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Mathematical Problems in Engineering
Volume 2012, Article ID 808913, 28 pages
Research Article

Integral and Variational Formulations for the Helmholtz Equation Inverse Source Problem

Nuclear/Coppe/Universidade Federal do Rio de Janeiro, 21945-970 Rio de Janeiro, RJ, Brazil

Received 27 July 2011; Revised 30 November 2011; Accepted 8 December 2011

Academic Editor: Cristian Toma

Copyright © 2012 Marcelo L. Rainha and Nilson C. Roberty. 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.


The purpose of this paper is to explore the Hilbert space functional structure of the Helmholtz equation inverse source problem. An integral equation for the sources reconstruction based on the composition of the trace and Green's function operators is introduced and compared with the reciprocity source reconstruction methodologies. An equivalence theorem comparing the integral inverse source equation with the variational weak reciprocity gap functional equation is then demonstrated. Some examples on applications to the unitary disk are presented.

1. Introduction

The inverse source problem for the Helmholtz Dirichlet equation is a basic tool for the investigation of transient source problems [14]. In order to investigate this class of problems, let be a bounded domain with smooth boundary . Let be a complex number, , and . The direct problem with the Helmholtz operator: to find a regular field that satisfy the system is well posed and has a unique solution when is not an eigenvalue of the Laplacian. In this paper we consider the trace operator and simplify the notation by calling the boundary data of problem (1.1) . The trace theorem [5] assures the existence of a function which is the normal trace, that is,

When is a real positive or an imaginary number we have, respectively, the proper or the modified Helmholtz equation. When , we obtain the Laplace equation. For the complete setting of complex values, we consider the problem as the Helmholtz equation direct problem.

The inverse source problem consist, in knowing the Cauchy data in the boundary , that is, the Dirichlet to Neumann map in at least one Dirichlet datum , to recover the source . It may be formally posed as follows:

The two problems, direct and inverse, can also be formulated with only one system of equations: to find such that

Since in the inverse problem the Cauchy data are known, we may associate these data with a fourth-order Dirichlet problem with the Bilaplacian operator where is an arbitrary given function. This problem is well posed and has a unique solution when is not an eigenvalue of the Bilaplacian. This motivates the following naive existence result

Remark 1.1. Suppose that a Cauchy data pair is more regular than normal case and is not an eigenvalue of the Bilaplacian, then there exists a solution to inverse source problem (1.3).

Proof. Since data is regular, we may consider the fourth-order direct Dirichlet problem (1.5) with the given Cauchy data and some . The inverse source solution will be .

To obtain the existence of global solution for problems (1.1), and (1.5), please see [6]. For more information about linear integral equations, see [7]. For the inverse source problems, see [8]. For functional analysis, please see [9].

In Section 2 we develop a Hilbert space functional framework to the problem based on special decomposition. The analysis of the Dirichlet to Newman map and of the Source to Neumann maps is done in Sections 2.1 and 2.2, respectively. It is based only on the analysis of the direct problem structure. The analysis of the Adjoint Source to Neumann map is done in Section 2.3. In Section 3 we use the Green function operator to put together the results found in Section 2. There in Section 3.1 we present an integral equation for the inverse problem based on the relative Dirichlet to Newman map. The reciprocity gap functional is introduced in Section 4, where an equivalence theorem between this and the integral formulation is proved. In Section 5 some particular results for the unitary disk in are presented. Finally, we conclude the paper in Section 6.

2. The Dirichlet and the Source to Neumann Map

For , the space is the Sobolev class of the functions of the spatial variable . For more information, see [5]. Let us consider for future use the following sets of eigenvalues:

Definition 2.1. One says that a function is metaharmonic when it is in the set where and .

2.1. The Dirichlet to Neumann Map

Definition 2.2. Consider problem (1.1) with zero source, that is, and . This problem has a solution . One defines the Dirichlet to Neumann map for the Helmholtz equation as the operator By the trace theorem [5], it is well defined, linear, and continuous.

Remark 2.3. We alternatively can define the Dirichlet to Neumann map (2.2) for the problem (1.1) as an operator with a nonzero source problem defined by the graph .

Definition 2.4. One has where is a Dirichlet to Neumann map.

Theorem 2.5. Let . A function is a solution to the inverse source problem (1.3) if and only if

Proof. Suppose . Consider the following fourth-order problem: Let us use this problem to define a source , where is solution of problem (2.5). Note that . Let be a solution of the inverse source problem (1.4). Then , where is the solution of the homogeneous source problem in the definition of the Dirichlet to Neumann map (2.2) and . The sufficiency is proved.
To proof necessity, suppose that there exists a which is solution of the inverse problem (1.4).
Consider the second-order problem with homogeneous boundary Dirichlet data with a unique solution . By the trace theorem, we have . Note that , where . So, .

