ⊥, and we set

Ωξy := t ∈ R : y + tξ ∈ Ω for all y ∈ Ωξ .

Furthermore, for any function u ∈ L1(Ω) and for Ln−1-a.a. y ∈ Ωξ we can defineuξy(t) := u(y + tξ) for a.a. t ∈ Ωξy.

One can show the following important results revealing the connection between afunction u ∈ SBV(Ω) and its sliced functions uξy. There are more general results forBV-functions, which are not needed in this context. The interested reader can findthe details in [8, Section 3.11].

Theorem 2.3. Let u ∈ L1(Ω). Then u ∈ SBV(Ω) if and only if for all ξ ∈ Sn−1

there holds uξy ∈ SBV(Ωξy) for Ln−1-a.a. y ∈ Ωξ and∫Ωξ

∣∣Duξy∣∣(Ωξy)dLn−1(y) <∞ .

Furthermore, if u ∈ BV(Ω) there holds for all ξ ∈ Sn−1, for Ln−1-a.a. y ∈ Ωξ andfor a.a. t ∈ Ωξy:

1. (uξy)′(t) =⟨∇u(y + tξ), ξ

⟩,

2. Suξy = (Su)ξy,3. (uξy)± (t) = u±(y + tξ),4.∣∣〈D∗u, ξ〉∣∣(Ω) =

∫Ωξ

∣∣D∗uξy∣∣(Ωξy) dLn−1(y) for ∗ = a, j, c.

The following corollary directly follows by a truncation argument.


Corollary 2.4. Let u ∈ L1(Ω). Then u ∈ GSBV(Ω) if and only if for all ξ ∈ Sn−1

there holds uξy ∈ SBV(Ωξy) for Ln−1-a.e. y ∈ Ωξ and∫Ωξ

∣∣D((−M) ∨ uξy ∧M)∣∣(Ωξy)dLn−1(y) <∞ for all M > 0 .

2.4. Convex functions. Especially, for the numerical part of this paper we alsoneed some theory about convex functions. A good reference for this topic is [33]and [27].

For Ω ⊂ Rn the characteristic function χΩ over Ω is given by χΩ = 0 on Ω andχΩ = +∞ on Rn \ Ω. It is a convex function if and only if Ω is a convex set. Forany function f : Ω→ R, bounded from below by some affine function, f∗ : Rn → Rdenotes its convex conjugate, i.e.

f∗(y) = supx∈R

(〈x, y〉 − f(x)

)for all s ∈ Rn

where f is set to +∞ outside of Ω. This definition directly yields Fenchel’s inequality,which says

〈x, y〉 ≤ f(x) + f∗(y) for all x, y ∈ Rn . (2.6)We remark that f∗ is always convex and lower semi-continuous and the biconjugatef∗∗ = (f∗)∗ is the lower semi-continuous convex hull of f . Furthermore, f is convexand lower semi-continuous if and only if f = f∗∗.

We will also make use of the subdifferential of a function f : Rn → (−∞,+∞],which we denote by ∂f . It is given by

∂f(x) = z ∈ Rn : f(x)− f(y) ≤ 〈z, x− y〉 for all y ∈ Rn for all x ∈ Rn .

If f is differentiable in x ∈ Rn, we have ∂f(x) = ∇f(x).

3. Main Result

For our main result we need several, quite technical assumptions. In order tokeep a better overview we first list them here.

Assumption 3.1. Let ε0 > 0. For each 0 < ε < ε0 let[A1] Wε : [0, 1]→ [0,∞) be continuous such that Wε →W in L1([0, 1]) as ε→ 0

for some W ∈ L1([0, 1]) with 1 ∈ suppW .[A2] ϕε : Wε([0, 1]) → R be a convex function such that ϕε(Wε(1)) → 0 and

ϕε(Wε(·)) → +∞ uniformly on [0, T ] for all 0 < T < 1, i.e. for all C > 0there exists 0 < ε < ε0 such that ϕε(Wε(t)) > C for all t ∈ [0, T ] and ε < ε.

[A3] ψε : [0,∞)→ [0,∞) be a convex function such that limt→∞ψε(t)t = cε <∞,

ψε(0)→ 0 and cε → c0 :=∫ 1

0 W (s) ds as ε→ 0 for W from [A1], ϕ∗ε ≤ ψεon [0,∞), where ϕ∗ε denotes the convex conjugate of ϕε (see Section 2.4),and ψε(t) ≥ ct+ d for all t ≥ 0 and some c > 0, d ∈ R independent of ε.

[A4] ηε > 0 such that ηεϕε(Wε(0))→ 0 as ε→ 0.Furthermore, assume that

[A5] f : [0, 1] → [0,∞) is a Lipschitz continuous, non-decreasing function withf(0) = 0 and f > 0 on (0, 1].

We are now ready to state our main theorem.


Theorem 3.2. Let Ω ⊂ Rn be a non-empty, open, bounded set with Lipschitzboundary, let Wε, ϕε, ψε, ηε, f and cε, c0 > 0 be given as in Assumption 3.1. Foreach ε > 0, we define the functional Fε : L1(Ω)× L1(Ω)→ R by

Fε(u, v) :=∫

Ω

(f(v) + ηε

)|∇u|2 dx+

∫Ωϕε(Wε(v)

)+ ψε

(|∇v|

)dx

+ cε(|Djv|(Ω) + |Dcv|(Ω)

)(3.1)

for all u ∈ H1(Ω), v ∈ BV(Ω; [0, 1]) and Fε(u, v) := +∞ otherwise.Moreover, define F : L1(Ω)× L1(Ω)→ R by

F (u, v) :=

∫

Ωf(1)|∇u|2 dx+ 2c0Hn−1(Su) for u ∈ GSBV2(Ω), v = 1 a.e.,

+∞ otherwise.

Then F = Γ-limε→0 Fε with respect to the strong topology in L1(Ω)× L1(Ω).

For our application in image segmentation we aim for a minimization of (1.2). Inthe following corollary we add the missing L2-penalization term in the functionals Fand Fε.

Corollary 3.3. Let Ω ⊂ Rn be a non-empty, open, bounded set with Lipschitzboundary, let Wε, ϕε, ψε, ηε, f and cε, c0 > 0 be given as in Assumption 3.1.Furthermore, let β > 0 and g ∈ L∞(Ω), and let Fε and F be given as in Theorem 3.2.We define for every ε > 0 the functional

Gε(u, v) :=

Fε(u, v) + β

2

∫Ω|u− g|2 dx for u ∈ H1(Ω), v ∈ BV(Ω; [0, 1]),

+∞ otherwise.

Moreover, we define G : L1(Ω)× L1(Ω)→ R by

G(u, v) :=

F (u, v) + β

2

∫Ω|u− g|2 dx for u ∈ SBV2(Ω) ∩ L∞(Ω), v = 1 a.e.,

+∞ otherwise.

Then, G = Γ-limε→0Gε with respect to the strong topology in L1(Ω)× L1(Ω).

Proof. Since u 7→∫

Ω|u − g|2 dx is lower semi-continuous, the lim inf-inequalityfollows directly from Theorem 3.2. Hence, we have

G(u, v) ≤ Γ-lim infε→0

Gε(u, v) for all u, v ∈ L1(Ω) . (3.2)

In order to show the lim sup-inequality it suffices to consider u ∈ SBV2(Ω)∩L∞(Ω)and v = 1 a.e., since otherwise, the left hand side of (3.2) is +∞ and there is nothingto show.

From Theorem 3.2 we know that there exists a sequence (uε, vε) in H1(Ω)×BV(Ω)converging to (u, v) as ε→ 0 in the strong L1(Ω)× L1(Ω)-topology such that

limε→0

Fε(uε, vε) = F (u, v) .

