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On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients
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.
Since the seminal paper of Santosa , 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 domain in subdomains , with , 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 , 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 quantity that represents all parameter profiles inside the individual subregions of , from data , where and are Banach spaces and 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 where is the forward operator.
Neither existence nor uniqueness of a solution to (1) is a guarantee. For simplicity, we assume that, for exact data , 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 are not known and we have only access to noise data , whose level of noise is assumed to be known a priori and satisfies
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 . 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 with distinct, but known, piecewise constant values were investigated.
In [12, 17, 24], level-set approaches were derived to solve inverse problems, assuming that is defined by several distinct constant values. In both cases, one needs only to identify the level sets of , that is, the inverse problem reduces to a shape identification problem. On the other hand, when the level values of 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 . 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 parameter 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 .
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 solution of (1). We obtain a regularized solution using a Tikhonov-type regularization method, since the level values of 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 Section 2, 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 parameter in (1) assumes two unknown functional values, that is, a.e. in , where is a bounded set. More specifically, we assume the existence of a mensurable set , with , such that if and if . With this framework, the inverse problem that we are interested in in this paper is the stable identification of both the shape of and the value function for belonging to and to , respectively, from observation of the data .
We remark that, if and with and unknown constants values, the problem of identifying was rigorously studied before in . Moreover, many other approaches to this case appear in the literature; see [2, 19, 23, 24] and references therein. Recently, in , an 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 . 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 . 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 that belongs to . Note that, we can identify the distinct level sets of the function with the definition of the Heaviside projector Now, from the framework introduced above, a solution of (1) can be represented as With this notation, we are able to determine the shapes of as and as .
The functional level values , are also assumed be unknown, and they should be determined as well.
Assumption 1. We assume that is measurable and , a.e.in, for some constant values .
Remark 1. We remark that implies that . Since is bounded, . Moreover, Hence, .
Note that in the case that and 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].
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 where and represent some a priori information about the true solution of (1). The parameter plays the role of a regularization parameter, and the values of , act as scaling factors. In other words, ,, needs to be chosen a priori, but independent of the noise level . In practical, , can be chosen in order to represent a priori knowledge of features of the parameter solution and/or to improve the numerical algorithm. A more complete discussion about how to choose , is provided in [17–19].
The regularization strategy in this context is based on penalization. The term on -norm acts simultaneously as a control on the size of the norm of the level-set function and a regularization on the space . The term on is a variational measure of . It is well known that the seminorm acts as a penalizing for the length of the Hausdorff measure of the boundary of the set (see [33, Chapter 5] for details). Finally, the last term on is a variational measure of that acts as a regularization term on the set . This Tikhonov functional extends the ones proposed in [16, 17, 19, 23, 24] (based on - penalization).
Existence of minimizers for the functional (7) in the topology does not follow by direct arguments, since the operator is not necessarily continuous in this topology. Indeed, if is a continuous function at the contact region, then is continuous and the standard Tikhonov regularization theory to the inverse problem holds true . On the other hand, in the interesting case where and represent the level of discontinuities of the parameter , 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 , if with , where denotes the ()-dimensional Hausdorff-measure of the set , then the Heaviside operator maps into the set Therefore, the operator in (4) maps into the admissible parameter set where
Consider the model problem described in Section 1. In this paper, we assume the following.(A1) is bounded with piecewise boundary .(A2)The operator is continuous on with respect to the topology.(A3), , and , denote positive parameters.(A4)Equation (1) has a solution, that is, there exists satisfying and a function satisfying , in the neighborhood of such that , for some . Moreover, there exist functional values such that . For each , we define a smooth approximation to the operator by where is the smooth approximation to described by
Remark 2. It is worth noting that, for any , belongs to and satisfies a.e. in , for all . Moreover, taking into account that , it follows that the operators and , as above, are well defined.
In order to guarantee the existence of a minimizer of defined in (7) in the space , we need to introduce a suitable topology such that the functional has a closed graphic. Therefore, the concept of generalized minimizers (compare with [17, 25]) in this paper is as follows.
Definition 3. Let the operators , , , and be defined as above and the positive parameters , and satisfy the Assumption (A3).
A quadruple is called admissible when(a)there exists a sequence of functions satisfying ,(b)there exists a sequence converging to zero such that , (c)there exist sequences and belonging to such that (d)a generalized minimizer of is considered to be any admissible quadruple minimizing on the set of admissible quadruples. Here the functional is defined by and the functional is defined as The infimum in (16) is taken over all sequences and characterizing as an admissible quadruple.
The convergence 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 -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 (see Lemma 11). The identification of nonconstant level values 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. For , let , be such that in the neighborhood of the level-set and . For each , set and . Then, for all sequences of of positive numbers converging to zero, we have
Here, we use the fact that in the neighborhood of implies that is a local diffeomorphism together with a coarea formula [33, Chapter 4]. Moreover, in satisfies Definition 3, item (c).
