# On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients

Authors: A. De Cezaro
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

We investigate a level-set-type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem, we propose a Tikhonov-type regularization approach coupled with a level-set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability for regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level-set method in some interesting inverse problems arising in elliptic PDE models.


---

## Body

## 1. Introduction

Since the seminal paper of Santosa [1], level-set techniques have been successfully developed and have recently become a standard technique for solving inverse problems with interfaces (e.g., [2–10]).In many applications, interfaces represent interesting physical parameters (inhomogeneities, heat conductivity between materials with different heat capacity, and interface diffusion problems) across which one or more of these physical parameters change value in a discontinuous manner. The interfaces divide the domainΩ⊂ℝn in subdomains  Ωj, with  j=1,…,k,  of different regions with specific internal parameter profiles. Due to the different physical structures of each of these regions, different mathematical models might be the most appropriate for describing them. Solutions of such models represent a free boundary problem, that is, one in which interfaces are also unknown and must be determined in addition to the solution of the governing partial differential equation. In general such solutions are determined by a set of data obtained by indirect measurements [2–4, 11–15]. Applications include image segmentation problems [12–15], optimal shape designer problems [2, 16], Stefan’s type problems [2], inverse potential problems [17–19], inverse conductivity/resistivity problems [4, 5, 10, 11, 20], among others [2–4, 6, 16].There is often a large variety of priors information available for determining the unknown physical parameter, whose characteristic depends on the given application. In this paper, we are interested in inverse problems that consist in the identification of an unknown quantityu∈D(F)⊂X that represents all parameter profiles inside the individual subregions of Ω, from data  y∈Y, where  X  and Y are Banach spaces and  D(F) will be adequately specified in Section 3. In this particular case, only the interfaces between the different regions and, possibly, the unknown parameter values need to be reconstructed from the gathered data. This process can be formally described by the operator equation
(1)F(u)=y,
where  F:D(F)⊂X→Y is the forward operator.Neither existence nor uniqueness of a solution to (1) is a guarantee. For simplicity, we assume that, for exact data y∈Y, the operator equation (1) admits a solution and we do not strive to obtain results on uniqueness. However, in practical applications, data are obtained only by indirect measurements of the parameter. Hence, in general, exact data y∈Y are not known and we have only access to noise data  yδ∈Y, whose level of noise δ>0 is assumed to be known a priori and satisfies
(2)∥yδ-y∥Y≤δ.We assume that the inverse problem associated with the operator equation (1) is ill-posed. Indeed, it is the case in many interesting problems [4, 6, 10, 16, 20–22]. Therefore, accuracy of an approximated solution calls for a regularization method [21]. In this paper, we propose a Tikhonov-type regularization method coupled with a level-set approach to obtain a stable approximation of the unknown level sets and values of the piecewise (not necessarily constant) solution of (1).Many approaches, in particular level-set type approaches, have previously been suggested for such problems. In [1, 11, 23–26], level-set approaches for identification of the unknown parameter u with distinct, but known, piecewise constant values were investigated.In [12, 17, 24], level-set approaches were derived to solve inverse problems, assuming that u is defined by several distinct constant values. In both cases, one needs only to identify the level sets of u, that is, the inverse problem reduces to a shape identification problem. On the other hand, when the level values of u are also unknown, the inverse problem becomes harder, since we have to identify both the level sets and the level values of the unknown parameter u. In this situation, the dimension of the parameter space increases by the number of unknown level values. Level-set approaches to ill-posed problems with unknown constant level values appeared before in [14, 16, 18, 19, 27]. Level-set regularization properties of the approximated solution for inverse problems are described in [17–19, 25, 28].However, regularization theory for inverse problems where the components of the parameteru are variable and have discontinuities has not been well investigated. Indeed, level-set regularization theory applied to inverse problems [17–19] that recover the shape and the values of variable discontinuous coefficients is unknown to the author. Some early results in the numerical implementation of level-set type methods were previously used to obtain solutions of elliptic problems with discontinuous and variable coefficients in [4].In this paper, we propose a level-set type regularization method to ill-posed problems whose solution is composed by piecewise components which are not necessarily constants. In other words, we introduce a level-set type regularization method to recover the shape and the values of variable discontinuous coefficients. In this framework, a level-set function is used to parameterize the solutionu of (1). We obtain a regularized solution using a Tikhonov-type regularization method, since the level values of u are not constant and also unknown.In the theoretical point of view, the advantage of our approach in relation to [2, 17–19, 25, 29] is that we are able to obtain regularized solutions to inverse problems with piecewise solutions that are more general than those covered by the regularization methods proposed before. We still prove regularization properties for the approximated solution of the inverse problem model (1), where the parameter is a nonconstant piecewise solution. The topologies needed to guarantee the existence of a minimizer (in a generalized sense) of the Tikhonov functional (defined in (7)) are quite complicated and differ in some key points from [18, 19, 25]. In this particular approach, the definition of generalized minimizers is quite different from other works [17, 19, 25] (see Definition 3). As a consequence, the arguments used to prove the well-posedness of the Tikhonov functional, the stability, and convergence of the regularized solutions of the inverse problem (1) are quite complicated and need significant improvements (see Section 3).The main applicability advantage of the proposed level-set type method compared to that in the literature is that we are able to apply this method to problems whose solutions depend of nonconstant parameters. This implies that we are able to handle more general and interesting physical problems, where the components of the desired parameter are not necessarily homogeneous, as those presented before in the literature [4, 6, 14, 16, 18, 19, 27, 30–32]. Examples of such interesting physical problems are heat conduction between materials of different heat capacity and conductivity, interface diffusion processes, and many other types of physical problems where modeling components are related with embedded boundaries. See, for example, [3, 4, 6, 19, 30, 32] and references therein. As a benchmark problem, we analyze two inverse problems modeled by elliptic PDEs with discontinuous and variable coefficients.In contrast, the nonconstant characteristics of the level values impose different types of theoretical problems, since the topologies where we are able to provide regularization properties of the approximated solution are more complicated than the ones presented before [14, 16, 18, 19, 27]. As a consequence, the numerical implementations become harder than the other approaches in the literature [18, 19, 29, 32].This paper is outlined as follows: in Section2, we formulate the Tikhonov functional based on the level-set framework. In Section 3, we present the general assumptions needed in this paper and the definition of the set of admissible solutions. We prove relevant properties about the admissible set of solutions, in particular convergence in suitable topologies. We also present relevant properties of the penalization functional. In Section 4, we prove that the proposed method is a regularization method to inverse problems, that is, we prove that the minimizers of the proposed Tikhonov functional are stable and convergent with respect to the noise level in the data. In Section 5, a smooth functional is proposed to approximate minimizers of the Tikhonov functional defined in the admissible set of solutions. We provide approximation properties and the optimality condition for the minimizers of the smooth Tikhonov functional. In Section 6, we present an application of the proposed framework to solve some interesting inverse elliptic problems with variable coefficients. Conclusions and future directions are presented in Section 7.


## 2. The Level-Set Formulation

Our starting point is the assumption that the parameteru in (1) assumes two unknown functional values, that is,  u(x)∈{ψ1(x),ψ2(x)} a.e. in Ω⊂ℝd, where Ω is a bounded set. More specifically, we assume the existence of a mensurable set D⊂⊂Ω, with 0<|D|<|Ω|, such that u(x)=ψ1(x) if x∈D and  u(x)=ψ2(x) if x∈Ω/D. With this framework, the inverse problem that we are interested in in this paper is the stable identification of both the shape of D and the value function ψj(x) for x belonging to D and to Ω/D, respectively, from observation of the data yδ∈Y.We remark that, ifψ1(x)=c1 and  ψ2(x)=c2 with c1 and c2 unknown constants values, the problem of identifying u was rigorously studied before in [19]. Moreover, many other approaches to this case appear in the literature; see [2, 19, 23, 24] and references therein. Recently, in [18], an L2 level-set approach to identify the level and constant contrast was investigated.Our approach differs from the level-set methods proposed in [18, 19], by considering also the identification of variable unknown levels of the parameter u. In this situation, many topological difficulties appear in order to have a tractable definition of an admissible set of parameters (see Definition 3). Generalization to problems with more than two levels is possible applying this approach and following the techniques derived in [17]. As observed before, the present level-set approach is a rigorous derivation of a regularization strategy for identification of the shape and nonconstant levels of discontinuous parameters. Therefore, it can be applied to physical problems modeled by embedded boundaries whose components are not necessarily piecewise constant [2, 17–19, 25].In many interesting applications, the inverse problem modeled by (1) is ill-posed. Therefore a regularization method must be applied in order to obtain a stable approximate solution. We propose a regularization method by, first, introducing a parameterization on the parameter space, using a level-set function  ϕ that belongs to  H1(Ω). Note that, we can identify the distinct level sets of the function  ϕ∈H1(Ω) with the definition of the Heaviside projector
(3)H:H1(Ω)→L∞(Ω),ϕ↦H(ϕ)∶={1ifϕ(x)>0,0otherelse.
Now, from the framework introduced above, a solution u of (1) can be represented as
(4)u(x)=ψ1(x)H(ϕ)+ψ2(x)(1-H(ϕ))=∶P(ϕ,ψ1,ψ2)(x).
With this notation, we are able to determine the shapes of  D  as  {x∈Ω;ϕ(x)>0} and Ω/D as  {x∈Ω;ϕ(x)<0}.The functional level valuesψ1(x), ψ2(x) are also assumed be unknown, and they should be determined as well.Assumption 1.
We assume thatψ1,ψ2∈𝔹∶={f:f is measurable and f(x)∈[m,M],  a.e.inΩ}, for some constant values m,M.Remark 1.
We remark thatf∈𝔹 implies that  f∈L∞(Ω). Since Ω is bounded,  f∈L1(Ω). Moreover,
(5)∫Ω‍f(x)∇·φ(x)dx≤|M|∫Ω‍|∇·(φ)(x)|dx≤|M|∥∇·φ∥L1(Ω),∀φ∈C01(Ω,ℝn).
Hence, f∈BV(Ω).Note that in the case thatψ1 and ψ2 assume two distinct constant values (as covered by the analysis done in [2, 18, 19] and references therein) the assumptions above are satisfied. Hence, the level-set approach proposed here generalizes the regularization theory developed in [18, 19].From (4), the inverse problem in (1), with data given as in (2), can be abstractly written as the operator equation
(6)F(P(ϕ,ψ1,ψ2))=yδ.Once an approximate solution(ϕ,ψ1,ψ2) of (6) is obtained, a corresponding solution of (1) can be computed using (4).Therefore, to obtain a regularized approximated solution to (6), we will consider the least square approach combined with a regularization term, that is, minimizing the Tikhonov functional
(7)𝒢^α(ϕ,ψ1,ψ2)∶=∥F(P(ϕ,ψ1,ψ2))-yδ∥Y2+α{∑j=12‍β1|H(ϕ)|BV+β2∥ϕ-ϕ0∥H1(Ω)2+β3∑j=12‍|ψj-ψ0j|BV},
where ϕ0 and ψ0j represent some a priori information about the true solution u* of (1). The parameter α>0 plays the role of a regularization parameter, and the values of βi,i=1,2,3, act as scaling factors. In other words,  βi,i=1,2,3, needs to be chosen a priori, but independent of the noise level δ. In practical,  βi,i=1,2,3, can be chosen in order to represent a priori knowledge of features of the parameter solution u and/or to improve the numerical algorithm. A more complete discussion about how to choose  βi,i=1,2,3, is provided in [17–19].The regularization strategy in this context is based onTV-H1-TV penalization. The term on H1-norm acts simultaneously as a control on the size of the norm of the level-set function and a regularization on the space H1. The term on BV is a variational measure of H(ϕ). It is well known that the BV seminorm acts as a penalizing for the length of the Hausdorff measure of the boundary of the set  {x:ϕ(x)>0} (see [33, Chapter 5] for details). Finally, the last term on BV is a variational measure of ψj that acts as a regularization term on the set 𝔹. This Tikhonov functional extends the ones proposed in [16, 17, 19, 23, 24] (based on TV-H1 penalization).Existence of minimizers for the functional (7) in the H1×𝔹2 topology does not follow by direct arguments, since the operator P is not necessarily continuous in this topology. Indeed, if  ψ1=ψ2=ψ is a continuous function at the contact region, then  P(ϕ1,ψ2,ψ)=ψ is continuous and the standard Tikhonov regularization theory to the inverse problem holds true [21]. On the other hand, in the interesting case where ψ1 and ψ2 represent the level of discontinuities of the parameter u, the analysis becomes more complicated and we need a definition of generalized minimizers (see Definition 3) in order to handle these difficulties.