We consider the truncated function sequence uMε with M = ‖u‖L∞ , and note thatuMε → u in L2(Ω) as ε→ 0. Therefore, we also have

limε→0

∫Ω|uMε − g|2 dx =

∫Ω|u− g|2 dx .


Furthermore, one can easily verify that Fε(uMε , vε) ≤ Fε(uε, vε), so that

lim supε→0

Gε(uMε , vε) ≤ lim supε→0

Fε(uε, vε) + β

2 lim supε→0

∫Ω|uMε − g|2 dx = G(u, v) ,

which is the required lim sup-inequality.

In view of Proposition 2.1 the existence and compactness of minimizers of theapproximating functionals Gε needs to be shown in order to obtain their convergenceto a minimizer of the functional G. We give a rigorous proof in the following theorem.

Theorem 3.4. In the setting of Corollary 3.3 a minimizer of Gε exists for everyε > 0. Furthermore, let εj be an infinitesimal sequence, and let (uεj , vεj ) ∈ H1(Ω)×BV(Ω; [0, 1]) be a minimizer of Gεj for every j ∈ N. Then, vεj → 1 in L1(Ω), andthere exists u ∈ SBV2(Ω) ∩ L∞(Ω), such that, up to a subsequence, uεj → u inL1(Ω), and (u, 1) minimizes G.

Proof. In order to show the existence of minimizers of Gε we fix ε > 0 and take aminimizing sequence (uj , vj) of Gε, i.e.

limj→∞

Gε(uj , vj) = infL1(Ω)×L1(Ω)

Gε = infH1(Ω)×BV(Ω;[0,1])

Gε .

In view of ψε(t) ≥ ct+ d for all t ≥ 0 and some c > 0 as stated in Assumption [A3],it is easy to see that (|Dvj |(Ω)) is bounded. Further, (uj) is bounded in H1(Ω),since ηε > 0. By the compactness properties of BV(Ω) (see [8, Theorem 3.23])and H1(Ω) there exist subsequences of (uj) and (vj) (not relabeled) and functionsv ∈ BV(Ω) and u ∈ H1(Ω), such that vj → v in L1(Ω), Dvj converges sequentiallyweakly* (in the space of Radon measures) to Dv and uj u weakly in H1(Ω).

From Fatou’s Lemma and [8, Theorems 5.4 and 5.8] we get the lower semi-continuity of Gε so that

Gε(u, v) ≤ lim infj→∞

Gε(uj , vj) ≤ infH1(Ω)×BV(Ω;[0,1])

Gε .

Hence, the pair (u, v) minimizes Gε.Now let (εj) be a sequence converging to 0, and let the pair (uεj , vεj ) be a

minimizer of Gεj for every j ∈ N. Then we simply have∫Ωϕεj(Wεj (vεj )

)dx ≤ min

L1(Ω)×L1(Ω)Gεj ≤ Gεj (0, 0) = β

∫Ω|g|2 dx

which implies, together with Assumption [A2], vεj → 1 in L1(Ω) as εj → 0.From a simple cut-off argument we get that ‖uεj‖L∞(Ω) ≤ ‖g‖L∞(Ω). Since f

is Lipschitz continuous according to Assumption [A5], we have f(vεj ) ∈ BV(Ω)with Df(vεj ) obeying the chain rule for BV-functions (see [8, Theorem 3.99]).Further, the multiplication operation (s, t) 7→ st is continuously differentiableand Lipschitz continuous on bounded sets, thus, since both uεj and f(vεj ) area.e. bounded, the product rule for BV-functions holds (see [8, Theorem 3.99]), givingwεj := uεjf(vεj ) ∈ BV(Ω). We moreover have

|Dwεj |(Ω) ≤∫

Ω|∇uεj |f(vεj ) dx+

∫Ω|uεj |f ′(vεj )|∇vεj |dx

+∫Svεj

|uεj |(f(v+

εj )− f(v−εj ))

dHn−1 +∫

Ω|uεj |f ′(vεj ) d|Dcvεj | .


Since uεj , f and f ′ are bounded, we can estimate, employing a weighted version ofthe Cauchy–Schwarz inequality and Young’s inequality,

|Dwεj |(Ω) ≤ C(

1 +∫

Ωf(vεj )|∇uεj |2 dx

)+ ‖g‖L∞(Ω)‖f ′‖L∞([0,1])

(∫Ω|∇vεj |dx+ |Djvεj |(Ω) + |Dcvεj |(Ω)

)where C > 0 depends on f and Ω.

By Assumption [A3], t ≤ c−1(ψεj (t)− d) for all t ≥ 0 and some c > 0, d ∈ R, so∫Ω|∇vεj |dx ≤ C

(1 +

∫Ωψεj(|∇vεj |

)dx)

with C > 0 suitably chosen. Altogether, since (cε) is bounded, we obtain

|Dwεj |(Ω) ≤ C(1 +Gεj (uεj , vεj )

)≤ C

(1 + β

∫Ω|g|2 dx

),

where here C > 0 is a constant depending on Ω, g, f and c0. Hence, |Dwεj |(Ω) isbounded.

Clearly, wεj is pointwise a.e. bounded independent of j, so by the compactnessproperties of BV(Ω) (see [8, Theorem 3.23]) there exists a subsequence of εj (notrelabeled) converging to 0, such that wεj converges to some w in L1(Ω). Sincevεj → 1 a.e. and f is continuous, we also have that uεj = wεj/f(vεj ) → w/f(1)a.e. as εj → 0. Since ‖uεj‖L∞(Ω) ≤ ‖g‖L∞(Ω), the Dominated Convergence Theoremyields uεj → w/f(1) in L1(Ω).

The assertion now follows from Proposition 2.1 and Corollary 3.3.

Remark 3.5. Note that Theorem 3.2 and Corollary 3.3 also holds for ηε = 0. However,for the existence of minimizers of Gε we require ηε > 0 as indicated in the proof ofTheorem 3.4.

The following corollary represents a special case of the previous results, whichrepresents the version that is relevant for our numerical computation in Section 5.

Corollary 3.6. Let Ω ⊂ Rn be a non-empty, open, bounded set with Lipschitzboundary and let α, β, γ > 0. For each ε > 0 let ηε > 0 such that ηε

ε → 0 as ε→ 0and define the functionals Gε : L1(Ω)× L1(Ω)→ R by

Gε(u, v) := α

2

∫Ω

(v2 + ηε)|∇u|2 dx+ β

2

∫Ω|g − u|2 dx

+ γ

2ε

∫Ω

(1− v) dx+ γ

2 |Dv|(Ω)

if u ∈ H1(Ω), v ∈ BV(Ω; [0, 1]) and Gε(u, v) := +∞ otherwise. Moreover, defineG : L1(Ω)× L1(Ω)→ R by

G(u, v) := α

2

∫Ω|∇u|2 dx+ β

2

∫Ω|g − u|2 dx+ γHn−1(Su)

for u ∈ SBV2(Ω) ∩ L∞(Ω), v = 1 a.e., and G(u, v) = +∞ otherwise.Then, for every infinitesimal sequence (εj) a minimizer (uεj , vεj ) of Gεj exists

for every j ∈ N. Furthermore, vεj → 1 in L1(Ω), and up to a subsequence uεj → u

in L1(Ω) with (u, 1) being a minimizer of G.


Proof of Corollary 3.6. We define Gε := 2γGε and, choose the functions f , Wε, ϕε

and ψε in the following way:

f(t) = α

γt2, Wε(t) = (1− t)ε, ϕε(t) = 1

εt

1ε , ψε(s) = s

for all t ∈ [0, 1], s ∈ [0,∞) and 0 < ε < 1. Note that in this setting we have

ϕ∗ε(s) =

(1− ε)(ε2εs)1

1−ε for s ∈ [0, ε−2] ,

s− 1ε

for s > ε−2 ,

and hence, one can simply verify that Assumption 3.1 is fulfilled with c0 = 1.From Theorem 3.2 we, therefore, get that Gε Γ-converges to G := 2

γG. SinceΓ-convergence is preserved under constant multiplication we get the result bymultiplying Gε and G with γ

2 .