Hence, is an admissible quadruple. In particular, we conclude from the general assumption above that the set of admissible quadruple satisfying is not empty.
3.1. Relevant Properties of Admissible Quadruples
Our first result is the proof of the continuity properties of operators , , and 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., ). We present it here for the sake of completeness.
Lemma 5. Let be a measurable subset of with finite measure.
If is a convergent sequence in for some , , then it is a convergent sequence in for all .
In particular, Lemma 5 holds for the sequence .
Proof. See [35, Lemma ].
The next two lemmas are auxiliary results in order to understand the definition of the set of admissible quadruples.
Lemma 6. Let be as in assumption (A1) and .(i)Let be a sequence in with a.e. converging in the -norm to some element and a sequence in converging in the -norm to some . Then converges to in .(ii)Let be such that in as , and let . Then in as .(iii)Given , let be a sequence in converging to in the -norm. Then in , as . Moreover, if are sequences in , converging to some in , with respect to the -norm, then in , as .
Proof. Since is assumed to be bounded, we have and is continuous embedding in . To prove (i), notice that
Here we use Lemma 5 in order to guarantee the convergence of to in .
Assertion (ii) follows with similar arguments and the fact that for all .
As , 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).
Proof (sketch of the proof). Let and . Since satisfies Definition 3, . From [33, Theorem 2, p. 172], there exist sequences in such that In particular, for the subsequence , it follows that Moreover, by assumption . From the lower semicontinuity of variational measure (see [33, Theorem 1, p. 172]), (20), and the definition of space, it follows that .
In the following lemma we prove that the set of admissible quadruples is closed with respect to the topology.
Lemma 8. Let be a sequence of admissible quadruples converging in to some , with . Then, is also an admissible quadruple.
Proof (sketch of the proof). Let . Since is an admissible quadruple, it follows from Definition 3 that there exist sequences , in , , in and a correspondent sequence converging to zero such that
Define the monotone increasing function such that, for every , it holds Hence, for each , From (22), Moreover, with the same arguments as Lemma 7, it follows that and . Therefore, it remains to prove that is an admissible quadruple. From Definition 3 and Lemma 7, it is enough to prove that . If this is not the case, there would exist a with and such that in (the other case, is analogous). Since a.e. in for (see remark after Definition 3), we would have contradicting the second limit in (24).
3.2. Relevant Properties of the Penalization Functional
In the following lemmas, we verify properties of the functional which are fundamental for the convergence analysis outlined in Section 4. In particular, these properties imply that the level sets of are compact in the set of admissible quadruple, that is, assumes a minimizer on this set. First, we prove a lemma that simplifies the functional in (15). Here we present the sketch of the proof. For more details, see the arguments in [19, Lemma 3].
Lemma 9. Let be an admissible quadruple. Then, there exist sequences , , and , as in the Definition 3, such that
Proof (sketch of the proof). For each , the definition of (see Definition 3) guaranties the existence of sequences , , and such that 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 functional in (15) is coercive on the set of admissible quadruples. In other words, given any admissible quadruple , one has
Lemma 11. The functional in (15) is weak lower semicontinuous on the set of admissible quadruples, that is, given a sequence of admissible quadruples such that in , in , and in , for some admissible quadruple , then
Proof. The functional is weak lower semicontinuous (cf. [17, Lemma 5]). As , it follows from [33, Theorem 2, p. 172] that there exist sequences such that . From a diagonal argument, we can extract a subsequence of such that in as . Let . Then, from [33, Theorem 1, p. 167], it follows that Thus, form the definition of (see ), we have Now, the lemma follows from the fact that the functional in (15) is a linear combination of lower semicontinuous functionals.
4. Convergence Analysis
In the following, we consider any positive parameter , as in the general assumption to this paper. First, we prove that the functional in (14) is well posed.
Theorem 12 (well-posedness). The functional in (14) attains minimizers on the set of admissible quadruples.
Proof. Notice that the set of admissible quadruples is not empty, since is admissible. Let be a minimizing sequence for , that is, a sequence of admissible quadruples satisfying . Then, is a bounded sequence of real numbers. Therefore, is uniformly bounded in . Thus, from the Sobolev Embedding Theorem [33, 36], we guarantee the existence of a subsequence (denoted again by ) and the existence of such that in , in , in , and . Moreover, , and . See [33, Theorem 4, p. 176].
From Lemma 8, we conclude that is an admissible quadruple. Moreover, from the weak lower semicontinuity of (Lemma 11), together with the continuity of (Lemma 6) and continuity of (see the general assumption), we obtain proving that minimizes .
In what follows, we will denote a minimizer of by . In particular the functional 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 is called an -minimizing solution if it satisfies (i), (ii) is an admissible quadruple and .
Theorem 14 (-minimizing solutions). Under the general assumptions of this paper, there exists a -minimizing solution.