## 3. Generalized Minimizers

As already observed in [25], if D⊂Ω with  ℋn-1(∂D)<∞, where ℋn-1(S) denotes the (n-1)-dimensional Hausdorff-measure of the set S, then the Heaviside operator H maps H1(Ω) into the set
(8)𝒱∶={χD;D⊂Ωmeasurable,:ℋn-1(∂D)<∞}.
Therefore, the operator P in (4) maps H1(Ω)×𝔹2 into the admissible parameter set
(9)D(F)∶={u=q(v,ψ1,ψ2);v∈𝒱,ψ1,ψ2∈𝔹},
where
(10)q:𝒱×𝔹2∋(v,ψ1,ψ2)↦ψ1v+ψ2(1-v)∈BV(Ω).Consider the model problem described in Section1. In this paper, we assume the following.(A1)
Ω⊆ℝn is bounded with piecewise C1 boundary ∂Ω.(A2)
The operatorF:D(F)⊂L1(Ω)→Y is continuous on D(F) with respect to the L1(Ω) topology.(A3)
ε, α, and βj,j=1,2,3, denote positive parameters.(A4)
Equation (1) has a solution, that is, there exists  u*∈D(F) satisfying F(u*)=y and a function ϕ*∈H1(Ω) satisfying  |∇ϕ*|≠0, in the neighborhood of {ϕ*=0} such that H(ϕ*)=z*,  for some z*∈𝒱. Moreover, there exist functional values ψ*1,ψ*2∈𝔹 such that q(z*,ψ*1,ψ*2)=u*. For each ε>0, we define a smooth approximation to the operator P by
(11)Pε(ϕ,ψ1,ψ2)∶=ψ1Hε(ϕ)+ψ2(1-Hε(ϕ)),
where Hε is the smooth approximation to H  described by
(12)Hε(t)∶={1+tεfort∈[-ε,0],H(t)fort∈ℝ[-ε,0].Remark 2.
It is worth noting that, for anyϕk∈H1(Ω), Hε(ϕk) belongs to L∞(Ω) and satisfies 0≤Hε(ϕk)≤1 a.e. in Ω, for all ε>0. Moreover, taking into account that ψj∈𝔹, it follows that the operators q and Pε, as above, are well defined.In order to guarantee the existence of a minimizer of𝒢α defined in (7) in the space H1(Ω)×𝔹2, we need to introduce a suitable topology such that the functional 𝒢α has a closed graphic. Therefore, the concept of generalized minimizers (compare with [17, 25]) in this paper is as follows.Definition 3.
Let the operatorsH, P, Hε, and Pε be defined as above and the positive parameters α,βj, and ε satisfy the Assumption (A3).
Aquadruple  (z,ϕ,ψ1,ψ2)∈L∞(Ω)×H1(Ω)×BV(Ω)2 is called admissible when(a)
there exists a sequence{ϕk} of H1(Ω) functions satisfying limk→∞∥ϕk-ϕ∥L2(Ω)=0,(b)
there exists a sequence{εk}∈ℝ+ converging to zero such that limk→∞∥Hεk(ϕk)-z∥L1(Ω)=0,(c)
there exist sequences{ψk1}k∈ℕ  and  {ψk2}k∈ℕ belonging to BV∩C∞(Ω) such that
(13)|ψkj|BV→|ψj|BV,j=1,2,(d)
ageneralized minimizer of 𝒢^α is considered to be any admissible quadruple (z,ϕ,ψ1,ψ2) minimizing
(14)𝒢α(z,ϕ,ψ1,ψ2)∶=∥F(q(z,ψ1,ψ2))-yδ∥Y2+αR(z,ϕ,ψ1,ψ2)
on the set of admissible quadruples. Here the functional R is defined by
(15)R(z,ϕ,ψ1,ψ2)=ρ(z,ϕ)+β3∑j=12‍|ψj-ψ0j|BV,
and the functional ρ is defined as
(16)ρ(z,ϕ)∶=inf{liminfk→∞[β1|Hεk(ϕk)|BV+β2∥ϕk-ϕ0∥H1(Ω)2]}. The infimum in (16) is taken over all sequences {εk} and {ϕk} characterizing (z,ϕ,ψ1,ψ2) as an admissible quadruple.The convergence|ψkj|BV→|ψj|BV in item (c) in Definition 3 is in the sense of variation measure [33, Chapter 5]. The incorporation of item (c) in the Definition 3 implies the existence of the Γ-limit of sequences of admissible quadruples [25, 34]. This appears in the proof of Lemmas 7, 8, and 11, where we prove that the set of admissible quadruples is closed in the defined topology (see Lemmas 7 and 8) and in the weak lower semicontinuity of the regularization functional R (see Lemma 11). The identification of nonconstant level values ψj implies in a different definition of admissible quadruples.As a consequence, the arguments in the proof of regularization properties of the level-set approach are the principal theoretical novelty and the difference between our definition of admissible quadruples and the ones in [18, 19, 25].Remark 4.
Forj=1,2, let  ψj∈𝔹∩C∞(Ω), ϕ∈H1(Ω) be such that |∇ϕ|≠0 in the neighborhood of the level-set {ϕ(x)=0} and  H(ϕ)=z∈𝒱. For each k∈ℕ, set  ψkj=ψj and  ϕk=ϕ. Then, for all sequences of {εk}k∈ℕ of positive numbers converging to zero, we have
(17)∥Hεk(ϕk)-z∥L1(Ω)=∥Hεk(ϕk)-H(ϕ)∥L1(Ω)=∫(ϕ)-1[-εk,0]‍|1-ϕεk|dx≤∫-εk0‍∫(ϕ)-1(τ)‍1dτ≤meas{(ϕ)-1(τ)}∫-εk0‍1dt→0.
Here, we use the fact that |∇ϕ|≠0 in the neighborhood of {ϕ=0} implies that ϕ is a local diffeomorphism together with a coarea formula [33, Chapter 4]. Moreover, {ψkj}k∈ℕ in 𝔹∩C∞(Ω) satisfies Definition 3, item (c).
Hence,(z,ϕ,ψ1,ψ2) is an admissible quadruple. In particular, we conclude from the general assumption above that the set of admissible quadruple satisfying F(u)=y is not empty.


### 3.1. Relevant Properties of Admissible Quadruples

Our first result is the proof of the continuity properties of operatorsPε, Hε, and q in suitable topologies. Such result will be necessary in the subsequent analysis.We start with an auxiliary lemma that is well known (see e.g., [35]). We present it here for the sake of completeness.Lemma 5.
LetΩ be a measurable subset of ℝn with finite measure.
If(fk)∈𝔹 is a convergent sequence in Lp(Ω) for some p, 1≤p<∞, then it is a convergent sequence in Lp(Ω) for all 1≤p<∞.
In particular, Lemma5 holds for the sequence zk∶=Hε(ϕk).Proof.
See [35, Lemma 2.1].The next two lemmas are auxiliary results in order to understand the definition of the set of admissible quadruples.Lemma 6.
LetΩ be as in assumption (A1) and j=1,2.(i)
Let{zk}k∈ℕ be a sequence in L∞(Ω) with zk∈[m,M] a.e. converging in the L1(Ω)-norm to some element z and {ψkj}k∈ℕ a sequence in 𝔹 converging in the BV-norm to some ψj∈𝔹. Then q(zk,ψk1,ψk2) converges to  q(z,ψ1,ψ2) in L1(Ω).(ii)
Let(z,ϕ)∈L1(Ω)×H1(Ω) be such that Hε(ϕ)→z in L1(Ω) as ε→0, and let ψ1,ψ2∈𝔹. Then  Pε(ϕ,ψ1,ψ2)→q(z,ψ1,ψ2) in L1(Ω) as ε→0.(iii)
Givenε>0, let  {ϕk}k∈ℕ be a sequence in H1(Ω) converging to ϕ∈H1(Ω) in the L2-norm. Then Hε(ϕk)→Hε(ϕ)  in  L1(Ω), as k→∞. Moreover, if  {ψkj}k∈ℕ are sequences in 𝔹, converging to some ψj in 𝔹, with respect to the L1(Ω)-norm, then  q(Hε(ϕk),ψk1,ψk2)→q(Hε(ϕ),ψ1,ψ2) in  L1(Ω), as  k→∞.Proof.
SinceΩ is assumed to be bounded, we have L∞(Ω)⊂L1(Ω) and BV(Ω) is continuous embedding in L2(Ω) [33]. To prove (i), notice that
(18)∥q(zk,ψk1,ψk2)-q(z,ψ1,ψ2)∥L1(Ω)=∥ψk1zk+ψk2(1-zk)-ψ1z-ψ2(1-z)∥L1(Ω)≤∥zk∥L∞(Ω)∥ψk1-ψ1∥L1(Ω)+∥ψ1∥L2(Ω)∥zk-z∥L2(Ω)+∥1-zk∥L∞(Ω)∥ψk2-ψ2∥L1(Ω)+∥ψ2∥L2(Ω)∥zk-z∥L2(Ω)→k→∞0.
Here we use Lemma 5 in order to guarantee the convergence of  zk to  z  in L2(Ω).
Assertion (ii) follows with similar arguments and the fact thatHε(ϕ)∈L∞(Ω)  for all  ε>0.
As∥Hε(ϕk)-Hε(ϕ)∥L1(Ω)≤ε-1meas(Ω)∥ϕk-ϕ∥L2(Ω), the first part of assertion (iii) follows. The second part of the assertion (iii) holds by a combination of the inequality above and inequalities in the proof of assertion (i).Lemma 7.
Let{ψkj}k∈ℕ be a sequence of functions satisfying Definition 3 converging in L1(Ω) to some ψj, for j=1,2. Then ψj also satisfies Definition 3.Proof (sketch of the proof).
Letk∈ℕ and j=1,2. Since ψkj satisfies Definition 3,  ψkj∈BV. From [33, Theorem 2, p. 172], there exist sequences  {ψk,lj}l∈ℕ in  BV×C∞(Ω) such that
(19)ψk,lj→l→∞ψkjinL1(Ω),|ψk,lj|BV→l→∞|ψkj|BV.
In particular, for the subsequence  {ψk,l(k)j}k∈ℕ, it follows that
(20)ψk,l(k)j→k→∞ψjinL1(Ω),|ψk,l(k)j|BV→k→∞|ψj|BV.
Moreover, by assumption ψj∈L1(Ω). From the lower semicontinuity of variational measure (see [33, Theorem 1, p. 172]), (20), and the definition of BV space, it follows that ψj∈BV.In the following lemma we prove that the set of admissible quadruples is closed with respect to theL1(Ω)×L2(Ω)×(L1(Ω))2 topology.Lemma 8.
Let(zk,ϕk,ψk1,ψk2) be a sequence of admissible quadruples converging in L1(Ω)×L2(Ω)×(L1(Ω))2 to some (z,ϕ,ψ1,ψ2), with ϕ∈H1(Ω). Then, (z,ϕ,ψ1,ψ2) is also an admissible quadruple.Proof (sketch of the proof).
Letk∈ℕ. Since (zk1,ϕk1,ψk1,ψk2) is an admissible quadruple, it follows from Definition 3 that there exist sequences {ϕk,l}l∈ℕ, in H1(Ω), {ψk,l1}l∈ℕ, {ψk,l2}l∈ℕ in BV×C∞(Ω) and a correspondent sequence {εkl}l∈ℕ converging to zero such that
(21)ϕk,l→l→∞ϕkinL2(Ω),Hεkl(ϕk,l)→l→∞zkinL1(Ω),|ψk,lj|BV→l→∞|ψkj|BV,j=1,2.
Define the monotone increasing functionτ:ℕ→ℕ such that, for every k∈ℕ, it holds
(22)εkτ(k)≤12εk-1τ(k-1),∥ϕk,τ(k)-ϕk∥L2(Ω)≤1k,∥Hεkτ(k)(ϕk,τ(k))-zk∥L1(Ω)≤1k,|ψk,τ(k)j|BV→|ψkj|BV,j=1,2.
Hence, for each k∈ℕ,
(23)∥ϕ-ϕk,τ(k)∥L2(Ω)≤∥ϕ-ϕk∥L2(Ω)+∥ϕk,τ(k)-ϕk∥L2(Ω),∥z-Hεkτ(k)(ϕk,τ(k))∥L1(Ω)≤∥z-zk∥L1(Ω)+∥Hεkτ(k)(ϕk,τ(k))-zk∥L1(Ω).
From (22),
(24)limk→∞∥ϕ-ϕk,τ(k)∥L2(Ω)=0,limk→∞∥z-Hεkτ(k)(ϕk,τ(k))∥L1(Ω)=0.
Moreover, with the same arguments as Lemma 7, it follows that
(25)|ψk,τ(k)j|BV→|ψj|BV,j=1,2,
and ψj∈BV(Ω). Therefore, it remains to prove that (z,ϕ,ψ1,ψ2) is an admissible quadruple. From Definition 3 and Lemma 7, it is enough to prove that z∈L∞(Ω). If this is not the case, there would exist a Ω′⊂Ω with |Ω′|>0 and γ>0 such that z(x)>1+γ in Ω′ (the other case, z(x)<-γ is analogous). Since (Hεkτ(k)(ϕk,τ(k)))(x)∈[0,1] a.e. in Ω for k∈ℕ (see remark after Definition 3), we would have
(26)∥z-Hεkτ(k)(ϕk,τ(k))∥L1(Ω)≥∥z-Hεkτ(k)(ϕk,τ(k))∥L1(Ω′)≥γ|Ω′|,k∈ℕ,
contradicting the second limit in (24).