4. Proof of Theorem 3.2

The proof of Theorem 3.2 follows the usual strategy that has been used forthe classical Ambrosio-Tortorelli approximation and various generalizations (see[9, 10, 16, 25, 34, 35]). Firstly, we show the lim inf-inequality on the real line (seeProposition 4.1). The generalization to the multi-dimensional case, stated inProposition 4.2, is then shown by a slicing argument.

The lim sup-inequality is shown with the help of the density result, Theorem 2.2.Here, we exploit the fact that the phase field variable is allowed to have jumps,which enables the construction of a much simpler recovery sequence than when thephase field needs to be smooth.

Proposition 4.1. In the setting of Theorem 3.2 with Ω ⊂ R we redefine F : L1(Ω)×L1(Ω)→ R by

F (u, v) :=

∫

Ωf(1)|u′|2 dx+ 2c0#Su for u ∈ SBV2(Ω), v = 1 a.e.

+∞ otherwise

Then there holds F ≤ Γ-lim infε→0 Fε.

Proof. First of all, for each open set I ⊂ Ω we define the localized functionals

Fε(u, v; I) :=∫I

(f(v) + ηε

)|u′|2 + ϕε

(Wε(v)

)+ ψε

(|v′|)

dx

+ cε(|Djv|(I) + |Dcv|(I)

)for all u ∈ H1(I) and v ∈ BV(I; [0, 1]), and Fε(u, v; I) := +∞ otherwise.

Now, let (εj) be a sequence greater than zero with εj → 0 as j → ∞, and let(uj) and (vj) be sequences in L1(Ω) such that uj → u and vj → v as j →∞. Bypossibly extracting a subsequence, we can assume that

lim infj→∞

Fεj (uj , vj) = limj→∞

Fεj (uj , vj) <∞ .

Therefore, we must have∫

Ω ϕεj (Wεj (vj)) dx < ∞, and because of to the uniformconvergence of ϕεj (Wεj (·)) to +∞ as ε→ 0 (see [A2]), we obtain that v = 1 a.e. onΩ.


We first show that #Su is finite and2c0#Su ≤ lim inf

j→∞Fεj(uj , vj ;Bδ(Su)

)for all δ > 0 sufficiently small . (4.1)

For that let y0 ∈ Su, and let δ > 0 sufficiently small such that Bδ(y0) ⊂ Ω.Set M := lim infj→∞ ess infB δ

2(y0)(f vj) and assume that M > 0. Furthermore,

let 0 < κ < M and choose j0 > 0 such that up to a subsequence, there holdsM < ess infB δ

2(y0)(f vj) + κ for all j > j0. Then there holds∫ y0+ δ

2

y0− δ2|u′j |2 dx ≤ 1

M − κ

∫ y0+ δ2

y0− δ2f(vj)|u′j |2 dx ≤ C

M − κfor all j > j0

so that u′j converges weakly to u′ in L2(B δ2(y0)) and consequently y0 /∈ Su. Hence,

we must have M = 0, and we can find a sequence (yj) such that f(vj(yj)) → 0,where vj is a precise representative of vj . The assumptions on f in [A5] implyvj(yj) → 0 as j → ∞. Since vj → 1 a.e. we can, therefore, find y+, y− ∈ Bδ(y0)with y− < y0 < y+ such that vj(y−)→ 1 as well as vj(y+)→ 1.

With this at hand we get from the L1-convergence of Wε (see [A1]),

2c0 = limj→∞

[∫ vj(y+)

vj(yj)Wεj (s) ds+

∫ vj(y−)

vj(yj)Wεj (s) ds

]. (4.2)

Defining

Φε(t) :=∫ t

0Wε(s) ds for all t ∈ [0, 1], ε > 0 (4.3)

we get, for j large enough,∫ vj(y+)

vj(yj)Wεj (s) ds+

∫ vj(y−)

vj(yj)Wεj (s) ds

=∣∣Φεj(vj(y+)

)− Φεj

(vj(yj)

)∣∣+∣∣Φεj(vj(y−)

)− Φεj

(vj(yj)

)∣∣and together with (2.5)∫ vj(y+)

vj(yj)Wεj (s) ds+

∫ vj(y−)

vj(yj)Wεj (s) ds ≤

∣∣D(Φεj vj)∣∣(Bδ(y0)

). (4.4)

Applying Lemma A.2 yields∣∣D(Φεj vj)∣∣(Bδ(y0)

)≤∫ y0+δ

y0−δϕε(Wε(v)

)dx+

∫ y0+δ

y0−δψε(|v′|) dx

+ cε

(|Djv|

(Bδ(y0)

)+ |Dcv|

(Bδ(y0)

))(4.5)

By merging (4.2), (4.4) and (4.5) and since cε → c0 as ε→ 0 (see [A3]) we deduce2c0 ≤ lim inf

j→∞Fεj(uj , vj ;Bδ(y0)

).

For every N ≤ #Su we can repeat the preceding arguments for each element ina set y1, . . . , yN ⊂ Su with δ > 0 sufficiently small such that Bδ(yk) ∩Bδ(y`) = ∅for k 6= ` in order to obtain

2c0N ≤N∑k=1

lim infj→∞

Fεj(uj , vj ;Bδ(yk)

)≤ lim inf

j→∞Fεj

(uj , vj ;

N⋃k=1

Bδ(yk)).


By assumption the right hand side is finite; hence, there must hold #Su <∞ andwe deduce (4.1).

In the next step we show that for all δ > 0,∫Ω\Bδ(Su)

f(1)|u′|2 dx ≤ lim infj→∞

Fεj(uj , vj ; Ω \Bδ(Su)

). (4.6)

Let I := (a, b) ⊂ Ω be an open interval such that I ∩ Su = ∅. For k ∈ N and` ∈ 1, . . . , k we define the intervals

Ik` :=(a+ `− 1

k(b− a), a+ `

k(b− a)

),

and we extract a subsequence of (vj) (not relabeled) such that limj→∞ ess infIk`vj

exists for all `. Moreover, for 0 < z < 1 we define the setT kz := ` ∈ 1, . . . , k : lim

j→∞ess infIk`

vj ≤ z .

For every ` ∈ T kz there exists a sequence (xj) in Ik` and y ∈ Ik` such thatlimj→∞

vj(xj) = limj→∞

ess infIk`

vj and vj(y)→ 1 .

Thus, analogously to the above it follows that∫ 1

z

W (s) ds ≤ limj→∞

∫ vj(y)

vj(xj)Wεj (s) ds ≤ lim inf

j→∞Fεj (uj , vj ; Ik` ) ≤ C

for some C > 0 by assumption.Repeating this argument for every ` ∈ T kz we get(

#T kz) ∫ 1

z

W (s) ds ≤ lim infj→∞

Fεj (uj , vj ; I) ≤ C .

Note that in view of [A1] there holds∫ 1zW (s) ds > 0, and hence, #T kz is bounded

independently of k. Because #T kz is also non-decreasing with respect to k it remainsconstant for k large enough. As a consequence, we can pick `k1 < `k2 < · · · < `kNz ∈ T

kz

with Nz := maxk∈N(#T kz

), such that each `ki /k converges to some θi ∈ [0, 1] as

k → ∞. Define Tz := y1, . . . , yN with yi := a + θi(b − a). Let ρ > 0, choosek > 2(b− a)/ρ large enough, and let ` ∈ T kz . Then we have Ik` ⊂ Bρ(Tz). Therefore,

lim infj→∞

f(z)∫I\Bρ(Tz)

|u′j |2 dx ≤ lim infj→∞

∫I

f(vj)|u′j |2 dx

≤ lim infj→∞

Fεj (uj , vj ; I) .(4.7)

From [A5] we have f(z) > 0, and thus, we obtain u′j u′ in L2(I \ Bρ(Tz))up to a subsequence, and consequently u ∈ H1(I \ Bρ(Tz)). By the weak lowersemi-continuity of the norm we have

f(z)∫I\Bρ(Tz)

|u′|2 dx ≤ lim infj→∞

Fεj (uj , vj ; I) .

