Abstract
In this paper, we are interested in the inverse problem of the determination of the unknown part of the boundary of a uniformly Lipschitzian domain included in from the measurement of the normal derivative on suitable part of its boundary, where is the solution of the wave equation in and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part of . From necessary conditions, we estimate a Lagrange multiplier which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.
1. Introduction and Main Result
The inverse problem in this paper means the problem of reconstructing object from observation data. We restrict ourselves to the case when the observation data are given as a boundary of the Cauchy data of a solution of a wave equation and the unknown object is a boundary. Let and let be a bounded domain with smooth boundary . Moreover, let us consider a partition of this boundary , where is the accessible regular part, for example, , and it satisfies the interior sphere condition (see [1]) and is the unknown part of boundary. Throughout this paper, let us take the functional with . We use the following notations:
We consider the following wave equation:
First of all, assume that , and are given and verified the compatibility condition for all . The Cauchy problem (2) is known to be well posed and one can also prove the solution ; this result can be found in [2].
Our inverse problem consists of determining , the unknown part of boundary from Cauchy data of a weak solution of the following problem (3) with a given potential .where is a given function, and the corresponding Neumann data measured on and is outer normal vector unit. In this case, and are unknowns and we assume that the normal derivatives of function can be measured by .
In Section 1, we present the inverse problem which consists of finding a formula reconstructing the part of boundary from the Cauchy data. The remainder of the paper is organized as follows.
In Section 2, we establish the shape optimization problem and prove the existence results. In Section 3, we study the derivation with respect to the domain and we prove the necessary conditions of optimality, that is, the existence of a Lagrange multiplier. Section 4 is devoted to auxiliary lemmas based on maximum principle theory for hyperbolic equations; see [3]. In Section 5, we give by a monotonicity result and under geometrical assumptions a uniqueness result of our inverse problem.
The questions for the wave equation have all already received positive answers since the uniqueness result for the linear inverse problem has been proved by Klibanov (see [4]) and Lipschitz stability results (for both linear and nonlinear inverse problems) of Yamamoto (see [5]). Many results, to which we can refer concerning the wave equation, are related to the same type of inverse problem of determining a potential . Some of them can be found in [6], for example. These references are all based upon local Carleman estimates for the wave operator (see [5]) or global Carleman estimates for Schrödinger equation (see [7]) to prove uniqueness and stability estimate solution. Nevertheless, in our approach, for a given potential, the reconstruction of from the Cauchy data is one of our aims and the estimation of the Lagrange multiplier which appears by derivation with respect to the domain of the energy of system in an admissible set of domains is another interesting one.
In [8], Isakov and Friedman studied the inverse spectral problems. This domain problem was formulated already by Sir A. Shuster who in 1882 introduced spectroscopy as a way to find a shape of a bell by means of the sounds which it is capable of sending out. More rigorously, it has been posed by Bochner in the 1950s and then in the well-known lecture of Kac (see [9]) “Can one hear the shape of a drum?” in 1966. He also studied inverse problem of gravimetry, inverse conductivity problem, tomography, and the inverse seismic problem and indicated their applications.
In [10], using conformal mapping technique, Kress studied mathematical modelling of electrostatic or thermal imaging methods in nondestructive testing and evaluation. In these applications, an unknown inclusion within a conducting host medium with constant conductivity is assessed from overdetermined Cauchy data on the accessible exterior boundary of the medium.
2. Study of the Shape Optimization Problem
2.1. Auxiliary Results
We describe some fundamental properties which will be useful in the following. We consider a fixed and bounded domain in which contains all open subsets we used.
Definition 1. Let and be two compact subsets of . LetNote thatLetand we call Hausdorff distance of and , the following positive number, denoted by .
Let be a sequence of open subsets of and let be an open subset of . We say that the sequence converges on in the Hausdorff sense and we denote it by if .
Let be a sequence of open sets of and let be an open set of . We say that the sequence converges on in the sense of , if converges on in being the characteristic functions of .
Remark 1. Let be a sequence of compact sets included in a fixed and bounded set of ; then there are a compact set and such that converges on in the sense of Hausdorff.
We have the following lemmas.
Lemma 1. Let be a sequence of open set in having the -cône property with , with being a compact set. Then there exists an open set including which satisfies the -cône property and a subsequence such thatFor detailed proof, see [11].
Lemma 2. Let be a sequence of open sets having the cône property and converging to in the sense of Hausdorff.
If is the solution of problem 2 in for all and is the solution of this problem in , then converges to .
For detailed proof, see [11].
Lemma 3. Let be a Lipschitz domain with in ; if in , then in .
For detailed proof, see [11].
For all solution ofat time , the energy of is defined byand it verifiesLet be the functional defined bywhere is the solution:We study the existence of the result of the following optimization problem: , where the class of admissible domains is defined bywhere is a bounded domain of containing all and is positive real.
Remark 2. Remark that being uniformly Lipschitz means satisfies the -cone property; for details, see [11].