### 3.2. Relevant Properties of the Penalization Functional

In the following lemmas, we verify properties of the functionalR which are fundamental for the convergence analysis outlined in Section 4. In particular, these properties imply that the level sets of 𝒢α are compact in the set of admissible quadruple, that is, 𝒢α assumes a minimizer on this set. First, we prove a lemma that simplifies the functional R in (15). Here we present the sketch of the proof. For more details, see the arguments in [19, Lemma 3].Lemma 9.
Let(z,ϕ,ψ1,ψ2) be an admissible quadruple. Then, there exist sequences {εk}k∈ℕ, {ϕk}k∈ℕ, and {ψkj}k∈ℕ, as in the Definition 3, such that
(27)R(z,ϕ,ψ1,ψ2)=limk→∞{∑j=12‍β1|Hεk(ϕk)|BV+β2∥ϕk-ϕ0∥H1(Ω)2+β3∑j=12‍|ψkj-ψ0j|BV}.Proof (sketch of the proof).
For eachl∈ℕ, the definition of R (see Definition 3) guaranties the existence of sequences εkl, {ϕk,lj}∈H1(Ω), and {ψk,lj}∈𝔹 such that
(28)R(z,ϕ,ψ1,ψ2)=liml→∞{∑j=12‍liminfk→∞{β1|Hεkl(ϕk,l)|BV+β2∥ϕk,l-ϕ0∥H1(Ω)2}+β3∑j=12‍|ψk,lj-ψ0j|BV}.
Now a similar extraction of subsequences as in Lemma 8 complete the proof.In the following, we prove two lemmas that are essential to the proof of well posedness of the Tikhonov functional (7).Lemma 10.
The functionalR in (15) is coercive on the set of admissible quadruples. In other words, given any admissible quadruple (z,ϕ,ψ1,ψ2), one has
(29)R(z,ϕ,ψ1,ψ2)≥(β1|z|BV+β2∥ϕ-ϕ0∥H1(Ω)2+β2∑j=12‍|ψj-ψ0j|BV).Proof (sketch of the proof).
Let(z,ϕ,ψ1,ψ2) be an admissible quadruple. From [17, Lemma 4], it follows that
(30)ρ(z,ϕ)≥(β1|z|BV+β2∥ϕ-ϕ0∥H1(Ω)2).
Now, from (30) and the definition of R in (15), we have
(31)(β1|z|BV+β2∥ϕ-ϕ0∥H1(Ω)2+β3∑j=12‍|ψj-ψ0j|BV)≤ρ(z,ϕ)+β3∑j=12‍|ψj-ψ0j|BV=R(z,ϕ,ψ1,ψ2),
concluding the proof.Lemma 11.
The functionalR in (15) is weak lower semicontinuous on the set of admissible quadruples, that is, given a sequence {(zk,ϕk,ψk1,ψk2)} of admissible quadruples such that zk→z in L1(Ω), ϕk⇀ϕ in H1(Ω), and ψkj→ψj in L1(Ω), for some admissible quadruple (z,ϕ,ψ1,ψ2), then
(32)R(z,ϕ,ψ1,ψ2)≤liminfk∈ℕR(zk,ϕk,ψk1,ψk2).Proof.
The functionalρ(z,ϕ) is weak lower semicontinuous (cf. [17, Lemma 5]). As ψkj∈BV, it follows from [33, Theorem 2, p. 172] that there exist sequences {ψk,lj}∈BV∩C∞(Ω) such that ∥ψk,lj-ψkj∥L1(Ω)≤1/l. From a diagonal argument, we can extract a subsequence {ψk,l(k)j} of {ψk,lj} such that {ψk,l(k)j}→ψj in L1(Ω) as k→∞. Let ξ∈Cc1(Ω,ℝn),|ξ|≤1. Then, from [33, Theorem 1, p. 167], it follows that
(33)∫Ω‍ψj∇·ξdx=limk→∞∫Ω‍ψk,l(k)j∇·ξdx=limk→∞[∫Ω‍(ψk,l(k)j-ψkj)∇·ξdx+∫Ω‍ψkj∇·ξdx]≤limk→∞[∫Ω‍∥ψk,l(k)j-ψkj∥L1(Ω)∥∇·ξ∥L∞(Ω)|Ω|-∫Ω‍ξ·σkd|ψkj|BV]≤liminfk→∞|ψkj|BV.
Thus, form the definition of |·|BV (see [33]), we have
(34)|ψj|BV=sup{∫Ω‍ψj∇·ξdx;ξ∈Cc1(Ω,ℝn),|ξ|≤1}≤liminfk→∞|ψkj|BV.
Now, the lemma follows from the fact that the functional R in (15) is a linear combination of lower semicontinuous functionals.


## 3.1. Relevant Properties of Admissible Quadruples

Our first result is the proof of the continuity properties of operatorsPε, Hε, and q in suitable topologies. Such result will be necessary in the subsequent analysis.We start with an auxiliary lemma that is well known (see e.g., [35]). We present it here for the sake of completeness.Lemma 5.
LetΩ be a measurable subset of ℝn with finite measure.
If(fk)∈𝔹 is a convergent sequence in Lp(Ω) for some p, 1≤p<∞, then it is a convergent sequence in Lp(Ω) for all 1≤p<∞.
In particular, Lemma5 holds for the sequence zk∶=Hε(ϕk).Proof.
See [35, Lemma 2.1].The next two lemmas are auxiliary results in order to understand the definition of the set of admissible quadruples.Lemma 6.
LetΩ be as in assumption (A1) and j=1,2.(i)
Let{zk}k∈ℕ be a sequence in L∞(Ω) with zk∈[m,M] a.e. converging in the L1(Ω)-norm to some element z and {ψkj}k∈ℕ a sequence in 𝔹 converging in the BV-norm to some ψj∈𝔹. Then q(zk,ψk1,ψk2) converges to  q(z,ψ1,ψ2) in L1(Ω).(ii)
Let(z,ϕ)∈L1(Ω)×H1(Ω) be such that Hε(ϕ)→z in L1(Ω) as ε→0, and let ψ1,ψ2∈𝔹. Then  Pε(ϕ,ψ1,ψ2)→q(z,ψ1,ψ2) in L1(Ω) as ε→0.(iii)
Givenε>0, let  {ϕk}k∈ℕ be a sequence in H1(Ω) converging to ϕ∈H1(Ω) in the L2-norm. Then Hε(ϕk)→Hε(ϕ)  in  L1(Ω), as k→∞. Moreover, if  {ψkj}k∈ℕ are sequences in 𝔹, converging to some ψj in 𝔹, with respect to the L1(Ω)-norm, then  q(Hε(ϕk),ψk1,ψk2)→q(Hε(ϕ),ψ1,ψ2) in  L1(Ω), as  k→∞.Proof.
SinceΩ is assumed to be bounded, we have L∞(Ω)⊂L1(Ω) and BV(Ω) is continuous embedding in L2(Ω) [33]. To prove (i), notice that
(18)∥q(zk,ψk1,ψk2)-q(z,ψ1,ψ2)∥L1(Ω)=∥ψk1zk+ψk2(1-zk)-ψ1z-ψ2(1-z)∥L1(Ω)≤∥zk∥L∞(Ω)∥ψk1-ψ1∥L1(Ω)+∥ψ1∥L2(Ω)∥zk-z∥L2(Ω)+∥1-zk∥L∞(Ω)∥ψk2-ψ2∥L1(Ω)+∥ψ2∥L2(Ω)∥zk-z∥L2(Ω)→k→∞0.
Here we use Lemma 5 in order to guarantee the convergence of  zk to  z  in L2(Ω).
Assertion (ii) follows with similar arguments and the fact thatHε(ϕ)∈L∞(Ω)  for all  ε>0.
As∥Hε(ϕk)-Hε(ϕ)∥L1(Ω)≤ε-1meas(Ω)∥ϕk-ϕ∥L2(Ω), the first part of assertion (iii) follows. The second part of the assertion (iii) holds by a combination of the inequality above and inequalities in the proof of assertion (i).Lemma 7.
Let{ψkj}k∈ℕ be a sequence of functions satisfying Definition 3 converging in L1(Ω) to some ψj, for j=1,2. Then ψj also satisfies Definition 3.Proof (sketch of the proof).
Letk∈ℕ and j=1,2. Since ψkj satisfies Definition 3,  ψkj∈BV. From [33, Theorem 2, p. 172], there exist sequences  {ψk,lj}l∈ℕ in  BV×C∞(Ω) such that
(19)ψk,lj→l→∞ψkjinL1(Ω),|ψk,lj|BV→l→∞|ψkj|BV.
In particular, for the subsequence  {ψk,l(k)j}k∈ℕ, it follows that
(20)ψk,l(k)j→k→∞ψjinL1(Ω),|ψk,l(k)j|BV→k→∞|ψj|BV.
Moreover, by assumption ψj∈L1(Ω). From the lower semicontinuity of variational measure (see [33, Theorem 1, p. 172]), (20), and the definition of BV space, it follows that ψj∈BV.In the following lemma we prove that the set of admissible quadruples is closed with respect to theL1(Ω)×L2(Ω)×(L1(Ω))2 topology.Lemma 8.
Let(zk,ϕk,ψk1,ψk2) be a sequence of admissible quadruples converging in L1(Ω)×L2(Ω)×(L1(Ω))2 to some (z,ϕ,ψ1,ψ2), with ϕ∈H1(Ω). Then, (z,ϕ,ψ1,ψ2) is also an admissible quadruple.Proof (sketch of the proof).
Letk∈ℕ. Since (zk1,ϕk1,ψk1,ψk2) is an admissible quadruple, it follows from Definition 3 that there exist sequences {ϕk,l}l∈ℕ, in H1(Ω), {ψk,l1}l∈ℕ, {ψk,l2}l∈ℕ in BV×C∞(Ω) and a correspondent sequence {εkl}l∈ℕ converging to zero such that
(21)ϕk,l→l→∞ϕkinL2(Ω),Hεkl(ϕk,l)→l→∞zkinL1(Ω),|ψk,lj|BV→l→∞|ψkj|BV,j=1,2.
Define the monotone increasing functionτ:ℕ→ℕ such that, for every k∈ℕ, it holds
(22)εkτ(k)≤12εk-1τ(k-1),∥ϕk,τ(k)-ϕk∥L2(Ω)≤1k,∥Hεkτ(k)(ϕk,τ(k))-zk∥L1(Ω)≤1k,|ψk,τ(k)j|BV→|ψkj|BV,j=1,2.
Hence, for each k∈ℕ,
(23)∥ϕ-ϕk,τ(k)∥L2(Ω)≤∥ϕ-ϕk∥L2(Ω)+∥ϕk,τ(k)-ϕk∥L2(Ω),∥z-Hεkτ(k)(ϕk,τ(k))∥L1(Ω)≤∥z-zk∥L1(Ω)+∥Hεkτ(k)(ϕk,τ(k))-zk∥L1(Ω).
From (22),
(24)limk→∞∥ϕ-ϕk,τ(k)∥L2(Ω)=0,limk→∞∥z-Hεkτ(k)(ϕk,τ(k))∥L1(Ω)=0.
Moreover, with the same arguments as Lemma 7, it follows that
(25)|ψk,τ(k)j|BV→|ψj|BV,j=1,2,
and ψj∈BV(Ω). Therefore, it remains to prove that (z,ϕ,ψ1,ψ2) is an admissible quadruple. From Definition 3 and Lemma 7, it is enough to prove that z∈L∞(Ω). If this is not the case, there would exist a Ω′⊂Ω with |Ω′|>0 and γ>0 such that z(x)>1+γ in Ω′ (the other case, z(x)<-γ is analogous). Since (Hεkτ(k)(ϕk,τ(k)))(x)∈[0,1] a.e. in Ω for k∈ℕ (see remark after Definition 3), we would have
(26)∥z-Hεkτ(k)(ϕk,τ(k))∥L1(Ω)≥∥z-Hεkτ(k)(ϕk,τ(k))∥L1(Ω′)≥γ|Ω′|,k∈ℕ,
contradicting the second limit in (24).


