Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2472060, 16 pages

https://doi.org/10.1155/2017/2472060

## On Inverse Problems for Characteristic Sources in Helmholtz Equations

^{1}CEMAT-IST, Instituto Superior Técnico, Lisboa, Portugal^{2}Universidade Federal do Rio de Janeiro, Campus Macaé, Macaé, RJ, Brazil^{3}Programa de Engenharia Nuclear, COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil^{4}CEMAT-IST, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal

Correspondence should be addressed to Roberto Mamud; rb.jrfu.eacam@dumamr

Received 8 December 2016; Accepted 24 January 2017; Published 27 February 2017

Academic Editor: Guido Ala

Copyright © 2017 Carlos J. S. Alves et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the inverse problem that consists in the determination of characteristic sources, in the modified and classical Helmholtz equations, based on external boundary measurements. We identify the location of the barycenter establishing a simple formula for symmetric shapes, which also holds for the determination of a single source point. We use this for the reconstruction of the characteristic source, based on the Method of Fundamental Solutions (MFS). The MFS is also applied as a solver for the direct problem, using an equivalent formulation as a jump or transmission problem. As a solver for the inverse problem, we may apply minimization using an equivalent reciprocity functional formulation. Numerical experiments with the barycenter and the boundary reconstructions are presented.

#### 1. Introduction

In this paper we focus on the inverse problem that consists in the determination of characteristic sources. Inverse source problems have been addressed since 1938, in the context of gravimetry, with the seminal work of Novikov [1], where he was able to prove uniqueness in the two-dimensional Laplace equation model, for star-shaped characteristic sources. For more general geometries, it has been shown by Sakai [2] that there are connected domains for which the problem of reconstruction from the potential outside the source region presents no uniqueness.

Here we consider the inverse problem for characteristic sources, but in the case of Helmholtz equations. When the center is known, uniqueness was proven for a convex characteristic source (see Isakov [3]) with no boundary restrictions. However the proof of Novikov is no longer available by the lack of minimum-maximum principles, and uniqueness has not been established for star-shaped characteristic sources.

While considering Helmholtz equations in a bounded domain this will change the problem in two different ways. If we consider the modified Helmholtz equation, useful for the transient heat problem, the associated boundary value problem has a unique solution, but the decay is exponential. This compromises the quality of the recovery in the inverse problem. On the other hand, considering the classical Helmholtz equation, the models wave phenomena such as acoustic waves, we must avoid values associated with the resonance frequencies, in which the direct problem has no unique solution.

In this paper we focus on two main issues related with the inverse characteristic source problem, that is, the determination of the barycenter of the characteristic source and the recovery of its geometry from a class of star-shaped characteristic sources. As a solver for the inverse problem, we may apply a minimization algorithm using an equivalent reciprocity functional formulation.

In general, there are restrictions on the number and type of sources that can be identified from boundary data. Several particular classes of problems have been considered: combinations of point sources—see [4–6]; linear/affine classes as in [7, 8]; classes of characteristic sources (e.g., [9–16]); and in particular for the Helmholtz equation we refer to the papers [17, 18], where a full identification result was established, but using instead an interval of frequencies.

In this paper, we only consider boundary measurements given by a single frequency.

The paper is organized as follows. In Section 2, we present the direct and inverse problems for Helmholtz equations. In Section 2.1 the inverse jump problem is also presented and we show the equivalence between this problem and the inverse characteristic source problem, which motivates the adoption of the Method of Fundamental Solutions to solve the direct problem (see Section 2.2). The MFS has been applied to solve several inverse problems in recent years (see [19] for an account of this research).

In Section 3 we obtain a new formula for the determination of the barycenter for characteristic sources with symmetric star-shaped support and show that it is the same as obtained for the point source inverse problem. In Section 3.3 the boundary reconstruction is presented as a minimization problem, where the classical Levenberg-Marquardt method is used but the minimization is not considered in terms of the boundary data directly, but in terms of the reciprocity functional.

Numerical experiments related to the barycenter and boundary reconstruction are presented in Section 4. In simulations, we solve the direct problems for different numerical parameters and generate the synthetic experimental data, avoiding the inverse crimes. Beside this, a barycenter formula presented in (46) is tested in Section 4.1. Section 4.2 is entirely dedicated to numerical experiments related to the boundary reconstruction, for the modified and classical Helmholtz equations.

#### 2. The Helmholtz Inverse Source Problem

Let be an open, simply connected and bounded set with boundary, . Given the source term and the Dirichlet data , consider the following problem:where we are considering the cases (Laplace equation), (modified Helmholtz equation), and the usual Helmholtz equation , with denoting the wave number.

Problem (1) has unique solution, except for some (eigenvalues of the Dirichlet-Laplace operator for ) [20], and we can define the* direct problem* as the problem of finding , with , from the source term and Dirichlet data , where is the normal outward unity vector. On the other hand, the* inverse source problem* consists in, given the Cauchy data , finding the source term .