2.2. Existence of Solution of Shape Optimization Problem
We study the existence result of the following shape optimization problem.
Proposition 1. “Find belonging to such that has a solution.”
Proof 1. For the proof, we take the following: , which implies Let . Therefore, there is a minimizing sequence such that converges to .
The fact that sequence is bounded ensures the existence of a subsequence and a domain such that converges to in the sense of Hausdorff according to Lemma 1.
Therefore, we consider and .
Sequence is bounded in . If not, converges to , which is a contradiction.
Space is reflexive; then there is a subsequence et such that converges weakly to in and always according to Lemma 3..
Thus, we obtain . Therefore, ; then
Remark 3. It is easy to verify that equals according to Lemma 1 and satisfies problem (12). On the other hand, we have a regularity of solution to problem (12) (see [12, 13]).
3. Derivation with respect to the Domain
These results would allow us to assume regularity on solution of the shape optimization problem to proceed with the derivation with respect to the domain and to show the result of monotony.
Let , with being the set of domains having the cône property and being a normal vector space.
Let us consider , where varies around 0 in a normalized vector space of applications from to . We can introduce the differentiability in the classic sense of Frechet for the application . This is efficient in proving the regularity properties of the shape functional, in using the derivation calculations, and in clearly identifying the derivatives structures so-called “of shape”.
We will use a numerical variable to be comfortable in the calculations.
Let us choose function with being a regular vector field from to ; . We analyse the derivative of and the expression of , where .
Let us consider . The question is, how can we derive the function where is a variable domain?(1)We know that is extended by 0 because (2)We know that function is always defined on the fixed domain ; it belongs to space
To derive function , it suffices to “transport” it by , because has more regularity than . Therefore, it is more strategic to study this problem.
We fix as measurable. It is easy to verify that is measurable and that if is open, then is also measurable.
3.1. Notations
Let denote the bounded space, and Lipschitzian applications from in itself provide with the normwhere is provided with the Euclidean norm . Let denote the identity of .
We recall that this space is identified with the subspace of , whose partial derivatives in the sense of distributions are functions of . In addition, the functions of are a.e. differentiable, and we havewhere the norms of differential are understood as linear operators of . However, the consideration of is interesting for deformations of the Lipschitzian domain.
If , by the theorem of fixed point, is inversible such that and we have (see [11])
Thus, is continuous in 0 is differentiable in 0, and its differentiability is the opposite of identity
Let us consider derivable in 0 with
Because is close to the identity in for close to 0, it is inversible and even if it decreases according to 7.
We write independently or (same for all other functions). Let be the Jacobian of in (which is, therefore, a.e. defined in ).
For all , we take and consider .
Subsequently, we note .
3.2. Derivation Formula
To calculate derivation, we use the following theorem.
Theorem 3.1. Let us consider verifying (19). We suppose thatThen, is derivable in 0, and we haveIf, in addition, is a Lipschitzian domain, thenFor detailed proof, see [11].
3.3. Optimality Conditions of the Problem
Again, verifying (19) , is open bounded Lipschitzian domain, and . If is close to 0, is a solution of the problem defined by its variational formulation (3). As in (11), we are interested in the following functional:
We set(i) with , a vector field with compact support and sufficiently small such that defines a diffeomorphism(ii)
We look for the derivative of with respect to domain in direction , that is, .
As regards derivative of , for a choice of , as mentioned above, we deform only the boundary part that we will denote by .
To calculate the derivative , it is useful to derive in the appropriate direction .
For the functional derivative, we have
This gives, according to Hadamard and the boundary conditions,
Therefore,
We suppose that it is possible to estimate the normal derivative of on ; that is, there exists such that on , where is the exterior normal unit vector defined on . We have the following necessary conditions of optimality.
Proposition 2. If is the solution of the shape optimization problem , there exists a Lagrange multiplier not depending on such that on .
Proof 2. As , using the derivative with respect to domain, in the direction of vector field, we show that there exists Lagrange multiplier such thatwhere . Note that we perturb only on is fixed.This gives us the following relationship according to (27):For more details on the expression in (29), see [14].
Let us take . To estimate , it suffices to recognize and if we suppose that is of class , since , then we havewhere is the outer normal vector. Note that is an optimality condition and if we situate , we will be able to estimate on . Therefore, we deduce an approximation of Lagrange multiplier .
4. Auxiliary Lemmas
In this section, we sum up some fundamental lemmas for the algorithm which we will present in the next sections. These lemmas are based only on maximum principle theory in wave equation in high dimension. We assume also that .
In sequel, we need some hypothesis for the operator and the initial value problem in order to apply maximum principle for hyperbolic problem to obtain additional information about functions which satisfy
If the hypothesis holds, then the solution satisfies ; see [3], page 234.
Lemma 4. Let be two open sets such that and . We consider andwhereThen
Proof 3. Considering , we haveBy maximum principle (see [3]), in .
Then, using again the maximum principle (see [3]), for all . Let , and where is exterior normal on and . We suppose that ; in this case, by the Hopf lemma, we have
Lemma 5. If is satisfied and are two solutions of the free boundary problemsuch that is the Lagrangian multiplier Then .