## 3.2. Relevant Properties of the Penalization Functional

In the following lemmas, we verify properties of the functionalR which are fundamental for the convergence analysis outlined in Section 4. In particular, these properties imply that the level sets of 𝒢α are compact in the set of admissible quadruple, that is, 𝒢α assumes a minimizer on this set. First, we prove a lemma that simplifies the functional R in (15). Here we present the sketch of the proof. For more details, see the arguments in [19, Lemma 3].Lemma 9.
Let(z,ϕ,ψ1,ψ2) be an admissible quadruple. Then, there exist sequences {εk}k∈ℕ, {ϕk}k∈ℕ, and {ψkj}k∈ℕ, as in the Definition 3, such that
(27)R(z,ϕ,ψ1,ψ2)=limk→∞{∑j=12‍β1|Hεk(ϕk)|BV+β2∥ϕk-ϕ0∥H1(Ω)2+β3∑j=12‍|ψkj-ψ0j|BV}.Proof (sketch of the proof).
For eachl∈ℕ, the definition of R (see Definition 3) guaranties the existence of sequences εkl, {ϕk,lj}∈H1(Ω), and {ψk,lj}∈𝔹 such that
(28)R(z,ϕ,ψ1,ψ2)=liml→∞{∑j=12‍liminfk→∞{β1|Hεkl(ϕk,l)|BV+β2∥ϕk,l-ϕ0∥H1(Ω)2}+β3∑j=12‍|ψk,lj-ψ0j|BV}.
Now a similar extraction of subsequences as in Lemma 8 complete the proof.In the following, we prove two lemmas that are essential to the proof of well posedness of the Tikhonov functional (7).Lemma 10.
The functionalR in (15) is coercive on the set of admissible quadruples. In other words, given any admissible quadruple (z,ϕ,ψ1,ψ2), one has
(29)R(z,ϕ,ψ1,ψ2)≥(β1|z|BV+β2∥ϕ-ϕ0∥H1(Ω)2+β2∑j=12‍|ψj-ψ0j|BV).Proof (sketch of the proof).
Let(z,ϕ,ψ1,ψ2) be an admissible quadruple. From [17, Lemma 4], it follows that
(30)ρ(z,ϕ)≥(β1|z|BV+β2∥ϕ-ϕ0∥H1(Ω)2).
Now, from (30) and the definition of R in (15), we have
(31)(β1|z|BV+β2∥ϕ-ϕ0∥H1(Ω)2+β3∑j=12‍|ψj-ψ0j|BV)≤ρ(z,ϕ)+β3∑j=12‍|ψj-ψ0j|BV=R(z,ϕ,ψ1,ψ2),
concluding the proof.Lemma 11.
The functionalR in (15) is weak lower semicontinuous on the set of admissible quadruples, that is, given a sequence {(zk,ϕk,ψk1,ψk2)} of admissible quadruples such that zk→z in L1(Ω), ϕk⇀ϕ in H1(Ω), and ψkj→ψj in L1(Ω), for some admissible quadruple (z,ϕ,ψ1,ψ2), then
(32)R(z,ϕ,ψ1,ψ2)≤liminfk∈ℕR(zk,ϕk,ψk1,ψk2).Proof.
The functionalρ(z,ϕ) is weak lower semicontinuous (cf. [17, Lemma 5]). As ψkj∈BV, it follows from [33, Theorem 2, p. 172] that there exist sequences {ψk,lj}∈BV∩C∞(Ω) such that ∥ψk,lj-ψkj∥L1(Ω)≤1/l. From a diagonal argument, we can extract a subsequence {ψk,l(k)j} of {ψk,lj} such that {ψk,l(k)j}→ψj in L1(Ω) as k→∞. Let ξ∈Cc1(Ω,ℝn),|ξ|≤1. Then, from [33, Theorem 1, p. 167], it follows that
(33)∫Ω‍ψj∇·ξdx=limk→∞∫Ω‍ψk,l(k)j∇·ξdx=limk→∞[∫Ω‍(ψk,l(k)j-ψkj)∇·ξdx+∫Ω‍ψkj∇·ξdx]≤limk→∞[∫Ω‍∥ψk,l(k)j-ψkj∥L1(Ω)∥∇·ξ∥L∞(Ω)|Ω|-∫Ω‍ξ·σkd|ψkj|BV]≤liminfk→∞|ψkj|BV.
Thus, form the definition of |·|BV (see [33]), we have
(34)|ψj|BV=sup{∫Ω‍ψj∇·ξdx;ξ∈Cc1(Ω,ℝn),|ξ|≤1}≤liminfk→∞|ψkj|BV.
Now, the lemma follows from the fact that the functional R in (15) is a linear combination of lower semicontinuous functionals.


## 4. Convergence Analysis

In the following, we consider any positive parameterα,βj,j=1,2,3, as in the general assumption to this paper. First, we prove that the functional 𝒢α in (14) is well posed.Theorem 12 (well-posedness).
The functional𝒢α in (14) attains minimizers on the set of admissible quadruples.Proof.
Notice that the set of admissible quadruples is not empty, since(0,0,0,0) is admissible. Let {(zk,ϕk,ψk1,ψk2)} be a minimizing sequence for 𝒢α, that is, a sequence of admissible quadruples satisfying 𝒢α(zk,ϕk,ψk1,ψk2)→inf𝒢α≤𝒢α(0,0,0,0)<∞. Then, {𝒢α(zk,ϕk,ψk1,ψk2)} is a bounded sequence of real numbers. Therefore, {(zk,ϕk,ψk1,ψk2)} is uniformly bounded in BV×H1(Ω)×BV2. Thus, from the Sobolev Embedding Theorem [33, 36], we guarantee the existence of a subsequence (denoted again by {(zk,ϕk,ψk1,ψk2)}) and the existence of (z,ϕ,ψ1,ψ2)∈L1(Ω)×H1(Ω)×BV2 such that ϕk→ϕ in L2(Ω), ϕk⇀ϕ in H1(Ω), zk→z in L1(Ω), and ψkj→ψjinL1(Ω). Moreover, z,ψ1, and  ψ2∈BV. See [33, Theorem 4, p. 176].
From Lemma8, we conclude that (z,ϕ,ψ1,ψ2) is an admissible quadruple. Moreover, from the weak lower semicontinuity of R (Lemma 11), together with the continuity of q (Lemma 6) and continuity of F (see the general assumption), we obtain
(35)inf𝒢α=limk→∞𝒢α(zk,ϕk,ψk1,ψk2)=limk→∞{∥F(q(zk,ψk1,ψk2))-yδ∥Y2+αR(zk,ϕk,ψk1,ψk2)∥F(q(zk,ψk1,ψk2))-yδ∥Y2}≥∥F(q(z,ψ1,ψ2))-yδ∥Y2+αR(z,ϕ,ψ1,ψ2)=𝒢α(z,ϕ,ψ1,ψ2),
proving that (z,ϕ,ψ1,ψ2) minimizes 𝒢α.In what follows, we will denote a minimizer of𝒢α by (zα,ϕα,ψα1,ψα2). In particular the functional 𝒢^α in (50) attains a generalized minimizer in the sense of Definition 3. In the following theorem, we summarize some convergence results for the regularized minimizers. These results are based on the existence of a generalized minimum norm solutions.Definition 13.
An admissible quadruple(z†,ϕ†,ψ1,†,ψ2,†) is called an R-minimizing solution if it satisfies (i)
F(q(z†,ψ1,†,ψ2,†))=y,(ii)
R(z†,ϕ†,ψ1,†,ψ2,†)=ms∶=inf{R(z,ϕ,ψ1,ψ2);(z,ϕ,ψ1,ψ2) is an admissible               quadruple and  F(q(z,ψ1,ψ2))=y}.Theorem 14 (R-minimizing solutions).
Under the general assumptions of this paper, there exists aR-minimizing solution.Proof.
From the general assumption on this paper and Remark4, we conclude that the set of admissible quadruple satisfying F(q(z,ψ1,ψ2))=y is not empty. Thus, ms in (ii) is finite and there exists a sequence {(zk,ϕk,ψk1,ψk2)}k∈ℕ  of admissible quadruple satisfying
(36)F(q(zk,ψk1,ψk2))=y,R(zk,ϕk,ψk1,ψk2)→ms<∞.
Now, form the definition of R, it follows that the sequences {ϕk}k∈ℕ, {zk}k∈ℕ, and {ψkj}k∈ℕj=1,2 are uniformly bounded in H1(Ω) and BV(Ω), respectively. Then, from the Sobolev Compact Embedding Theorem [33, 36], we have (up to subsequences) that
(37)ϕk→ϕ†inL2(Ω),zk→z†inL1(Ω),ψkj→ψj,†inL1(Ω),j=1,2.
Lemma 8 implies that (z†,ϕ†,ψ1,†,ψ2,†) is an admissible quadruple. Since R is weakly lower semicontinuous (cf. Lemma 11), it follows that
(38)ms=liminfk→∞R(zk,ϕk,ψk1,ψk2)≥R(z†,ϕ†,ψ1,†,ψ2,†).
Moreover, we conclude from Lemma 6 that
(39)q(z†,ψ1,†,ψ1,†)=limk→∞q(zk,ψk1,ψk2),F(q(z†,ψ1,†,ψ2,†))=limk→∞F(q(zk,ψk1,ψk2))=y.
Thus, (z†,ϕ†,ψ1,†,ψ2,†) is an R-minimizing solution.Using classical techniques from the analysis of Tikhonov regularization methods (see [21, 37]), we present in the following the main convergence and stability theorems of this paper. The arguments in the proof are somewhat different of those presented in [18, 19]. But, for sake of completeness, we present the proof.Theorem 15 (convergence for exact data).
Assume that one has exact data, that is,yδ=y. For every α>0, let (zα,ϕα,ψα1,ψα2) denote a minimizer of 𝒢α on the set of admissible quadruples. Then, for every sequence of positive numbers {αk}k∈ℕ converging to zero, there exists a subsequence, denoted again by {αk}l∈ℕ, such that (zαk,ϕαk,ψαk1,ψαk2) is strongly convergent in L1(Ω)×L2(Ω)×(L1(Ω))2. Moreover, the limit is a solution of (1).Proof.
Let(z†,ϕ†,ψ1,†,ψ2,†) be an R-minimizing solution of (1)—its existence is guaranteed by Theorem 14. Let {αk}k∈ℕ be a sequence of positive numbers converging to zero. For each k∈ℕ, denote (zk,ϕk,ψk1,ψk2)∶=(zαk,ϕαk,ψαk1,ψαk2) to be a minimizer of Gαk. Then, for each k∈ℕ, we have
(40)Gαk(zk,ϕk,ψk1,ψk2)≤∥F(q(z†,ψ1,†,ψ2,†))-y∥+αkR(z†,ϕ†,ψ1,†,ψ2,†)=αkR(z†,ϕ†,ψ1,†,ψ2,†).
Since αkR(zk,ϕk,ψk1,ψk2)≤Gαk(zk,ϕk,ψk1,ψk2), it follows from (40) that
(41)R(zk,ϕk,ψk1,ψk2)≤R(z†,ϕ†,ψ1,†,ψ2,†)<∞.
Moreover, from the assumption on the sequence {αk}, it follows that
(42)limk→∞αkR(z†,ϕ†,ψ1,†,ψ1,†)=0.
From (41) and Lemma 10, we conclude that sequences {ϕk}, {zk}, and {ψkj} are bounded in H1(Ω) and BV, respectively, for j=1,2. Using an argument of extraction of diagonal subsequences (see proof of Lemma 8), we can guarantee the existence of an admissible quadruple (z~,ϕ~,ψ~1,ψ~2) such that
(43)(zk,ϕk,ψk1,ψk2)→(z~,ϕ~,ψ~1,ψ~2)inL1(Ω)×L2(Ω)×(L1(Ω))2.
Now, from Lemma 6(i), it follows that q(z~,ψ~1,ψ~2)=limk→∞q(zk,ψk1,ψk2) in L1(Ω). Using the continuity of the operator F together with (40) and (42), we conclude that
(44)y=limk→∞F(q(zk,ψk1,ψk2))=F(q(z~,ψ~1,ψ~2)).
On the other hand, from the lower semicontinuity of R and (41), it follows that
(45)R(z~,ϕ~,ψ~1,ψ~2)≤liminfk→∞R(zk,ϕk,ψk1,ψk2)≤limsupk→∞R(zk,ϕk,ψk1,ψk2)≤(z†,ϕ†,ψ~1,ψ~2),
concluding the proof.Theorem 16 (stability).
Letα=α(δ) be a function satisfying limδ→0α(δ)=0 and limδ→0δ2α(δ)-1=0. Moreover, let {δk}k∈ℕ be a sequence of positive numbers converging to zero and yδk∈Y corresponding noisy data satisfying (2). Then, there exists a subsequence, denoted again by {δk} and a sequence {αk∶=α(δk)} such that (zαk,ϕαk,ψαk1,ψαk2) converges in L1(Ω)×L2(Ω)×(L1(Ω))2 to solution of (1).Proof.
Let(z†,ϕ†,ψ1,†,ψ1,†) be an R-minimizer solution of (1) (such existence is guaranteed by Theorem 14). For each k∈ℕ, let (zk,ϕk,ψk1,ψk2)∶=(zα(δk),ϕα(δk),ψα(δk)1,ψα(δk)2) be a minimizer of Gα(δk). Then, for each k∈ℕ, we have
(46)Gαk(zk,ϕk,ψk1,ψk2)≤∥F(q(z†,ψ1,†,ψ1,†))-yδk∥Y2+α(δk)R(z†,ϕ†,ψ1,†,ψ2,†)≤δk2+α(δk)R(z†,ϕ†,ψ1,†,ψ2,†).
From (46) and the definition of  Gαk, it follows that
(47)R(zk,ϕk,ψk1,ψk2)≤δk2α(δk)+R(z†,ϕ†,ψ1,†,ψ2,†).
Taking the limit ask→∞ in (47), it follows from theorem assumptions on α(δk) that
(48)limk→∞∥F(q(zk,ψk1,ψk2))-yδk∥≤limk→∞(δk2+α(δk)R(z†,ϕ†,ψ1,†,ψ2,†))=0,limsupk→∞R(zk,ϕk,ψk1,ψk2)≤(z†,ϕ†,ψ1,†,ψ2,†).
With the same arguments as in the proof of Theorem 15, we conclude that at least a subsequence that we denote again by (zk,ϕk,ψk1,ψk2) converges in L1(Ω)×L2(Ω)×(L1(Ω))2 to some admissible quadruple (z,ϕ,ψ1,ψ2). Moreover, by taking the limit as k→∞ in (46), it follows from the assumption on F and Lemma 6 that
(49)F(q(z,ϕ,ψ1,ψ2))=limk→∞F(q(zk,ψk1,ψk2))=y.The functional𝒢α defined in (14) is not easy to be handled numerically, that is, we are not able to derive a suitable optimality condition to the minimizers of 𝒢α. In the following section, we work in sight to surpass such difficulty.