Since this inequality holds for all ρ > 0, we have u ∈ H1(I\Tz), and since u ∈ SBV(I)with I ∩ Su = ∅, we deduce that u ∈ H1(I). Taking the limit for ρ→ 0 results in

f(z)∫I

|u′|2 dx ≤ lim infj→∞



Finally, we take the limit for z → 1 and obtain (note that f is continuous from [A5])∫I

f(1)|u′|2 dx ≤ lim infj→∞


Since I ⊂ Ω was chosen arbitrarily such that I ∩ Su = ∅ we end up with (4.6).Together with (4.1) we obtain∫

Ω\Bδ(Su)f(1)|u′|2 dx+ 2c0#Su ≤ lim inf

j→∞Fεj (uj , vj) .

and we conclude the proof by taking the limit for δ → 0 .

Proposition 4.2. In the setting of Theorem 3.2 there holds

F (u, v) ≤ Γ-lim infε→0

Fε(u, v) for all u, v ∈ L1(Ω) .

Proof. For the proof we use the usual notation in the setting of slicing, introducedin Section 2.3. In what follows let ξ ∈ Sn−1 and y ∈ Ωξ, let A ⊂ Ω be open andchoose u, v ∈ L1(Ω) arbitrarily. We define the localized version of (3.1) by

Fε(u, v;A) :=∫A

(f(v) + ηε

)|∇u|2 + ϕε

(Wε(v)

)+ ψε

(|∇v|

)dx

+ cε(|Djv|(A) + |Dcv|(A)

)if u ∈ H1(A), v ∈ BV(A; [0, 1]) and Fε(u, v;A) := +∞ otherwise. Furthermore, wedefine for I ⊂ R open

F ε(u, v; I) :=∫I

(f(v) + ηε

)|u′|2 + ϕε

(Wε(v)

)+ ψε

(|v′|)

dx

+ cε(|Djv|(I) + |Dcv|(I)

)if u ∈ H1(I), v ∈ BV(I; [0, 1]) and Fε(u, v; I) := +∞ otherwise. We additionally set

F ξε (u, v;A) :=∫Aξ

F ε(uξy, v

ξy;Aξy

)dLn−1(y) .

From Fubini’s theorem and Theorem 2.3 we therefore obtain

F ξε (u, v;A) =∫A

(f(v) + ηε

)∣∣〈∇u, ξ〉∣∣2 + ϕε(Wε(v)

)+ ψε

(|〈∇v, ξ〉|

)dx

+ cε∣∣〈Djv, ξ〉

∣∣(A) + cε∣∣〈Dcv, ξ〉

∣∣(A)

if |〈Du, ξ〉| is absolutely continuous with respect to Ln, and F ξε (u, v;A) = +∞otherwise. Thus, there clearly holds

F ξε (u, v;A) ≤ Fε(u, v;A) . (4.8)

From Proposition 4.1 we know that F (u, v; I) ≤ Γ-lim infε→0 F ε(u, v; I) with

F (u, v; I) :=

∫I

f(1)|u′|2 dx+ 2c0#Su for u ∈ SBV2(I), v = 1 a.e.,

+∞ otherwise.

Choosing

F ξ(u, v;A) :=∫Aξ

F (uξy, vξy;Aξy) dLn−1(y) ,


there holds for all sequences (uj) and (vj) with uj → u and vj → v in L1(Ω) asj →∞

F ξ(u, v;A) ≤∫Aξ

lim infj→∞

F ε((uj)ξy, (vj)ξy;Aξy

)dLn−1(y) .

Fatou’s Lemma and (4.8) yieldF ξ(u, v;A) ≤ Γ-lim inf

ε→0F ξε (u, v;A) ≤ Γ-lim inf

ε→0Fε(u, v;A) . (4.9)

Moreover, by construction, F ξ(u, v;A) is finite if and only if for a.a. y ∈ Aξ thereholds vξy = 1 a.e. on Aξy, uξy ∈ SBV2(Aξy) as well as∫

Aξ

∫Aξy

f(1)∣∣(uξy)′

∣∣2 dx+ 2c0#Suξy dLn−1(y) <∞ .

Since there holds for every M > 0 and every u ∈ L1(Ω) with uξy ∈ SBV2(Aξy) fora.a. y ∈ Aξ∫

Aξ

∣∣D((−M) ∨ uξy ∧M)∣∣(Aξy)dLn−1(y)

≤∫Aξ

14L

1(Aξy) +∫Aξy

∣∣((−M) ∨ uξy ∧M)′∣∣2 dx+ 2M#Suξy dLn−1(y)

≤ Ln(A) + C

∫Aξ

∫Aξy

f(1)∣∣((−M) ∨ uξy ∧M

)′∣∣2 dx+ 2c0#Suξy dLn−1(y) ,

we get by Corollary 2.4 that F ξ(u, v;A) is finite only if u ∈ GSBV2(A) and v = 1a.e. in A. Hence,

F ξ(u, v;A) =∫A

f(1)|〈∇u, ξ〉|2 dx+ 2c0∫Su

∣∣〈νu, ξ〉∣∣ dHn−1

if u ∈ GSBV2(A) and v = 1 a.e. in A, and F ξ(u, v;A) = +∞ otherwise.Since A and ξ were chosen arbitrarily, if v = 1 a.e. in A, then [16, Theorem 1.16]

and (4.9) imply

F (u, v;A) =∫A

f(1) supξ∈Sn−1

|〈∇u, ξ〉|2 dLn + 2c0∫Su

supξ∈Sn−1

∣∣〈νu, ξ〉∣∣Hn−1

≤ Γ-lim infε→0

Fε(u, v;A) .

Otherwise, the lim inf-inequality follows directly from (4.9) with ξ arbitrary.

The following proposition now shows the lim sup-inequality.

Proposition 4.3. In the setting of Theorem 3.2 there holdsΓ-lim sup

ε→0Fε(u, v) ≤ F (u, v) for all u, v ∈ L1(Ω) .

Proof. If u /∈ GSBV2(Ω) or v 6= 1 on some set with non-zero measure the assertion isobvious. We first show that the result holds for u replaced by w ∈ SBV2(Ω)∩L∞(Ω)for which 1.–3. in Theorem 2.2 (replacing wj by w) hold.

For this purpose choose for every ε > 0 some δε > 0 such that ηεδε→ 0 as ε→ 0

but still δεϕε(Wε(0))→ 0 as ε→ 0, for instance

δε =√ηε√

ϕε(Wε(0)).


Take some smooth cutoff function φ : R → [0, 1] with φ = 1 on B 12(0) and φ = 0

on Ω \ B1(0), and define τ(x) = dist(x, Sw) for all x ∈ Ω. Then, we set φε(x) =φ(τ(x)/δε) for all x ∈ Ω, and we fix for every ε > 0 the function wε = (1−φε)w, forwhich holds wε ∈ H1(Ω), wε = w on Ω \Bδε(Sw) and wε → w in L1(Ω) as ε→ 0.Furthermore we define

vε =0 on Bδε(Sw) ∩ Ω ,

1 elsewhere.

Since Sw is polyhedral there holds Hn−1(∂Bδε(Sw) ∩ Ω) < ∞. Consequently, wehave vε ∈ BV(Ω; [0, 1]) for all ε > 0.