For test functions , we introduce the reciprocity functionaland using Green’s second identity, we getfor all . Therefore,with .

In next theorem we establish an equivalence between the source reconstruction from the Cauchy data and from the reciprocity functional.

Theorem 1. *The Cauchy data uniquely determines the source if, and only if, is uniquely determined by , for all .*

*Proof. *Consider the sources and for problem (1). Then Imposing the same Dirichlet data on the boundary, for all . Since is homeomorphic to (e.g., [17]), we have Therefore, , in , is equivalent to , in .

##### 2.1. Equivalence to an Inverse Jump Problem

We now show that the inverse source problem for characteristic source is equivalent to an inverse jump problem.

Consider problem (1) with source , where is constant and is an admissible source set. Note that we can rewrite this problem as the transmission problemwhere denotes the jump between the inner and outer traces of the solution, respectively. Let be a particular solution of the equation Notice that if , we can take and if , we can take . So, we can split (8) into two problems andThus, consideringwe have that, up to , problem (8) is equivalent to the inverse jump problemThus, denoting the reciprocity functional of this problem as , we establish the following result.

Theorem 2. *If is connected, then . Therefore, the inverse source problem of characteristic source (1) is equivalent to inverse jump problem (13).*

*Proof. *The reciprocity functional for the inverse jump problem, with , is given by, applying the Green formula to the connected set , Notice that the orientation of the normal changes, by taking as the boundary of , instead of as a part of the boundary of . Also, we haveThen, for all , using (2).

*Remark 3. *If was not connected, this equivalence would not hold, in view of

This jump or transmission problem will be solved using the Method of Fundamental Solutions (MFS) that we present next.

##### 2.2. Solving the Direct Problem with MFS

The Method of Fundamental Solutions (MFS) is a meshless boundary method that has been widely studied in recent years. Its main goal is to approximate the solution of certain boundary value problems by a combination of fundamental solutions of the partial differential equation, in which the singularities lie outside the domain (e.g., [21]).

Furthermore, this method has become a popular tool because of its applicability in several areas where we can cite, as an example, [7, 8, 19, 22, 23], including inverse problems.

Consider the direct problemwhere the source term is , with constant and an open, connected, bounded subset with boundary , .

Using Section 2.1, problem (17) can be rewritten as the transmission problem for (13), whereand is given by (12). In this way, the functions and will be determined by the MFS. For this, consider the following expansions centered at the corresponding source points: with , and with and , where is a fundamental solution of Laplace or Helmholtz equation; that is, where is the Dirac distribution. Beside this, fictitious boundaries , and are chosen such that (i)the curve is outer of , with ;(ii)the curve is outer of , with ;(iii)the curve is outer of , with .

So, is a solution of (13) if it satisfies the following boundary conditions.(i), on ; that is, for , (ii), on ; that is, (iii), on ; that is,In this way, taking collocation points , with , and , with , we can write the above problem as a linear system: where and the matrix M, , of the linear system coefficients, is given by

In general, it is considered that and the system should be considered in the least squares sense. In the 2D case, the fundamental solution can be taken aswhere is a Hänkel function and are Bessel functions.

Therefore, solving the linear system, in the least squares sense, by Tikhonov regularization ( being a regularization parameter), we find a solution, , of (13), in which we may generate an approximation of the direct solution This approximation will be used to generate the synthetic data in Section 3.3.

*Remark 4. *In the numerical experiments of Section 4, the fictitious boundaries , , and were chosen as circles each one in the correspondent region. The choice of the location of source points is not unique and its best location is a research subject (see, e.g., [24–27]).

#### 3. Barycenter and Point Source Determination

We consider the special class of admissible sets.

*Definition 5. *A subset is said to be star-shaped with respect to the origin, if there exists a function , such that where is the positive radius function.

We say that a subset is a symmetric star-shaped set if it is star-shaped and the function satisfies When the barycenter is not at the origin, we can use the notation to refer to a shifted domain.

In this section we establish a result for the determination of the barycenter, in the case of symmetric star-shaped characteristic sources for the Helmholtz equation, with no restriction on the sign of , and we bound the reciprocity functional.

In Section 3.2 we see that the same formula also holds for the determination of a point source.

##### 3.1. Determination of the Barycenter

In the case (Laplace equation), since and are functions of , it is well known that the barycenter is determined from where and (for any weight function ).

Consider now , and (therefore ).

Using a test function , where and are arbitrary, the reciprocity functional is On the other hand, by (3), we have, denoting ,

In the following, we will suppose that the origin of the coordinate system, , is the barycenter of an admissible source set , in which we suppose symmetric star-shaped set, whose boundary, , is parametrized by a function .

Therefore, by (35), for all .