## 5. Numerical Solution

In this section, we introduce a functional which can be handled numerically and whose minimizers are “near” to the minimizers of𝒢α. Let 𝒢ε,α be the functional defined by
(50)𝒢ε,α(ϕ,ψ1,ψ2)∶=∥F(Pε(ϕ,ψ1,ψ2))-yδ∥Y2+α(∑j=12‍β1|Hε(ϕ)|BV+β2∥ϕ-ϕ0∥H12+β3∑j=12‍|ψj-ψ0j|BV),
where Pε(ϕ,ψ1,ψ2)∶=q(Hε(ϕ),ψ1,ψ2) is defined in (11). The functional 𝒢ε,α is well posed as the following lemma shows.Lemma 17.
Given positive constantsα, ε, βj as in the general assumption of this paper, ϕ0∈H1(Ω) and ψ0j∈𝔹, j=1,2. Then, the functional 𝒢ε,α in (50) attains a minimizer on H1(Ω)×(BV)2.Proof.
Sinceinf{𝒢ε,α(ϕ,ψ1,ψ2):(ϕ,ψ1,ψ2)∈H1(Ω)×(BV)2}≤𝒢ε,α(0,0,0)<∞, there exists a minimizing sequence {(ϕk,ψk1,ψk2)}  in  H1(Ω)×𝔹2  satisfying
(51)limk→∞𝒢ε,α(ϕk,ψk1,ψk2)=inf{𝒢ε,α(ϕ,ψ1,ψ2):(ϕ,ψ1,ψ2)∈H1(Ω)×𝔹2}.
Then, for fixed α>0, the definition of 𝒢ε,α in (50) implies that the sequences {ϕk} and {ψkj}j=1,2 are bounded in H1(Ω) and (BV)2, respectively. Therefore, from Banach-Alaoglu-Bourbaki Theorem [38] ϕk⇀ϕ in H1(Ω) and from [33, Theorem 4, p. 176], ψkj→ψj in L1(Ω), j=1,2. Now, a similar argument as in Lemma 7 implies that ψj∈𝔹, for j=1,2. Moreover, by the weak lower semicontinuity of the H1-norm [38] and |·|BV measure (see [33, Theorem 1, p. 172]), it follows that
(52)∥ϕ-ϕ0∥H12≤liminfk→∞∥ϕk-ϕ0∥H12,|ψj-ψ0j|BV≤liminfk→∞|ψkj-ψ0j|BV.
The compact embedding ofH1(Ω) into L2(Ω) [36] implies in the existence of a subsequence of {ϕk} (that we denote with the same index) such that ϕk→ϕ in L2(Ω). It follows from Lemma 6 and [33, Theorem 1, p. 172] that |Hε(ϕ)|BV≤liminfk→∞|Hε(ϕk)|BV. Hence, from continuity of F in L1, continuity of q (see Lemma 6), together with the estimates above, we conclude that
(53)𝒢ε,α(ϕ,ψ1,ψ2)≤limk→∞∥F(Pε(ϕk,ψk1,ψk2))-yδ∥Y2+α(∑j=12‍β1liminfk→∞|Hε(ϕk)|BV+β2liminfk→∞∥ϕk-ϕ0∥H1(Ω)2+β3liminfk→∞∑j=12‍|ψkj-ψ0j|BV)≤liminfk→∞𝒢ε,α(ϕk,ψk1,ψk2)=inf𝒢ε,α.
Therefore, (ϕ,ψ1,ψ2) is a minimizer of 𝒢ε,α.In the sequel, we prove that, whenε→0, the minimizers of 𝒢ε,α approximate a minimizer of the functional 𝒢α. Hence, numerically, the minimizer of 𝒢ε,α can be used as a suitable approximation for the minimizers of 𝒢α.Theorem 18.
Letα and βj be given as in the general assumption of this paper. For each ε>0, denote by (ϕε,α,ψε,α1,ψε,α2) a minimizer of 𝒢ε,α (that exists form Lemma 17). Then, there exists a sequence of positive numbers εk→0 such that (Hεk(ϕεk,α),ϕεk,α,ψεk,α1,ψεk,α2) converges strongly in L1(Ω)×L2(Ω)×(L1(Ω))2 and the limit minimizes 𝒢α on the set of admissible quadruples.Proof.
Let(zα,ϕα,ψα1,ψα2) be a minimizer of the functional 𝒢α on the set of admissible quadruples (cf. Theorem 12). From Definition 3, there exists a sequence {εk} of positive numbers converging to zero and corresponding sequences {ϕk} in H1(Ω) satisfying ϕk→ϕα in L2(Ω), Hεk(ϕk)→zα in L1(Ω) and, finally, sequences {ψkj} in BV×Cc∞(Ω) such that |ψkj|BV→|ψj|BV. Moreover, we can further assume (see Lemma 9) that
(54)R(zα,ϕα,ψα1,ψα2)=limk→∞(∑j=12‍β1|Hεk(ϕk)|BV+β2∥ϕk-ϕ0∥H1(Ω)2+β3∑j=12‍|ψkj-ψ0j|BV).
Let (ϕεk,ψεk1,ψεk2) be a minimizer of 𝒢εk,α. Hence, (ϕεk,ψεk1,ψεk2) belongs to H1(Ω)×𝔹2 (see Lemma 17). The sequences {Hεk(ϕεk)}, {ϕεk}, and {ψεkj} are uniformly bounded in BV(Ω), H1(Ω), and BV(Ω), for j=1,2, respectively. Form compact embedding (see Theorems [36] and [33, Theorem 4, p. 176]), there exist convergent subsequences whose limits are denoted by z~,  ϕ~, and ψ~j belonging to BV(Ω), H1(Ω), and BV(Ω), for j=1,2, respectively.
Summarizing, we haveϕεk→ϕ~ in L2(Ω), Hεk(ϕεk)→z~ in L1(Ω), and ψεkj→ψ~j in L1(Ω), j=1,2. Thus, (z~,ϕ~,ψ~1,ψ~2)∈L1(Ω)×H1(Ω)×L(Ω) is an admissible quadruple (cf. Lemma 8).
From the definition ofR, Lemma 6, and the continuity of F, it follows that
(55)∥F(q(z~,ψ~1,ψ~2))-yδ∥Y2=limk→∞∥F(Pεk(ϕεk,ψεk1,ψεk2))-yδ∥Y2,R(z~,ϕ~,ψ~1,ψ~2)≤liminfk→∞(∑j=12‍β1|Hεk(ϕεk)|BV+β2∥ϕεk-ϕ0∥H1(Ω)2+β3∑j=12‍|ψεkj-ψ0j|BV).
Therefore,
(56)𝒢α(z~,ϕ~,ψ~1,ψ~2)=∥F(q(z~,ψ~1,ψ~2))-yδ∥Y2+αR(z~,ϕ~,ψ~1,ψ~2)≤liminfk→∞𝒢εk,α(ϕεk,ψεk1,ψεk2)≤liminfk→∞𝒢εk,α(ϕk,ψk1,ψk2)≤limsupk→∞∥F(Pεk(ϕk,ψk1,ψk2))-yδ∥Y2+αlimsupk→∞(∑j=12‍β1|Hεk(ϕk)|BV+β2∥ϕk-ϕ0∥H1(Ω)2+αlimsupk→∞+β3∑j=12‍|ψkj-ψ0j|BV)=∥F(q(zα,ψα1,ψα2))-yδ∥Y2+αR(zα,ϕα,ψα1,ψα2)=𝒢α(zα,ϕα1,ψα1,ψα2),
characterizing (z~,ϕ~,ψα1,ψα2) as a minimizer of 𝒢α.