Proof 4. We haveThanks to maximum principle (see [3]), on and on . Let ; then and where is exterior normal on and . We suppose that ; in this case, by the Hopf lemma, we have
5. Uniqueness and Convergence Result
In this section, using results established in Section 4, we show the uniqueness of the domain under some hypothesis, by following the methods of I. Ly et al. (see [15]). Most of the time, in the inverse problems, it is a great challenge to get uniqueness results. Now we are not able to produce a uniqueness result in the case where the unknown domain is supposed to be star-shaped with respect to a fixed point . We think that it would be interesting to investigate this question. Our uniqueness result is obtained for any domains belonging to , a class of geometrical sets in the Beurling sense of admissible domains satisfying an inequality constraint on the accessible boundary (see [11, 16]). Another important hypothesis for our uniqueness result is the inclusion property in the following sense: one assumes that there are two domains in class and one of the two domains is included in the other. This is an interesting problem to weaken the inclusion property’s hypothesis. The uniqueness result is started in Proposition 3.
Using results in Section 4, we show, under geometrical assumptions, that the domain is unique. We have already assumed that one is able to estimate the normal derivative of , there existswhere is the exterior unit normal vector field defined on .
Let us define a class of geometrical sets in the Beurling sense (see, e.g., [15–17]).
Let be a bounded domain which is uniformly Lipschitz and let be a subset of that is regular. Let us takeand is solution to the following problem:
By construction, is a nonempty set. In fact, this is because of Proposition 1 and measure assumption. We use set in the proof of Proposition 3. At first, we get the following lemma.
Lemma 6. If is satisfied, let ; then .
Proof 5. Let be solution ofWe haveBy the maximum principle (see [3]), we show thatLet , and we getThen,Then we get .
Proposition 3. Let be a bounded domain and let be defined as in the Introduction. Consider the following Cauchy problem:Assume also that there are two domains for which (49) is verified. Then, we have .
Proof 6. Consider such thatAs , we obtain .
Since , we have , and, by Lemma 6, we have .
Consider the following problem:We have (see [3])As satisfies the interior sphere condition, supposing that , by the Hopf lemma, we getwhich is false; then .
6. Stability of the Inverse Problem
In this section, we establish the stability of solution under some hypothesis too. The stability issue for this inverse problem seems important and difficult to be ignored. Recall the following lemma solving the Cauchy problem for the usual wave equation and giving energy and trace estimates of the solution.
Lemma 7. Let be uniformly Lipschitzian included in , and a final time .
is given. Let be the unique weak solution of the wave equationThen there exists a constant (which depends only on and ) such that, for all ,The normal derivative belongs to and verifiesThis result is very classical; estimate (20) can be formally deduced from the multiplication of (19) by and then by part integrations of this equality on . Concerning estimate (56), we refer to [18]. This is a hidden regularity result that can be proved using the method of multipliers.
Proposition 4. Let be a bounded uniformly Lipschitz domain and let be subsets of that is regular such thatAssume that (49) is verified. ThenThis inequality (58) describes the Lipschitz stability of the inverse problem.
Proof. From (56), is bounded; then there exists such that
. Then we have
7. Numerical Simulations
In this part, we solve our inverse problem numerically using polar coordinates. Let be a function as defined in the Introduction; we seek to determine of class under the constraints given. We choose function defined bywith . The integer takes values from 0 to and is a multiple for depending on . We consider that is the normal derivative of a function unique solution of problem 1 in a domain of which a part of the boundary is unknown. We seek to determine its optimal shape of . With Matlab, by varying the parameters , we determine of class which we describe as a trajectory. Thus, we choose arbitrary values for and and a maximum value for time . By varying the angle and , we obtain the geometrical optimal shapes of in the below figures. For each given unit of time, we vary and . Thus, for each fixed time, we obtain an optimal shape when and is equal to the value indicated in the legend.
In Figure 1 where the observation time is 5 seconds for a maximum angle equal to , the length of is equal to 5.43 kilometers.
In Figure 2 where the observation time is 8 seconds for a maximum angle equal to , the length of is equal to 23.79 kilometers.
In Figure 3 where the observation time is 10 seconds for a maximum angle equal to , the length of is equal to 195.15 kilometers.
We notice that if the observation time is small, then the optimal shape of looks like an arc of circle. But if it is larger, the optimal shape of is a part of a hyperbola.
8. Conclusion
In this paper, we prove the existence result of the solution for our inverse problem by determining the optimal shape in Section 2 and we prove the existence of a Lagrange multiplier, which appears in the optimality condition of the problem. By maximum principle for hyperbolic equations, we prove the uniqueness and the Lipschitz stability of the solution for our inverse problem. We make some numerical simulations with Matlab to illustrate the theoretical results and then identify the optimal shape of . It would be interesting to study the problem with nonsmooth boundaries; topological optimization is the technique to use in future researches.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.