Proposition 6. *Let be a symmetrical star-shaped subset, whose boundary, , is parametrized by the function . Then *

*Proof. *In fact, let be an arbitrary direction. Consider the sets So, and thenOn the other hand, note thatTherefore, since , , and the star-shaped set is symmetrical, then and we obtain In this way, since is arbitrary, we have , for all .

So, by proposition 6 and by (35), Therefore, that is,

*Remark 7. *Note that in this case we have used the boundary parametrization associated with the symmetric star-shaped characteristic sources in the proof.

Thereby, we establish the following new result for the barycenter identification of symmetrical star-shaped supports in the inverse source problem posed on the modified Helmholtz operator.

Theorem 8. *Let be a symmetrical star-shaped subset and let be the source term for the inverse problem. Then the barycenter, , of is uniquely determined bywhere .*

##### 3.2. Determination of Point Sources

In this subsection we will study the inverse source problem for a single point source. Consider the inverse problem with a single point source of type where is the Dirac delta distribution centered at the point and is the intensity of this source. In this way, taking in (4), with , Thus, since , the source intensity is determined in the following way:On the other hand, the source position is obtained fromwhere . The coordinates of the point source are simply given by , with being the canonical basis vector.

*Remark 9. *Note that, by (50) and (46), we obtain the same expression for the barycenter of symmetric star-shaped sources and for the position of a single point source.

##### 3.3. Determination of the Source Support Boundary from Cauchy Data

Consider the subsets , with same barycenter, whose boundaries, and , are parametrized, respectively, by and by a truncated Fourier serieswith . Here, we suppose that the barycenter of is the origin.

*Remark 10. *In an arbitrary coordinate system, where the barycenter of is not the origin, the boundary points are given bywith . The barycenter position will be calculated using the reconstruction formula (46); see Section 3.1.

*Remark 11. *In , is parametrized, in the barycenter coordinate system, by truncated -dimensional spherical harmonics series of the radius , which express the distance between the centroid and the correspondent source boundary point.

Using results from Section 3.1, we can assume that the sources have the same barycenter. Theorem 2 shows us how to implement the boundary reconstruction algorithm using solutions of direct problem (1) with or equivalently solving interface problem (13) with traces at the source boundary of equivalent particular solution.

*Remark 12. *The normal trace of trial solutions with parameters guess is calculated solving equivalent interface problem (13) with the corresponding trials. As we have pointed out before, the Method of Fundamental Solution is appropriated for the Helmholtz equation model.

Related to these subsets and , we have the associated reciprocity functionals and , for .

Considering, in , test functions and taking different directions , , we define the following discrete functional to be minimized:

Discrete functional (53) represents a possible misfit between the reciprocity functional related to the sources and . Furthermore, note that(i) the given depends only on Cauchy data;(ii) the guessed can be calculated by solving Laplace-Helmholtz direct problem (17) with a trial source term guessed for parameters .We present a numerical optimization method to approximate the boundary support which is based on a Levenberg-Marquardt algorithm implementation for the nonlinear least squares formulation given by (53).

*Remark 13. *In problems, we can define the functional to be minimized asvalid for test functions , with

In the next sections we present some bidimensional numerical experiments for the minimization of functional (53), defined above, using the Levenberg-Marquardt algorithm.

#### 4. Numerical Experiments

When we study inverse source problems with characteristic sources defined by star-shaped sets, there are two pieces of information that we aim to retrieve: the barycenter of the source support and the boundary parametrization.

##### 4.1. Numerical Experiments for Barycenter Determination

The barycenter reconstruction problem was studied in Section 3.1, and the main objective is to verify the accuracy of the new formula proposed in Theorem 8.

In the numerical Examples 1, 2; and 3 we considered . In Example 4, was considered.

Beside this, relative noise over the collocation points of domain was imposed; that is, the boundary was discretized and the reciprocity functional was determined with the Dirichlet data, in each element of this point list. This value was multiplied by , where was given by a random function with uniform probability distribution.

*Example 1. *In this bidimensional experiment we consider circular geometries, with . In this case and , using the notation , where stands for the radius and for the center.

The direct problem was solved using the Method of Fundamental Solutions (MFS), where the projection of a solution on the boundary was taken as Dirichlet data, considering collocation points and point sources in the MFS. For the determination of the barycenter, the reciprocity functional was calculated considering the Cauchy data from the solution of the direct problem, . The Dirichlet and Neumann data were given by the trace and the normal trace , on

Thus, the influence of relative noise of , and in the Cauchy data was studied, and the results are shown in Table 1. In this case, it was considered that Dirichlet and Neumann data have the same noise level.

In Table 2, noise in the Dirichlet data was fixed, and the noise in the Neumann data ranges up to . On the other hand, in Table 3, noise in the Dirichlet data was fixed and the noise in the Neumann data ranges up to . The results were worse than when noise was considered in the Dirichlet data.