### 5.1. Optimality Conditions for the Stabilized Functional

For numerical purposes it is convenient to derive first-order optimality conditions for minimizers of functional𝒢α. Since P is a discontinuous operator, it is not possible. However, thanks to Theorem 16, the minimizers of the stabilized functionals 𝒢ε,α can be used for approximate minimizers of the functional 𝒢α. Therefore, we consider 𝒢ε,α in (50), with Y a Hilbert space, and we look for the Gâteaux directional derivatives with respect to ϕ and the unknown ψj for  j=1,2.SinceHε′(ϕ) is self-adjoint (note that Hε′(t)={1/εt∈(-ε,0)0other  else.), we can write the optimality conditions for the functional 𝒢ε,α in the form of the system
(57)α(Δ-I)(ϕ-ϕ0)=Lε,α,β(ϕ,ψ1,ψ2),inΩ,(58)(ϕ-ϕ0)·ν=0,at∂Ω,(59)α∇·[∇(ψj-ψ0j)|∇(ψj-ψ0j)|]=Lε,α,βj(ϕ,ψ1,ψ2),j=1,2.
Here ν(x) represents the external unit normal quadruple at x∈∂Ω and
(60)Lε,α,β(ϕ,ψ1,ψ2)=(ψ1-ψ2)β2-1Hε′(ϕ)*F′×(Pε(ϕ,ψ1,ψ2))*×(F(Pε(ϕ,ψ1,ψ2))-yδ)-β1(2β2)-1Hε′(ϕ)∇·[∇Hε(ϕ)|∇Hε(ϕ)|],(61)Lε,α,β1(ϕ,ψ1,ψ2)=(2β3)-1(F′(Pε(ϕ,ψ1,ψ2))Hε(ϕ))*×(F(Pε(ϕ,ψ1,ψ2))-yδ),(62)Lε,α,β2(ϕ,ψ1,ψ2)=(2β3)-1×(F′(Pε(ϕ,ψ1,ψ2))(1-Hε(ϕ)))*×(F(Pε(ϕ,ψ1,ψ2))-yδ).It is worth noticing that the derivation of (57), (58), and (59) is purely formal, since the BV seminorm is not differentiable. Moreover the terms |∇Hε(ϕ)| and |∇(ψj-ψ0j)| appear in the denominators of (57), (58), (59), (60), (61), and (62), respectively.In Section6, systems (57), (58), (59), (60), (61), and (62), are used as starting point for the derivation of a level-set-type method.


## 5.1. Optimality Conditions for the Stabilized Functional

For numerical purposes it is convenient to derive first-order optimality conditions for minimizers of functional𝒢α. Since P is a discontinuous operator, it is not possible. However, thanks to Theorem 16, the minimizers of the stabilized functionals 𝒢ε,α can be used for approximate minimizers of the functional 𝒢α. Therefore, we consider 𝒢ε,α in (50), with Y a Hilbert space, and we look for the Gâteaux directional derivatives with respect to ϕ and the unknown ψj for  j=1,2.SinceHε′(ϕ) is self-adjoint (note that Hε′(t)={1/εt∈(-ε,0)0other  else.), we can write the optimality conditions for the functional 𝒢ε,α in the form of the system
(57)α(Δ-I)(ϕ-ϕ0)=Lε,α,β(ϕ,ψ1,ψ2),inΩ,(58)(ϕ-ϕ0)·ν=0,at∂Ω,(59)α∇·[∇(ψj-ψ0j)|∇(ψj-ψ0j)|]=Lε,α,βj(ϕ,ψ1,ψ2),j=1,2.
Here ν(x) represents the external unit normal quadruple at x∈∂Ω and
(60)Lε,α,β(ϕ,ψ1,ψ2)=(ψ1-ψ2)β2-1Hε′(ϕ)*F′×(Pε(ϕ,ψ1,ψ2))*×(F(Pε(ϕ,ψ1,ψ2))-yδ)-β1(2β2)-1Hε′(ϕ)∇·[∇Hε(ϕ)|∇Hε(ϕ)|],(61)Lε,α,β1(ϕ,ψ1,ψ2)=(2β3)-1(F′(Pε(ϕ,ψ1,ψ2))Hε(ϕ))*×(F(Pε(ϕ,ψ1,ψ2))-yδ),(62)Lε,α,β2(ϕ,ψ1,ψ2)=(2β3)-1×(F′(Pε(ϕ,ψ1,ψ2))(1-Hε(ϕ)))*×(F(Pε(ϕ,ψ1,ψ2))-yδ).It is worth noticing that the derivation of (57), (58), and (59) is purely formal, since the BV seminorm is not differentiable. Moreover the terms |∇Hε(ϕ)| and |∇(ψj-ψ0j)| appear in the denominators of (57), (58), (59), (60), (61), and (62), respectively.In Section6, systems (57), (58), (59), (60), (61), and (62), are used as starting point for the derivation of a level-set-type method.


## 6. Inverse Elliptic Problems

In this section, we discuss the proposed level-set approach and its application in some physical problems modeled by elliptic PDEs. We also discuss briefly the numerical implementations of the iterative method based on the level-set approach. We remark that in the case of noise data the iterative algorithm derived by the level-set approach needs an early stooping criteria [21].


### 6.1. The Inverse Potential Problem

In this subsection, we apply the level-set regularization framework in an inverse potential problem [18, 22, 32]. Differently from [10, 18, 19, 25, 29, 32, 39], we assume that the source u is not necessarily piecewise constant. For relevant applications of the inverse potential problem, see [20, 22, 32, 39] and references therein.The forward problem consists of solving the Poisson boundary value problem(63)-∇·(σ∇w)=u,inΩ,γ1w+γ2wν=gon∂Ω,
on a given domain Ω⊂ℝn with ∂Ω Lipschitz, for a given source function u∈L2(Ω) and a boundary function g∈L2(∂Ω). In (71), ν represents the outer normal vector to ∂Ω, and σ is a known sufficient smooth function. Note that, depending of γ1,γ2∈{0,1}, we have Dirichlet, Neumann, or Robin boundary condition. In this paper, we only consider the case of Dirichlet boundary condition that corresponds to γ1=1 and γ2=0 in (71). Therefore, it is well known that there exists a unique solution w∈H1(Ω) of (71) with w-g∈H01(Ω), [40].Assuming homogeneous Dirichlet boundary condition in (63), the problem can be modeled by the operator equation
(64)F1:L2(Ω)→L2(∂Ω)u↦F1(u)∶=wν|∂Ω.The corresponding inverse problem consists in recovering theL2 source function u, from measurements of the Cauchy data of its corresponding potential w on the boundary of Ω.Using this notation, the inverse potential problem can be written in the abbreviated formF1(u)=yδ, where the available noisy data yδ∈L2(∂Ω) has the same meaning as in (2). It is worth noticing that this inverse problem has, in general, nonunique solution [22]. Therefore, we restrict our attention to minimum-norm solutions [21]. Sufficient conditions for identifiability are given in [20]. Moreover, we restrict our attention to solve the inverse problem (64) in D(F), that is, we assume the unknown parameter u∈D(F), as defined in Section 3. Note that, in this situation, the operator F1 is linear. However, the inverse potential problem is well known to be exponentially ill-posed [20]. Therefore, the solution calls for a regularization strategy [20–22].The following lemma implies that the operatorF1 satisfies the assumption (A2).Lemma 19.
The operatorF1:D(F)⊂L1(Ω)→L2(∂Ω) is continuous with respect to the L1(Ω) topology.Proof.
It is well known from the elliptic regularity theory [40] that ∥w∥H1(Ω)≤c1∥u∥L2(Ω). Let un,u0∈D(F) and wn,w0 the respective solution of (63). Then, from the linearity and continuity of the trace operator from H1(Ω) to L2(∂Ω) [40], we have that
(65)∥F1(un)-F1(u0)∥L2(∂Ω)≤C∥wn-w0∥H1(Ω)≤C~∥un-u0∥L2(Ω)≤C~1∥un-u0∥L1(Ω),
where we use Lemma 5 to obtain the last inequality. Therefore, F1 is sequentially continuous on the L1(Ω) topology. Since L1(Ω) is a metrizable spaces [38], the proof is complete.


#### 6.1.1. A Level-Set Algorithm for the Inverse Potential Problem

We propose an explicit iterative algorithm derived from the optimality conditions (57), (58), (59), (60), (61), and (62), for the Tikhonov functional 𝒢ε,α.For the inverse potential problem with Dirichlet boundary condition (γ1=1  and  γ2=0), the algorithm reads as  shown in Algorithm 20.Algorithm 20 (iterative algorithm based on the level-set approach for the inverse potential problem).
Givenσ and g,(1)
evaluate the residualrk:=F1(Pε(ϕk,ψk1,ψk2))-yδ=(wk)ν|∂Ω-yδ, where wk solves
(66)-∇·(σ∇wk)=Pε(ϕk,ψk1,ψk2),inΩ;wk=g,at∂Ω.(2)
evaluatehk:=F1′(Pε(ϕk,ψk1,ψk2))*(rk)∈L2(Ω), solving
(67)Δhk=0,inΩ;hk=rk,at∂Ω.(3)
calculateδϕk:=Lε,α,β(ϕk,ψk1,ψk2) and δψkj:=Lε,α,βj·(ϕk,ψk1,ψk2), as in  (60), (61), and (62),(4)
update the level-set functionϕk and the level values ψkj, j=1,2,
(68)ϕk+1=ϕk+1αδϕk,ψk+1j=ψkj+1αδψkj.Each step of this iterative method consists of three parts (see Algorithm20): (1) the residual rk∈L2(∂Ω) of the iterate (ϕk,ψkj) is evaluated (this requires solving one elliptic BVP of Dirichlet type); (2) the L2-solution hk of the adjoint problem for the residual is evaluated (this corresponds to solving one elliptic BVP of Dirichlet type); (3) the update δϕk for the level-set function and the updates δψkj for the level values are evaluated (this corresponds to multiplying two functions).In [29], a level-set method was proposed, where the iteration is based on an inexact Newton type method. The inner iteration is implemented using the conjugate gradient method. Moreover, the regularization parameter α>0 is kept fixed. In contrast to [29], in Algorithm 20, we define δt=1/α (as a time increment) in order to derive an evolution equation for the level-set function. Therefore, we are looking for a fixed-point equation related to the system of optimality conditions for the Tikhonov functional. Moreover, the iteration is based on a gradient-type method as in [18].