With this at hand, recalling [A5], we get

Fε(wε, vε) ≤∫

Ωf(1)|∇w|2 dx+ ηε

∫Ω|∇wε|2 dx

+ Ln(Ω)(ϕε(Wε(1)) + ψε(0)

)+ Ln

(Bδε(Sw)

)ϕε(Wε(0))

+Hn−1(∂Bδε(Sw))cε . (4.10)

By the choice of wε, the fact that ‖w‖L∞(Ω) ≤M and that |∇τ(x)| = 1 a.e. on Ω(see [29, Lemma 3.2.34]) we get on Bδε(Sw)

|∇wε| ≤ |w∇φε|+ |(1− φε)∇w| ≤M

δε‖φ′‖L∞(Ω) + |∇w| ,

which implies

ηε

∫Ω|∇wε|2 dx ≤ ηε

∫Ω\Bδε (Sw)

|∇w|2 dx+ Cηεδ2ε

Ln(Bδε(Sw)

)+ 2ηε

∫Bδε (Sw)

|∇w|2 dx

with C = 2M2‖φ′‖2L∞(Ω) independent of ε. The first and the last term obviouslyconverge to 0 as ε → 0. For the second term we remark that for a polyhedralset, the Hausdorff measure coincides with the Minkowski content (see, e.g., [29,Theorem 3.2.29]), so that

Ln(Bδε(Sw)

)2δε

→ Hn−1(Sw) = Hn−1(Sw) <∞ as ε→ 0 . (4.11)

As a consequence, recalling that ηεδε→ 0 we get

Cηεδ2ε

Ln(Bδε(Sw)

)→ 0 as ε→ 0 ,

and thereforeηε

∫Ω|∇wε|2 dx→ 0 as ε→ 0.

Additionally, (4.11) and δεϕε(Wε(0))→ 0 as ε→ 0 imply

Ln(Bδε(Sw)

)ϕε(Wε(0))→ 0 as ε→ 0 .

Furthermore, there holds

Hn−1(∂Bδε(Sw))→ 2Hn−1(Sw) as ε→ 0 ,

which is again due to Sw being a polyhedral set.


Applying the previous three convergence statements in (4.10) together with thelimit behaviour of ϕε(Wε(1)), ψε(0) and cε from [A2] and [A3], we get

lim supε→0

Fε(wε, vε) ≤ F (w, 1) . (4.12)

If u ∈ GSBV2(Ω) we have for every M > 0 that uM ∈ SBV2(Ω) ∩ L∞(Ω) withuM := (−M) ∨ u ∧M , and we can find a sequence (wj) in SBV2(Ω) ∩ L∞(Ω) suchthat 1.–6. in Theorem 2.2 (replacing u by uM ) holds. Together with the lowersemi-continuity of Γ-lim supFε in L1(Ω)× L1(Ω) and (4.12) we deduce

Γ-lim supε→0

Fε(uM , 1) ≤ lim infj→∞

Γ-lim supε→0

Fε(wj , 1) ≤ lim infj→∞

F (wj , 1) = F (uM , 1) .

Obviously, there holds ‖∇uM‖L2(Ω) ≤ ‖∇u‖L2(Ω), and from Su =⋃M>0 SuM

(see Section 2.3) follows that Hn−1(SuM ) ≤ Hn−1(Su). Thus, using again the lowersemi-continuity of Γ-lim supFε we get

Γ-lim supε→0

Fε(u, 1) ≤ lim infM→∞

Γ-lim supε→0

Fε(uM , 1) ≤ lim infM→∞

F (uM , 1) ≤ F (u, 1) ,

which concludes the proof.

The proof of Theorem 3.2 is now a direct consequence of Proposition 4.2 andProposition 4.3.

5. Numerical Examples

The aim of this section is to numerically compare our new approximation fromCorollary 3.6 with the classical Ambrosio-Tortorelli approach. We aim for a simpleand easy to implement algorithm in order to illustrate the differences between thosetwo models and justify our theory. As an application for the numerical computationswe choose the image segmentation problem already described in the introduction.

Thus, for Ω ⊂ Rn being non-empty, open, bounded and with Lipschitz boundary,we seek to minimize the following functional with respect to u ∈ SBV2(Ω) ∩ L∞(Ω)

E(u) = α

2

∫Ω|∇u|2 dx+ β

2

∫Ω|u− g|2 dx+ γH1(Su) , (5.1)

where g ∈ L∞(Ω) is the original image and α, β, γ > 0 are the parameters influencingthe smoothing and segment detection in the solution. They have, of course, to bechosen with care in order to get a sensible result.

Using now Corollary 3.6 we can approximately minimize E by minimizing

Gε(u, v) := α

2

∫Ω

(v2 + ηε)|∇u|2 dx+ β

2

∫Ω|u− g|2 dx

+ γ

2ε

∫Ω

(1− v) dx+ γ

2 |Dv|(Ω) , (5.2)

for small ε > 0, which we also refer to as the BV-model.On the other hand we consider the elliptic approximation (1.3), introduced in [10]:

AT ε(u, v) := α

2

∫Ω

(v2 + ηε)|∇u|2 dx+ β

2

∫Ω|u− g|2 dx

+ γ

∫Ω

14ε (1− v)2 + ε|∇v|2 dx (5.3)


for u ∈ H1(Ω) and v ∈ H1(Ω; [0, 1]), which we refer to as the H1-model (note thatwe “redefined” AT ε as in the following, we will only use (5.3) such that there is nochance of confusion).

For the discretization of these functionals we consider a 2-dimensional imagewith its natural pixel grid with pixel length h > 0. If the image is given by M ×Npixels, we use the discrete grid Ωh = h, . . . ,Mh × h, . . . Nh and we identifythe piecewise constant functions u, g, v as elements in the Euclidean space RM×N .Precisely, one sets u =

∑ij uij1[(i−1)h,ih)×[(j−1)h,jh) for (uij) ∈ RM×N , where 1A

denotes the characteristic function of A ⊂ R2, i.e., 1A = 1 on A and 1A = 0on R2 \A.

For the discretization of the appearing gradients and the total variation we use afinite difference scheme. For this purpose we define the finite difference operatorwith zero Neumann boundary condition ∇h : RM×N → R2×M×N by

(∇hu)ij = 1h

((∂+

1 u)ij , (∂+2 u)ij

)for u ∈ RM×N

with

(∂+1 u)ij :=

ui+1,j − uij for i ∈ 1, . . . ,M − 1 ,0 for i = M ,

(∂+2 u)ij :=

ui,j+1 − uij for j ∈ 1, . . . , N − 1 ,0 for j = N .

Furthermore, we denote the adjoint of ∇h by −divh, i.e. for w ∈ R2×M×N theoperator divh : R2×M×N → RM×N is defined by

(divh w)ij := 1h

((∂−1 w

(1))ij

+(∂−2 w

(2))ij

),

where for all u ∈ RM×N

(∂−1 u)ij :=

u1j for i = 1 ,uij − ui−1,j for i ∈ 2, . . . ,M − 1 ,−uM−1,j for i = M ,

(∂−2 u)ij :=

ui1 for j = 1 ,uij − ui,j−1 for j ∈ 2, . . . , N − 1 ,−ui,N−1 for j = N .

For functions u, v ∈ RM×N , operations such as the product uv (or u · v), theminimum u ∧ v, the maximum u ∨ v, and the square u2 are always meant to beelement-wise. With ‖u‖2, ‖u‖1 and ‖u‖∞ we respectively refer to the Frobeniusnorm, the `1-norm of u vectorized, and the maximum norm of u. The Frobeniusinner product of u and v is written as 〈u, v〉. For any field q = (q(1), q(2)) ∈ R2×M×N ,like ∇hu for u ∈ RM×N , we denote by |q| the Euclidean norm along the first axis,i.e. |q| ∈ RM×N

|q|ij =√(

q(1)ij

)2 +(q

(2)ij

)2.