Proof. From the general assumption on this paper and Remark 4, we conclude that the set of admissible quadruple satisfying is not empty. Thus, in (ii) is finite and there exists a sequence of admissible quadruple satisfying Now, form the definition of , it follows that the sequences , , and are uniformly bounded in and , respectively. Then, from the Sobolev Compact Embedding Theorem [33, 36], we have (up to subsequences) that Lemma 8 implies that is an admissible quadruple. Since is weakly lower semicontinuous (cf. Lemma 11), it follows that Moreover, we conclude from Lemma 6 that Thus, is an -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, . For every , let denote a minimizer of on the set of admissible quadruples. Then, for every sequence of positive numbers converging to zero, there exists a subsequence, denoted again by , such that is strongly convergent in . Moreover, the limit is a solution of (1).
Proof. Let be an -minimizing solution of (1)—its existence is guaranteed by Theorem 14. Let be a sequence of positive numbers converging to zero. For each , denote to be a minimizer of . Then, for each , we have Since , it follows from (40) that Moreover, from the assumption on the sequence , it follows that From (41) and Lemma 10, we conclude that sequences , , and are bounded in and , respectively, for . Using an argument of extraction of diagonal subsequences (see proof of Lemma 8), we can guarantee the existence of an admissible quadruple such that Now, from Lemma 6(i), it follows that in . Using the continuity of the operator together with (40) and (42), we conclude that On the other hand, from the lower semicontinuity of and (41), it follows that concluding the proof.
Theorem 16 (stability). Let be a function satisfying and . Moreover, let be a sequence of positive numbers converging to zero and corresponding noisy data satisfying (2). Then, there exists a subsequence, denoted again by and a sequence such that converges in to solution of (1).
Proof. Let be an -minimizer solution of (1) (such existence is guaranteed by Theorem 14). For each , let be a minimizer of . Then, for each , we have
From (46) and the definition of , it follows that
Taking the limit as in (47), it follows from theorem assumptions on that With the same arguments as in the proof of Theorem 15, we conclude that at least a subsequence that we denote again by converges in to some admissible quadruple . Moreover, by taking the limit as in (46), it follows from the assumption on and Lemma 6 that
The functional 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 . 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 of . Let be the functional defined by where is defined in (11). The functional is well posed as the following lemma shows.
Lemma 17. Given positive constants , , as in the general assumption of this paper, and , . Then, the functional in (50) attains a minimizer on .
Proof. Since , there exists a minimizing sequence in satisfying
Then, for fixed , the definition of in (50) implies that the sequences and are bounded in and , respectively. Therefore, from Banach-Alaoglu-Bourbaki Theorem  in and from [33, Theorem 4, p. 176], in , . Now, a similar argument as in Lemma 7 implies that , for . Moreover, by the weak lower semicontinuity of the -norm  and measure (see [33, Theorem 1, p. 172]), it follows that
The compact embedding of into  implies in the existence of a subsequence of (that we denote with the same index) such that in . It follows from Lemma 6 and [33, Theorem 1, p. 172] that . Hence, from continuity of in , continuity of (see Lemma 6), together with the estimates above, we conclude that Therefore, is a minimizer of .
In the sequel, we prove that, when , the minimizers of approximate a minimizer of the functional . Hence, numerically, the minimizer of can be used as a suitable approximation for the minimizers of .
Theorem 18. Let and be given as in the general assumption of this paper. For each , denote by a minimizer of (that exists form Lemma 17). Then, there exists a sequence of positive numbers such that converges strongly in and the limit minimizes on the set of admissible quadruples.
Proof. Let be a minimizer of the functional on the set of admissible quadruples (cf. Theorem 12). From Definition 3, there exists a sequence of positive numbers converging to zero and corresponding sequences in satisfying in , in and, finally, sequences in such that . Moreover, we can further assume (see Lemma 9) that
Let be a minimizer of . Hence, belongs to (see Lemma 17). The sequences , , and are uniformly bounded in , , and , for , respectively. Form compact embedding (see Theorems  and [33, Theorem 4, p. 176]), there exist convergent subsequences whose limits are denoted by , , and belonging to , , and , for , respectively.
Summarizing, we have in , in , and in , . Thus, is an admissible quadruple (cf. Lemma 8).
From the definition of , Lemma 6, and the continuity of , it follows that Therefore, characterizing as a minimizer of .
5.1. Optimality Conditions for the Stabilized Functional
For numerical purposes it is convenient to derive first-order optimality conditions for minimizers of functional . Since is a discontinuous operator, it is not possible. However, thanks to Theorem 16, the minimizers of the stabilized functionals can be used for approximate minimizers of the functional . Therefore, we consider in (50), with a Hilbert space, and we look for the Gâteaux directional derivatives with respect to and the unknown for .
Since is self-adjoint (note that ), we can write the optimality conditions for the functional in the form of the system Here represents the external unit normal quadruple at and