### 6.2. The Inverse Problem in Nonlinear Electromagnetism

Many interesting physical problems are model by quasilinear elliptic equations. Examples of applications include the identification of inhomogeneity inside nonlinear magnetic materials form indirect or local measurements. Electromagnetic nondestructive tests aim to localize cracks or inhomogeneities in the steel production process, where impurities can be described by a piecewise smooth function [2–4, 11].In this section, we assume thatD⊂⊂Ω is measurable and
(69)u={ψ1,x∈D,ψ2,x∈ΩD,
where ψ1,ψ2∈𝔹 and m>0.The forward problem consists of solving the Poisson boundary value problem(70)-∇·(u∇w)=f,inΩ,w=gon∂Ω,
where Ω⊂ℝn with ∂Ω Lipschitz, the source f∈H-1(Ω), and boundary condition g∈H1/2(∂Ω). It is well known that there exists a unique solution w∈H1(Ω) such that w-g∈H01(Ω) for the PDE (70), [40].Assuming that during the production process the workpiece is contaminated by impurities and that such impurities are described by piecewise smooth function, the inverse electromagnetic problem consists in the identification and the localization of the inhomogeneities as well as the function values of the impurities. The localization of support and the tabulation of the inhomogeneities values can indicate possible sources of contamination in the magnetic material.In other words, the inverse problem in electromagnetism consists in the identification of the support (shape) and the function values ofψ1,ψ2 of the coefficient function u(x) defined in (69). The voltage potential g is chosen such that it corresponds to the current measurement h∶=(w)ν|∂Ω, available as a set of continuous measurement in ∂Ω. This problem is known in the literature as the inverse problem for the Dirichlet-to-Neumann map [20].With this framework, the problem can be modeled by the operator equation(71)F2:D(F)⊂L1(Ω)→H1/2(∂Ω),u↦F2(u)∶=w|∂Ω,
where the potential profile  g=w|∂Ω∈H1/2(Ω)  is given.The authors in [4] investigated a level-set approach for solving an inverse problem of identification of inhomogeneities inside a nonlinear material, from local measurements of the magnetic induction. The assumption in [4] is that part of the inhomogeneities is given by a crack localized inside the workpiece and that, outside the crack region, magnetic conductivities are nonlinear and they depend on the magnetic induction. In other words, that ψ1=μ1 and ψ2=μ2(|∇w|2), where μ1 is the (constant) air conductivity and μ2=μ2(|∇w|2) is a nonlinear conductivity of the workpiece material, whose values are assumed to be known. In [4], they also present a successful iterative algorithm and numerical experiment. However, in [4], the measurements and therefore the data are given in the whole Ω. Such amount of measurements is not reasonable in applications. Moreover, the proposed level-set algorithm is based on an optimality condition of a least square functional with H1(Ω)-seminorm regularization. Therefore, there is no guarantee of existence of minimum for the proposed functional.Remark 21.
Note thatF2(u)=TD(w), where TD is the Dirichlet trace operator. Moreover, since TD:H1(Ω)→H1/2(∂Ω) is linear and continuous [40], we have ∥TD(w)∥H1/2(∂Ω)≤c∥w∥H1(Ω).In the following lemma, we prove that the operatorF2 satisfies the Assumption (A2).Lemma 22.
Let the operatorF2:D(F)⊂L1(Ω)→H1/2(∂Ω) as defined in (71). Then, F2 is continuous with respect to the L1(Ω) topology.Proof.
Letun,u0∈D(F) and wn,w0 denoting the respective solution of (63). The linearity of (70) implies that wn-w0∈H01(Ω), and it satisfies
(72)∇·(un∇wn)-∇·(u0∇w0)=0,
with homogeneous boundary condition. Therefore, using the weak formulation for (72), we have
(73)∫Ω‍(∇·(un∇wn)-∇·(u0∇w0))φdx=0,∀φ∈H01(Ω).
In particular, the weak formulation holds true for φ=wn-w0. From the Green formula [40] and the assumption that m>0 (that guarantee ellipticity of (70)), it follows that
(74)m∥∇wn-∇w0∥L2(Ω)2≤∫Ω‍un|∇wn-∇w0|2dx≤∫Ω‍|(un-u0)∥∇w0∥(∇wn-∇w0)|dx.
From [41, Theorem 1], there exists ε>0 (small enough) such that w0∈W1,p(Ω) for p=2+ε. Using the Hölder inequality [40] with 1/p+1/q=1/2 (note that q>2 in (74)), it follows that
(75)m∥∇wn-∇w0∥L2(Ω)2≤∥un-u0∥Lq(Ω)×∥∇w0∥Lp(Ω)∥∇wn-∇w0∥L2(Ω).
Therefore, using the Poincaré inequality [40] and (75), we have
(76)∥wn-w0∥H1(Ω)≤C∥un-u0∥Lq(Ω),
where the constant C depends only of m,Ω,∥∇w0∥ and the Poincaré constant. Now, the assertion follows from Lemma 5 and Remark 21.


#### 6.2.1. A Level-Set Algorithm for Inverse Problem in Nonlinear Electromagnetism

We propose an explicit iterative algorithm derived from the optimality conditions (57), (58), (59), (60), (61), and (62), for the Tikhonov functional 𝒢ε,α. Each iteration of this algorithm consists in the following steps: in the first step the residual vector r∈L2(∂Ω) corresponding to the iterate (ϕn,ψn1,ψn2) is evaluated. This requires the solution of one elliptic BVP of Dirichlet type. In the second step, the solutions v∈H1(Ω) of the adjoint problems for the residual components r are evaluated. This corresponds to solving one elliptic BVP of Neumann type and to computing the inner product ∇w·∇v in L2(Ω). Next, the computation of Lε,α,β(ϕn,ψn1,ψn2) and Lε,α,βj(ϕn,ψn1,ψn2) as in (60), (61), and (62), follows. The fourth step is the updates of the level-set function δϕn∈H1(Ω) and the level function values δψnj∈BV(Ω) by solving (57), (58), and (59).The algorithm is summarized in Algorithm23.Algorithm 23 (an explicit algorithm based on the proposed level-set iterative regularization method).
(1)
Evaluate the residualr:=F2(Pε(ϕn,ψn1,ψn2))-yδ=w|∂Ω-gδ, where w∈H1(Ω) solves
(77)∇·(Pε(ϕn,ψn1,ψn2)∇w)=f,inΩ;w=g,at∂Ω.
(2)
EvaluateF2′(Pε(ϕn,ψn1,ψn2))*r:=∇w·∇v∈L2(Ω), where w is the function computed in step (1) and v∈H1(Ω)  solves
(78)∇·(Pε(ϕn,ψn1,ψn2)∇v)=0inΩ;vν=r,at∂Ω.
(3)
CalculateLε,α,β(ϕn,ψn1,ψn2) and Lε,α,βj(ϕn,ψn1,ψn2) as in  (60), (61), and (62).
(4)
Evaluate the updatesδϕ∈H1(Ω), δψj∈BV(Ω) by solving  (57), (58), and (59).
(5)
Update the level-set functionsϕn+1=ϕn+(1/α)δϕ and the level function values ψn+1j=ψnj+(1/α)δψj.


## 6.1. The Inverse Potential Problem

In this subsection, we apply the level-set regularization framework in an inverse potential problem [18, 22, 32]. Differently from [10, 18, 19, 25, 29, 32, 39], we assume that the source u is not necessarily piecewise constant. For relevant applications of the inverse potential problem, see [20, 22, 32, 39] and references therein.The forward problem consists of solving the Poisson boundary value problem(63)-∇·(σ∇w)=u,inΩ,γ1w+γ2wν=gon∂Ω,
on a given domain Ω⊂ℝn with ∂Ω Lipschitz, for a given source function u∈L2(Ω) and a boundary function g∈L2(∂Ω). In (71), ν represents the outer normal vector to ∂Ω, and σ is a known sufficient smooth function. Note that, depending of γ1,γ2∈{0,1}, we have Dirichlet, Neumann, or Robin boundary condition. In this paper, we only consider the case of Dirichlet boundary condition that corresponds to γ1=1 and γ2=0 in (71). Therefore, it is well known that there exists a unique solution w∈H1(Ω) of (71) with w-g∈H01(Ω), [40].Assuming homogeneous Dirichlet boundary condition in (63), the problem can be modeled by the operator equation
(64)F1:L2(Ω)→L2(∂Ω)u↦F1(u)∶=wν|∂Ω.The corresponding inverse problem consists in recovering theL2 source function u, from measurements of the Cauchy data of its corresponding potential w on the boundary of Ω.Using this notation, the inverse potential problem can be written in the abbreviated formF1(u)=yδ, where the available noisy data yδ∈L2(∂Ω) has the same meaning as in (2). It is worth noticing that this inverse problem has, in general, nonunique solution [22]. Therefore, we restrict our attention to minimum-norm solutions [21]. Sufficient conditions for identifiability are given in [20]. Moreover, we restrict our attention to solve the inverse problem (64) in D(F), that is, we assume the unknown parameter u∈D(F), as defined in Section 3. Note that, in this situation, the operator F1 is linear. However, the inverse potential problem is well known to be exponentially ill-posed [20]. Therefore, the solution calls for a regularization strategy [20–22].The following lemma implies that the operatorF1 satisfies the assumption (A2).Lemma 19.
The operatorF1:D(F)⊂L1(Ω)→L2(∂Ω) is continuous with respect to the L1(Ω) topology.Proof.
It is well known from the elliptic regularity theory [40] that ∥w∥H1(Ω)≤c1∥u∥L2(Ω). Let un,u0∈D(F) and wn,w0 the respective solution of (63). Then, from the linearity and continuity of the trace operator from H1(Ω) to L2(∂Ω) [40], we have that
(65)∥F1(un)-F1(u0)∥L2(∂Ω)≤C∥wn-w0∥H1(Ω)≤C~∥un-u0∥L2(Ω)≤C~1∥un-u0∥L1(Ω),
where we use Lemma 5 to obtain the last inequality. Therefore, F1 is sequentially continuous on the L1(Ω) topology. Since L1(Ω) is a metrizable spaces [38], the proof is complete.


### 6.1.1. A Level-Set Algorithm for the Inverse Potential Problem

We propose an explicit iterative algorithm derived from the optimality conditions (57), (58), (59), (60), (61), and (62), for the Tikhonov functional 𝒢ε,α.For the inverse potential problem with Dirichlet boundary condition (γ1=1  and  γ2=0), the algorithm reads as  shown in Algorithm 20.Algorithm 20 (iterative algorithm based on the level-set approach for the inverse potential problem).
Givenσ and g,(1)
evaluate the residualrk:=F1(Pε(ϕk,ψk1,ψk2))-yδ=(wk)ν|∂Ω-yδ, where wk solves
(66)-∇·(σ∇wk)=Pε(ϕk,ψk1,ψk2),inΩ;wk=g,at∂Ω.(2)
evaluatehk:=F1′(Pε(ϕk,ψk1,ψk2))*(rk)∈L2(Ω), solving
(67)Δhk=0,inΩ;hk=rk,at∂Ω.(3)
calculateδϕk:=Lε,α,β(ϕk,ψk1,ψk2) and δψkj:=Lε,α,βj·(ϕk,ψk1,ψk2), as in  (60), (61), and (62),(4)
update the level-set functionϕk and the level values ψkj, j=1,2,
(68)ϕk+1=ϕk+1αδϕk,ψk+1j=ψkj+1αδψkj.Each step of this iterative method consists of three parts (see Algorithm20): (1) the residual rk∈L2(∂Ω) of the iterate (ϕk,ψkj) is evaluated (this requires solving one elliptic BVP of Dirichlet type); (2) the L2-solution hk of the adjoint problem for the residual is evaluated (this corresponds to solving one elliptic BVP of Dirichlet type); (3) the update δϕk for the level-set function and the updates δψkj for the level values are evaluated (this corresponds to multiplying two functions).In [29], a level-set method was proposed, where the iteration is based on an inexact Newton type method. The inner iteration is implemented using the conjugate gradient method. Moreover, the regularization parameter α>0 is kept fixed. In contrast to [29], in Algorithm 20, we define δt=1/α (as a time increment) in order to derive an evolution equation for the level-set function. Therefore, we are looking for a fixed-point equation related to the system of optimality conditions for the Tikhonov functional. Moreover, the iteration is based on a gradient-type method as in [18].