With this strategy we can define the discretized versions of (5.2) and (5.3), respec-tively, for all u, v ∈ RM×N by

Ghε (u, v) := α

2∥∥v|∇hu|∥∥2

2 + β

2 ‖u− g‖22


+ γ

2ε‖1− v‖1 + γ

2∥∥|∇hv|∥∥1 + χ0≤v≤1(v)

and

AT hε (u, v) := α

2∥∥v|∇hu|∥∥2

2 + β

2 ‖u− g‖22

+ γ

4ε‖1− v‖22 + γε

∥∥|∇hv|∥∥22 + χ0≤v≤1(v) .

The symbol 1 refers to the discretized function that is one almost everywhere. Notethat we neglected the factor h2 in the functionals since it does not change theirminimum. Moreover, we chose ηε = 0 here, because in the discrete setting, theproblem of finding a minimizer stays well-posed for this choice.

Remark 5.1. The choice of the recovery sequence in the proof of Proposition 4.3suggests that the width of the detected contours represented by the phase fieldvariable v correlates with the parameter ε. The precise relation between ε and thewidth of the phase field is, however, not known. Examining the structure of theapproximating functionals, we expect that it depends, in particular, on the trade-offbetween the two terms v2‖∇u‖2∞ and 1

4ε (1− v).Although, we would like to have the width of the phase field and therefore

ε extremely small, there is a limit of choice depending on the pixel size h. Tobe more precise, choosing hε > 0 depending on ε, it is well known that AT hεε Γ-converges as ε→ 0, provided that hε/ε→ 0 as ε→ 0 (see [12,15]). We believe that acorresponding statement is also true for the considered BV-phase field approximation.A study of this is, however, outside the scope of the present paper.

The difficulty in finding a minimizer lies in the non-convex, and for Ghε alsonon-smooth, structure. In previous works an alternating minimization schemehas been commonly used, exploiting the fact that the functionals are convex ineach variable separately (see [1, 11, 15]). However, in this work we choose a morerecent approach, which is the proximal alternating linearized minimization (in shortPALM) presented in [13]. This algorithm is a form of an alternating gradient descentprocedure, for which we do not have to solve any linear equation. This makes thealgorithm also faster than the alternating minimization scheme, especially for ratherlarge images. Our experience also showed no significant difference in the results.

For the PALM algorithm one uses the fact that the objective functional can bewritten as J(u, v) +K(u) +H(v). Then, for some initial value u0, v0 ∈ RM×N weset for each k ∈ N

uk = proxKtk(uk−1 − tk∇uJ(uk−1, vk−1)

), (5.4)

vk = proxHsk(vk−1 − sk∇vJ(uk, vk−1)

), (5.5)

where tk, sk > 0. By proxgt we denote the proximal operator with step size t > 0:

proxgt (w) = arg minu∈RM×N

(12t‖u− w‖

22 + g(u)

).

For the right choices of the step sizes tk and sk above one can show that thisscheme converges to a critical point of J(u, v) +K(u) +H(v) as k →∞ (see [13,Proposition 3.1]). Namely, we need to choose tk = θ1

L1(vk−1) and sk = θ2L2(uk) for some

θ1, θ2 ∈ (0, 1), where L1(v) and L2(u) are Lipschitz constants of u 7→ ∇uJ(u, v) andv 7→ ∇vJ(u, v), respectively. Unfortunately, convergence rates are not known, so


that as a stopping criterion, we are limited to measure the change of the variablesin each iteration. We stop the scheme when this change drops under a specifiedthreshold or if a certain number of iterations is reached.

We will now have a closer look on how the algorithm looks like for Ghε and AT hεseparately.

BV-model. We write Ghε (u, v) = J(u, v) +K(u) +H(v) with

J(u, v) = α

2∥∥v|∇hu|∥∥2

2 , K(u) = β

2 ‖u− g‖22 (5.6)

andH(v) = γ

2ε‖1− v‖1 + γ

2∥∥|∇hv|∥∥1 + χ0≤v≤1(v) .

We have∇uJ(u, v) = −α divh

(v2∇hu

)and ∇vJ(u, v) = αv|∇hu|2 .

Since the operator norm of ∇h is strictly below√

8h (see, e.g. [19,21]), we can choose

for some θ ∈ (0, 1)

tk = h2

8α and sk = θ

α∥∥|∇huk|2∥∥∞ , (5.7)

such that t = tk is constant throughout the algorithm.As a simple computation shows, solving (5.4) is then equivalent to

uk = uk + tβg

1 + tβwith uk = uk−1 + tα divh

((vk−1)2∇huk−1) . (5.8)

By completing squares and ignoring constant terms the problem (5.5) can beequivalently reformulated to

vk ∈ arg minv∈RM×N

(12

∥∥∥v − vk − γsk2ε 1

∥∥∥2

2+ γsk

2∥∥|∇hv|∥∥1 + χ0≤v≤1(v)

)(5.9)

with vk = vk−1 − skαvk−1|∇huk|2. Since the non-smooth term ‖|∇hv|‖1 is stillpresent, this minimization can not be solved directly. Instead we tackle the problemwith the algorithm introduced by A. Chambolle and T. Pock in [22], solving thecorresponding primal-dual problem. Therefore, we define for all v ∈ RM×N andw ∈ R2×M×N the functions

Pk(v) = 12

∥∥∥v − vk − γsk2ε 1

∥∥∥2

2+ χ0≤v≤1(v) and Qk(w) = γsk

2h∥∥|w|∥∥1 ,

such that (5.9) is equivalent tovk ∈ arg min

Pk(v) +Qk(∇1v) : v ∈ RM×N

. (5.10)

Here, ∇1 is the forward difference operator ∇h for h = 1.The corresponding primal-dual saddle point problem is given by

minp∈RM×N

maxq∈R2×M×N

(〈∇1p, q〉+ Pk(p)−Q∗k(q)

)(5.11)

where Q∗k denotes the convex conjugate of Qk, i.e., Q∗k = χ‖|·|‖∞≤γsk2h

. Clearly, forany solution (p, q) of (5.11) we have that vk = p is a solution of (5.10). We solve(5.11) with [22, Algorithm 1]. Namely, for 0 < τ2 ≤ 1

8 and for some p0k ∈ RM×N ,

q0k ∈ R2×M×N as well as p0

k := p0k we define for all ` ∈ N

q`k = proxQ∗k

τ

(q`−1k + τ∇1p

`−1k

), (5.12)


p`k = proxPkτ(p`−1k + τ div1 q

`k

), (5.13)

p`k = 2p`k − p`−1k . (5.14)

Then, [22, Theorem 1] guarantees the convergence of (p`k, q`k) as ` → ∞ to asolution of (5.11). For a stopping criterion of the primal-dual iteration we considerthe primal-dual gap which is for p ∈ RM×N and q ∈ R2×M×N given by

Gk(p, q) = Pk(p) +Qk(∇1p) + P ∗k (div1 q) +Q∗k(q) .

It vanishes if and only if (p, q) solves (5.11). For this reason, we stop iteration(5.12)–(5.14) if the corresponding primal-dual gap is smaller than a certain tolerance.

We now continue with the precise computations of the primal-dual steps for theBV-phase field approximation. Since Q∗k is the indicator function of a convex set,the update step (5.12) is the projection of q`−1

k + τ∇1p`−1k onto ‖|·|‖∞ ≤ γsk

2h (cf. [22, Section 6.2]). Thus, we simply get

q`k = q`k

1 ∨ 2h|q`k|

γsk

with q`k = q`−1k + τ∇1p

`−1k .

The proximal operator appearing in (5.13) can be solved directly. Namely, we get

0 ∈ 1 + τ

τp`k −

1τp`k − vk −

γsk2ε 1 + ∂χ0≤p≤1(p`k)

with p`k = p`−1k + τ div1 q

`k, which yields

p`k = 0 ∨(p`k + τ vk + τ γsk2ε 1

1 + τ

)∧ 1 .

The primal-dual gap for p`k and q`k can be computed explicitly. Taking into accountthat Q∗(q`k) = 0 and

P ∗k (div1 q`k) =

⟨(p`k)′,div1 q

`k

⟩− 1

2

∥∥∥(p`k)′ − vk − γsk2ε 1

∥∥∥2

2

with(p`k)′ = 0 ∨

(vk + γsk

2ε 1 + div1 q`k

)∧ 1 ,

it is given by

Gk(p`k, q`k) = γsk2h∥∥|∇1p

`k|∥∥

1 +⟨(p`k)′,div1 q

`k

⟩+ 1

2(‖p`k‖22 − ‖(p`k)′‖22

)−⟨p`k − (p`k)′, vk + γsk

2ε 1⟩.

Summing up all the previous computations for our BV-phase field model, we getAlgorithm 1 in the appendix, which is the numerical scheme as implemented.

H1-model (Ambrosio-Tortorelli). For the elliptic approximation we use J andK as in (5.6) and only redefine H by

H(v) := γ

4ε‖1− v‖22 + γε

∥∥|∇hv|∥∥22 + χ0≤v≤1(v)

in order to obtain AT hε (u, v) = J(u, v) +K(u) +H(v). Clearly, sk and t = tk canalso be chosen as before in (5.7). Hence, (5.4) results again in (5.8). The difference


Table 1. Numerical parameters

α β γ θ Tol1 Tol2 MaxIt

1.75 · 10−4 1 3 · 10−5 0.99 10−3 10−5 10000

of the algorithm compared to the one for the BV-phase field appears in (5.5), whichis now equivalent to

vk ∈ arg minv∈RM×N

(12

∥∥∥∥v − 2εvk + γsk1

2ε+ γsk

∥∥∥∥2

2+ 2γε2sk

2ε+ γsk

∥∥|∇hv|∥∥22 + χ0≤v≤1(v)

).

Since this problem is sufficiently smooth it could be easily solved directly, by solvinga linear system. Nevertheless, for a better comparability and for saving the effort ofsolving a large linear equation, we stay as close as possible to the algorithm usedfor the BV-model. Thus, we use again the primal-dual scheme as in (5.12)–(5.14),where this time we need to choose

Pk(v) = 12

∥∥∥∥v − 2εvk + γsk1

2ε+ γsk

∥∥∥∥2

2+ χ0≤v≤1(v)

for v ∈ RM×N and

Qk(w) = µ

2∥∥|w|∥∥2

2 with µ = 4γε2skh2(2ε+ γsk)

for w ∈ R2×M×N . Note, that we have Q∗k(w) = 12µ‖|w|‖

22 and thus (5.12) yields

q`k = µ

µ+ τq`k with q`k = q`−1

k + τ∇1p`−1k ,

and (5.13) results in

p`k = 0 ∨(

11 + τ

p`k + τ(2εvk + γsk1)(1 + τ)(2ε+ γsk)

)∧ 1 with p`k = p`−1

k + τ div1 q`k .

The primal-dual gap for this approximation is given by

Gk(p`k, q`k) = µ

2∥∥|∇1p

`k|∥∥2

2 +⟨div1 q

`k, (p`k)′

⟩+ 1

2µ∥∥|q`k|∥∥2

2

+ 12(‖p`k‖22 −

∥∥(p`k)′∥∥2

2

)−⟨p`k − (p`k)′, 2εvk + γsk1

2ε+ γsk

⟩with

(p`k)′ = 0 ∨(

2εvk + γsk1

2ε+ γsk+ div1 qk

)∧ 1 .

Altogether, this yields Algorithm 2 in the appendix, which is the numerical schemethat we use for computations.

Numerical Results. With the presented algorithms we perform computations fortwo different images. For all numerical examples we fix the width of the imagesto 1. The pixel size h then depends on the number of pixels and is given byh = L

number of horizontal pixels .For the first computation we use the noisy image from Figure 1. The latter

is generated by adding Gaussian noise of standard deviation 0.1 and clipping theresult to the original image range [0, 1]. In this computation, the input image gcorresponds to this noisy image and we only change the approximating variable


original imagea noisy image

Fig. 1. Input image with 256 × 256 pixels for the computations shown in Figure 2.

ε, in order to investigate its influence, while fixing the other parameters for thealgorithms as indicated in Table 1. The result can be observed in Figure 2.

One can clearly see that the BV-model produces almost binary phase fields, i.e.v takes only the values 0 (corresponding to a black pixel) and 1 (corresponding toa white pixel). In other words these phase fields are much sharper than the onesproduced by the H1-model. Moreover, we observe that ε can be chosen larger whenusing the BV-model in order to obtain a result that is comparable to the H1-model.

Besides the comparison of the two models one can also observe, that in bothapproximations of the Mumford-Shah functional, only few edges are detected if ε istoo small. Whereas, if ε is relatively large, the contours become rather wide. Theseeffects are well-known and have already been mentioned in Remark 5.1, from whichwe also expect that for small values of ε, the phase field may detect the edges again,when reducing h. Also this can be confirmed from Figure 3, where we use the sameimage but this time with 512 × 512 pixels keeping the width of the image domainfixed to 1 as above, resulting in the value of h being halved.

Figure 4 shows another picture with 512 × 512 pixel size. To the original imagewe again add Gaussian noise (noise level: 0.1). This noisy image serves as the inputdata g for our algorithms. Besides α and γ, the parameters have a been chosen likein Table 1.

aphoto credit: Irina Patrascu Gheorghita: alina’s eye https://www.flickr.com/photos/angel_ina/3201337190/ License: CC-BY 2.0 https://creativecommons.org/licenses/by/2.0/

https://www.flickr.com/photos/angel_ina/3201337190/

https://www.flickr.com/photos/angel_ina/3201337190/

https://creativecommons.org/licenses/by/2.0/


BV-model

ε = 5 · 10−4

ε = 1 · 10−3

ε = 2 · 10−3

ε = 3 · 10−3

ε = 5 · 10−3

H1-model

ε = 2 · 10−4

ε = 3 · 10−4

ε = 5 · 10−4

ε = 1 · 10−3

ε = 1.5 · 10−3

Fig. 2. Numerical result for different values of ε. The other parameters are given in Table 1.


BV-model with ε = 5 · 10−4

H1-model with ε = 2 · 10−4

Fig. 3. Result of a segmentally denoised image with 512 × 512 pixels using parameters fromTable 1.


original imageb noisy image

BV-model with ε = 1 · 10−3

H1-model with ε = 3 · 10−4

Fig. 4. Image with 512 × 512 pixels. Computation for α = 10−4, γ = 5 · 10−6 and the otherparameters as specified in Table 1.

bphoto credit: Phuketian.S: Sailing from Thailand to Malaysia. Our yacht at the seahttps://www.flickr.com/photos/124790945@N06/32397550408/ License: CC-BY 2.0 https://creativecommons.org/licenses/by/2.0/

https://www.flickr.com/photos/124790945@N06/32397550408/




Acknowledgements

This work was supported by the International Research Training Group IGDK1754 “Optimization and Numerical Analysis for Partial Differential Equations withNonsmooth Structures”, funded by the German Research Council (DFG) and theAustrian Science Fund (FWF):[W 1244-N18].