## 6.1.1. A Level-Set Algorithm for the Inverse Potential Problem

We propose an explicit iterative algorithm derived from the optimality conditions (57), (58), (59), (60), (61), and (62), for the Tikhonov functional 𝒢ε,α.For the inverse potential problem with Dirichlet boundary condition (γ1=1  and  γ2=0), the algorithm reads as  shown in Algorithm 20.Algorithm 20 (iterative algorithm based on the level-set approach for the inverse potential problem).
Givenσ and g,(1)
evaluate the residualrk:=F1(Pε(ϕk,ψk1,ψk2))-yδ=(wk)ν|∂Ω-yδ, where wk solves
(66)-∇·(σ∇wk)=Pε(ϕk,ψk1,ψk2),inΩ;wk=g,at∂Ω.(2)
evaluatehk:=F1′(Pε(ϕk,ψk1,ψk2))*(rk)∈L2(Ω), solving
(67)Δhk=0,inΩ;hk=rk,at∂Ω.(3)
calculateδϕk:=Lε,α,β(ϕk,ψk1,ψk2) and δψkj:=Lε,α,βj·(ϕk,ψk1,ψk2), as in  (60), (61), and (62),(4)
update the level-set functionϕk and the level values ψkj, j=1,2,
(68)ϕk+1=ϕk+1αδϕk,ψk+1j=ψkj+1αδψkj.Each step of this iterative method consists of three parts (see Algorithm20): (1) the residual rk∈L2(∂Ω) of the iterate (ϕk,ψkj) is evaluated (this requires solving one elliptic BVP of Dirichlet type); (2) the L2-solution hk of the adjoint problem for the residual is evaluated (this corresponds to solving one elliptic BVP of Dirichlet type); (3) the update δϕk for the level-set function and the updates δψkj for the level values are evaluated (this corresponds to multiplying two functions).In [29], a level-set method was proposed, where the iteration is based on an inexact Newton type method. The inner iteration is implemented using the conjugate gradient method. Moreover, the regularization parameter α>0 is kept fixed. In contrast to [29], in Algorithm 20, we define δt=1/α (as a time increment) in order to derive an evolution equation for the level-set function. Therefore, we are looking for a fixed-point equation related to the system of optimality conditions for the Tikhonov functional. Moreover, the iteration is based on a gradient-type method as in [18].


## 6.2. The Inverse Problem in Nonlinear Electromagnetism

Many interesting physical problems are model by quasilinear elliptic equations. Examples of applications include the identification of inhomogeneity inside nonlinear magnetic materials form indirect or local measurements. Electromagnetic nondestructive tests aim to localize cracks or inhomogeneities in the steel production process, where impurities can be described by a piecewise smooth function [2–4, 11].In this section, we assume thatD⊂⊂Ω is measurable and
(69)u={ψ1,x∈D,ψ2,x∈ΩD,
where ψ1,ψ2∈𝔹 and m>0.The forward problem consists of solving the Poisson boundary value problem(70)-∇·(u∇w)=f,inΩ,w=gon∂Ω,
where Ω⊂ℝn with ∂Ω Lipschitz, the source f∈H-1(Ω), and boundary condition g∈H1/2(∂Ω). It is well known that there exists a unique solution w∈H1(Ω) such that w-g∈H01(Ω) for the PDE (70), [40].Assuming that during the production process the workpiece is contaminated by impurities and that such impurities are described by piecewise smooth function, the inverse electromagnetic problem consists in the identification and the localization of the inhomogeneities as well as the function values of the impurities. The localization of support and the tabulation of the inhomogeneities values can indicate possible sources of contamination in the magnetic material.In other words, the inverse problem in electromagnetism consists in the identification of the support (shape) and the function values ofψ1,ψ2 of the coefficient function u(x) defined in (69). The voltage potential g is chosen such that it corresponds to the current measurement h∶=(w)ν|∂Ω, available as a set of continuous measurement in ∂Ω. This problem is known in the literature as the inverse problem for the Dirichlet-to-Neumann map [20].With this framework, the problem can be modeled by the operator equation(71)F2:D(F)⊂L1(Ω)→H1/2(∂Ω),u↦F2(u)∶=w|∂Ω,
where the potential profile  g=w|∂Ω∈H1/2(Ω)  is given.The authors in [4] investigated a level-set approach for solving an inverse problem of identification of inhomogeneities inside a nonlinear material, from local measurements of the magnetic induction. The assumption in [4] is that part of the inhomogeneities is given by a crack localized inside the workpiece and that, outside the crack region, magnetic conductivities are nonlinear and they depend on the magnetic induction. In other words, that ψ1=μ1 and ψ2=μ2(|∇w|2), where μ1 is the (constant) air conductivity and μ2=μ2(|∇w|2) is a nonlinear conductivity of the workpiece material, whose values are assumed to be known. In [4], they also present a successful iterative algorithm and numerical experiment. However, in [4], the measurements and therefore the data are given in the whole Ω. Such amount of measurements is not reasonable in applications. Moreover, the proposed level-set algorithm is based on an optimality condition of a least square functional with H1(Ω)-seminorm regularization. Therefore, there is no guarantee of existence of minimum for the proposed functional.Remark 21.
Note thatF2(u)=TD(w), where TD is the Dirichlet trace operator. Moreover, since TD:H1(Ω)→H1/2(∂Ω) is linear and continuous [40], we have ∥TD(w)∥H1/2(∂Ω)≤c∥w∥H1(Ω).In the following lemma, we prove that the operatorF2 satisfies the Assumption (A2).Lemma 22.
Let the operatorF2:D(F)⊂L1(Ω)→H1/2(∂Ω) as defined in (71). Then, F2 is continuous with respect to the L1(Ω) topology.Proof.
Letun,u0∈D(F) and wn,w0 denoting the respective solution of (63). The linearity of (70) implies that wn-w0∈H01(Ω), and it satisfies
(72)∇·(un∇wn)-∇·(u0∇w0)=0,
with homogeneous boundary condition. Therefore, using the weak formulation for (72), we have
(73)∫Ω‍(∇·(un∇wn)-∇·(u0∇w0))φdx=0,∀φ∈H01(Ω).
In particular, the weak formulation holds true for φ=wn-w0. From the Green formula [40] and the assumption that m>0 (that guarantee ellipticity of (70)), it follows that
(74)m∥∇wn-∇w0∥L2(Ω)2≤∫Ω‍un|∇wn-∇w0|2dx≤∫Ω‍|(un-u0)∥∇w0∥(∇wn-∇w0)|dx.
From [41, Theorem 1], there exists ε>0 (small enough) such that w0∈W1,p(Ω) for p=2+ε. Using the Hölder inequality [40] with 1/p+1/q=1/2 (note that q>2 in (74)), it follows that
(75)m∥∇wn-∇w0∥L2(Ω)2≤∥un-u0∥Lq(Ω)×∥∇w0∥Lp(Ω)∥∇wn-∇w0∥L2(Ω).
Therefore, using the Poincaré inequality [40] and (75), we have
(76)∥wn-w0∥H1(Ω)≤C∥un-u0∥Lq(Ω),
where the constant C depends only of m,Ω,∥∇w0∥ and the Poincaré constant. Now, the assertion follows from Lemma 5 and Remark 21.


### 6.2.1. A Level-Set Algorithm for Inverse Problem in Nonlinear Electromagnetism

We propose an explicit iterative algorithm derived from the optimality conditions (57), (58), (59), (60), (61), and (62), for the Tikhonov functional 𝒢ε,α. Each iteration of this algorithm consists in the following steps: in the first step the residual vector r∈L2(∂Ω) corresponding to the iterate (ϕn,ψn1,ψn2) is evaluated. This requires the solution of one elliptic BVP of Dirichlet type. In the second step, the solutions v∈H1(Ω) of the adjoint problems for the residual components r are evaluated. This corresponds to solving one elliptic BVP of Neumann type and to computing the inner product ∇w·∇v in L2(Ω). Next, the computation of Lε,α,β(ϕn,ψn1,ψn2) and Lε,α,βj(ϕn,ψn1,ψn2) as in (60), (61), and (62), follows. The fourth step is the updates of the level-set function δϕn∈H1(Ω) and the level function values δψnj∈BV(Ω) by solving (57), (58), and (59).The algorithm is summarized in Algorithm23.Algorithm 23 (an explicit algorithm based on the proposed level-set iterative regularization method).
(1)
Evaluate the residualr:=F2(Pε(ϕn,ψn1,ψn2))-yδ=w|∂Ω-gδ, where w∈H1(Ω) solves
(77)∇·(Pε(ϕn,ψn1,ψn2)∇w)=f,inΩ;w=g,at∂Ω.
(2)
EvaluateF2′(Pε(ϕn,ψn1,ψn2))*r:=∇w·∇v∈L2(Ω), where w is the function computed in step (1) and v∈H1(Ω)  solves
(78)∇·(Pε(ϕn,ψn1,ψn2)∇v)=0inΩ;vν=r,at∂Ω.
(3)
CalculateLε,α,β(ϕn,ψn1,ψn2) and Lε,α,βj(ϕn,ψn1,ψn2) as in  (60), (61), and (62).
(4)
Evaluate the updatesδϕ∈H1(Ω), δψj∈BV(Ω) by solving  (57), (58), and (59).
(5)
Update the level-set functionsϕn+1=ϕn+(1/α)δϕ and the level function values ψn+1j=ψnj+(1/α)δψj.


## 6.2.1. A Level-Set Algorithm for Inverse Problem in Nonlinear Electromagnetism

We propose an explicit iterative algorithm derived from the optimality conditions (57), (58), (59), (60), (61), and (62), for the Tikhonov functional 𝒢ε,α. Each iteration of this algorithm consists in the following steps: in the first step the residual vector r∈L2(∂Ω) corresponding to the iterate (ϕn,ψn1,ψn2) is evaluated. This requires the solution of one elliptic BVP of Dirichlet type. In the second step, the solutions v∈H1(Ω) of the adjoint problems for the residual components r are evaluated. This corresponds to solving one elliptic BVP of Neumann type and to computing the inner product ∇w·∇v in L2(Ω). Next, the computation of Lε,α,β(ϕn,ψn1,ψn2) and Lε,α,βj(ϕn,ψn1,ψn2) as in (60), (61), and (62), follows. The fourth step is the updates of the level-set function δϕn∈H1(Ω) and the level function values δψnj∈BV(Ω) by solving (57), (58), and (59).The algorithm is summarized in Algorithm23.Algorithm 23 (an explicit algorithm based on the proposed level-set iterative regularization method).
(1)
Evaluate the residualr:=F2(Pε(ϕn,ψn1,ψn2))-yδ=w|∂Ω-gδ, where w∈H1(Ω) solves
(77)∇·(Pε(ϕn,ψn1,ψn2)∇w)=f,inΩ;w=g,at∂Ω.
(2)
EvaluateF2′(Pε(ϕn,ψn1,ψn2))*r:=∇w·∇v∈L2(Ω), where w is the function computed in step (1) and v∈H1(Ω)  solves
(78)∇·(Pε(ϕn,ψn1,ψn2)∇v)=0inΩ;vν=r,at∂Ω.
(3)
CalculateLε,α,β(ϕn,ψn1,ψn2) and Lε,α,βj(ϕn,ψn1,ψn2) as in  (60), (61), and (62).
(4)
Evaluate the updatesδϕ∈H1(Ω), δψj∈BV(Ω) by solving  (57), (58), and (59).
(5)
Update the level-set functionsϕn+1=ϕn+(1/α)δϕ and the level function values ψn+1j=ψnj+(1/α)δψj.


## 7. Conclusions and Future Directions

In this paper, we generalize the results of convergence and stability of the level-set regularization approach proposed in [18, 19], where the level values of discontinuities are not piecewise constant inside each region. We analyze the particular case, where the set Ω is divided in two regions. However, it is easy to extend the analysis for the case of multiple regions adapting the multiple level-set approach in [17, 19].We apply the level-set framework for two problems: the inverse potential problem and in an inverse problem in nonlinear electromagnetism with piecewise nonconstant solution. In both cases, we prove that the parameter-to-solution map satisfies the assumption (A1). The inverse potential problem application is a natural generalization of the problem computed in [17–19]. We also investigate the applicability of an inverse problem in nonlinear electromagnetism in the identification of inhomogeneities inside a nonlinear magnetic workpiece. Moreover, we propose iterative algorithm based on the optimality condition of the smooth Tikhonov functional 𝒢ε,α.A natural continuation of this paper is the numerical implementation. Level-set numerical implementations for the inverse potential problem were done before in [17–19], where the level values are assumed to be constant. Implementations of level-set methods for resistivity/conductivity problem in elliptic equation have been intensively implemented recently, for example, [2, 4, 6, 10, 11, 42, 43].



---

*Source: 123643-2013-01-31.xml*