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Appendix A. Auxiliary statements

Lemma A.1. Let µ be a signed Radon measure on R, ψ : R → [0,∞] a proper,convex and lower semi-continuous function and η ∈ C∞c (R; [0,∞)) a mollifier, i.e.,∫R η dx = 1. Then,∫

Rψ((µ ∗ η)

)dx ≤

∫Rψ( dµ

dL1

)dx+

∫Rψ∞( dµs

d|µs|

)d|µs|

where µ = dµdL1L1 + µs denotes the Lebesgue decomposition of µ and ψ∞(s) =

limt→∞ψ(st)t is the recession function of ψ.

Proof. Fix x ∈ R, t > 0 and choose |µ|x,t = η(x − ·)(L1 + 1t |µ

s|) as well asµx = η(x− ·)µ. Then, |µ|x,t(R) = 1 + 1

t

∫R η(x− ·) d|µs| and µx = dµ

dL1 η(x− ·)L1 +t dµs

d|µs|1t η(x− ·)|µs|, such that Jensen’s inequality yields

ψ( (µ ∗ η)(x)|µ|x,t(R)

)= ψ

( 1|µ|x,t(R)

∫R

dµxd|µ|x,t

d|µ|x,t)

≤ 1|µ|x,t(R)

(∫Rψ( dµ

dL1 (y))η(x− y) dy +

∫R

1tψ(t

dµs

d|µs| (y))η(x− y) d|µs|(y)

).


Since dµsd|µs| is either 1 or −1 |µs|-almost everywhere, the rightmost integral reads as

1tψ(t)

∫I+η(x−·) d|µs|+ 1

tψ(−t)∫I−η(x−·) d|µs| where I+ =

x ∈ R : dµs

d|µs| (x) = 1

and I− =x ∈ R : dµs

d|µs| (x) = −1. Clearly, as t→∞, this expression converges to∫

R ψ∞( dµs

d|µs|)η(x − ·) d|µs| (possibly to ∞). Since limt→∞|µ|x,t(R) = 1, by lower

semi-continuity of ψ,

ψ((µ ∗ η)(x)

)≤ lim inf

t→∞ψ( (µ ∗ η)(x)|µ|x,t(R)

)≤∫Rψ( dµ

dL1 (y))η(x− y) dy +

∫Rψ∞( dµs

d|µs| (y))η(x− y) d|µs|(y).

Integrating both sides over R with respect to x and interchanging order on theright-hand side then yields the result.

Lemma A.2. Let I := (a, b) ⊂ R be a bounded open interval, v ∈ BV(I; [0, 1]) andε > 0. Then,

|D(Φε v)|(I) ≤∫ b

a

ϕε(Wε(v)

)dx+

∫ b

a

ψε(|v′|) dx+ cε(|Djv|(I) + |Dcv|(I)

)with Wε, ϕε and ψε according to [A1], [A2] and [A3], respectively, and Φε(s) =∫ s

0 Wε(t) dt for s ∈ [0, 1].

Proof. Denote by v(a) = limρ→01ρ

∫ a+ρa

v(x) dx and v(b) = limρ→01ρ

∫ bb−ρ v(x) dx

and extend v outside of I by v(x) = v(a) for x ≤ a and v(x) = v(b) for x ≥ b.Then, v ∈ BVloc(R) with Dv the zero extension of Dv on I. Choose a mollifierη ∈ C∞c (R; [0,∞)),

∫R η dx = 1 and denote by ηδ(x) = 1

δ η(xδ ) for δ > 0. Then, eachvδ = v ∗ ηδ is in C∞(I; [0, 1]) and by classical differentiation, the Fenchel inequalityand [A3],

|D(Φε vδ)|(I) =∫ b

a

Wε(vδ)|v′δ|dx ≤∫ b

a

ϕε(Wε(vδ)

)+ ϕ∗ε(|v′δ|) dx

≤∫ b

a

ϕε(Wε(vδ)

)dx+

∫Rψε(|v′δ|) dx.

We have vδ → v in L1(I) as δ → 0, so by continuity of Wε and ϕε (as a consequenceof convexity and finiteness onWε([0, 1])), one can conclude that

∫ baϕε(Wε(vδ)

)dx→∫ b

aϕε(Wε(v)

)dx as δ → 0. Denoting by ψε(t) = ψε(|t|) for t ∈ R yields a convex

function since ψε is increasing on [0,∞), so applying Lemma A.1 yields∫Rψε(|v′δ|) dx =

∫Rψε((Dv ∗ ηδ)

)dx ≤

∫Rψε(v′) dx+

∫Rψ∞ε

( dDsv

d|Dsv|

)d|Dsv|

=∫ b

a


).

By continuity of Φε, one can further conclude that Φε vδ → Φε v in L1(I) asδ → 0. The above then implies that D(Φε vδ) as a sequence of δ is boundedin the space of Radon measures on I, yielding a weak*-convergent subsequence.By strong-weak*-closedness of the weak derivative, the limit has to coincide with


D(Φε v). This holds for each subsequence, such that in fact, D(Φε vδ) convergesweakly* to D(Φε v) as δ → 0. Thus, using weak* lower semi-continuity, we obtain|D(Φε v)|(I) ≤ lim inf

δ→0|D(Φε vδ)|(I)

≤ limδ→0

∫ b

a

ϕε(Wε(vδ)

)dx∫ b

a


),

which, together with the above, yields the desired estimate.


Appendix B. Pseudo Codes

Algorithm 1 BV-model1: u← g, v ← 1, q ← 0

2: t← h2

8α , τ ←1√8

3: it← 04: repeat5: it← it+ 16: u0 ← u, v0 ← v

7: u← u+ tα divh(v2∇hu) + tβg

1 + tβ

8: s← θ

α‖|∇hu|2‖∞9: p← v, p← v

10: v ← v − sαv|∇hu|2

11: repeat12: p0 ← p

13: q ← q + τ∇1p

14: q ← q

1 ∨ 2hγs |q|

15: p← p+ τ div1 q

16: p← 0 ∨(p+ τ v + γτs

2ε 1

1 + τ

)∧ 1

17: p′ ← 0 ∨(v + γs

2ε1 + div1 q

)∧ 1

18: p← 2p− p0

19: gap← 12(‖p‖22 − ‖p′‖22

)−⟨p− p′, v + γs

2ε1⟩

20: until gap+ γs

2h∥∥|∇1p|

∥∥1 + 〈div1 q, p

′〉 ≤ Tol221: v ← p

22: until max‖v − v0‖∞, ‖u− u0‖∞

≤ Tol1 or it = MaxIt


Algorithm 2 H1-model1: u← g, v ← 1, q ← 0

2: t← h2

8α , τ ←1√8

3: it← 04: repeat5: it← it+ 16: u0 ← u, v0 ← v

7: u← u+ tα divh(v2∇hu) + tβg

1 + tβ

8: s← θ

α‖|∇hu|2‖∞9: p← v, p← v

10: v ← v − sαv|∇hu|2

11: µ← 4γε2s

h2(2ε+ γs)12: repeat13: p0 ← p

14: q ← q + τ∇1p

15: q ← µ

µ+ τq

16: p← 0 ∨(

11 + τ

p+ τ(2εv + γs1)(1 + τ)(2ε+ γs)

)∧ 1

17: p′ ← 0 ∨(

2εv + γs1

2ε+ γs+ div1 q

)∧ 1

18: p← 2p− p0

19: gap← 12(‖p‖22 − ‖p′‖22

)−⟨p− p′, 2εv + γs1

2εs+ γs

⟩20: until gap+ µ

2∥∥|∇1p|

∥∥22 + 〈div1 q, p

′〉+ 12µ∥∥|q|∥∥2

2 ≤ Tol221: v ← p

22: until max‖v − v0‖∞, ‖u− u0‖∞

≤ Tol1 or it = MaxIt

approximation of the mumford-shah functional by … · bv-phase field approximation for the...

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