Remark 2.6. We have proved the existence and uniqueness of solution to (1.4) in . However, this does not mean that when we do the search in a larger space , we will continue having uniqueness. In fact, we will prove in the next proposition that and consequently where and . This leads us to conclude that the solution of (1.4) is actually a class of functions, where we can “reconstruct” or “observe” only part of the solution in .

Proposition 2.7. If and , then Proof. Consequence of Lemmas 2.8 and 2.9.From now on, we are supposing always that and .

Lemma 2.8. One has .

Proof. For an arbitrarily given , the problem (1.1) with zero Dirichlet datum is well posed and has an with normal derivative trace . Also, since , the fourth-order problem (1.5) with Cauchy datum and zero source has solution and is well posed in . These two solutions may be used to define a function and since by problem (1.1) we obtain that arbitrary function is the sum of a function in and a function in . If we show that we prove that . For this, take a in the intersection . Then which means that is a solution of completely homogeneous fourth-order problem (1.5), with no source and zero Cauchy datum. Since , the unique solution is trivial , and . We have proved that .

Lemma 2.9. One has .

Proof. Using the second Green theorem with and for some , we obtain and since the scalar product with arbitrary is zero, the inclusion follows.
To prove the other inclusion, let , suppose is orthogonal to for all , and take a function such that is solution of (1.1) with source and Dirichlet datum .
By using the second Green formulas for , we have Note that, since the test functions are dense on , we may consider a variational formulation using the dual system . Because , by trace theorem and So, if an arbitrary function is orthogonal to all functions in , then it is in and the reverse inclusion follows.

2.2. The Source to Neumann Map

Let us consider the simultaneous solution of the direct and inverse source problem (1.4) and search for a solution . It follows from Theorem 2.5 that if we restrict the source search to the subspace , we will find a unique solution.

Definition 2.10. Consider problem (1.1) with zero Dirichlet data, that is, and and solution . One defines the Source to Neumann map for the Helmholtz equation as the operator By the trace theorem [5], it is well defined, linear, and continuous.

Theorem 2.11. ,  is an isomorphism.

Remark 2.12. If we consider problem (1.1) with Dirichlet data and and solution , we can define the Source to Neumann map for the Helmholtz equation as the operator
Note that this more general situation will produce results such as As consequence,(i)a functional such as a Source-Dirichlet to Neumann map may be defined (ii)and restrieted to be a functional such as a Dirichlet to Neumann map (iii)or to be a functional such as a Source to Neumann map It is important to note that in this more general definition it is not possible to prove that is an isomorphism, since the fact that the trace is used in the proof.

Lemma 2.13. is a Hilbert space with norm.

Proof. Let us consider the canonical projection Note that is continuous, is closed. Since for all , it follows that is closed. Consequently is closed subspace of the Banach space . So, it is Banach and, consequently, is a Hilbert space with the scalar product induced by .

Proof of Theorem 2.11. (i) is continuous.
Note that is a composition of the normal trace and the canonical embedding . The normal trace is continuous by trace theorem. The canonical embedding is also continuous since is closed by Lemma 2.13. So, is continuous.
(ii) is one to one.
Take some arbitrary . Then is the normal derivative of the problem (1.1) with . By hypotheses, and, consequently, . The fourth-order problem is well posed and has a unique . Then . Since is arbitrary, and the injectivity is proved.
(iii) is onto.
Consider an arbitrary and , where does not necessarily satisfy the compatibility condition. Let be a solution of the well-posed Neumann data problem, Note that with we obtain . So, is surjective.
It remains to prove the following.
(iv) is continuous.
In fact, this is a consequence of the Banach open map theorem, since. is a linear continuous bijective application between Banach spaces.

2.3. The Adjoint Source to Neumann Operator

Definition 2.14. Consider again the problem (1.1) with zero Dirichlet data, that is,  and and  solution . One defines the extension of the Source to Neumann map for the Helmholtz equation as the operator

Remark 2.15. By the trace theorem [5], it is well defined, linear, and continuous. As an extension of , surjectivity is preserved. Also .

Corollary 2.16. The quotient of by is a copy of .

Proof. Consider the following chain of embeddings: 808913.fig.002(2.28) where is a canonic embedding and is an isomorphism by the Banach isomorphism theorem. Since also is an isomorphism by Theorem 2.11, the corollary is proved, that is, .

Corollary 2.17. is a closed subspace of .

Remark 2.18. Since is bounded, then its adjoint operator is well defined and continuous and one to one.

Corollary 2.19.   is an isomorphism.

Proof. (i) is well defined.
We know that where and denote the operator range and kernel, respectively. It is well known that if is a Banach space and , that is, a subspace of its dual , then is the weak star closure of in . Applying this result to our case, we have in . Particularly, assures that is well defined.
(ii) is one to one.
Note that, is onto and is a closed subspace of ,
(iii) is onto.
Note that since, is closed in , Since is linear, continuous, and bijective from to , by the open mapping Banach theorem, is an isomorphism.

Proposition 2.20. If , then .

Proof. Suppose that . Then and, consequently, .

Remark 2.21. If we substitute by in problem (1.1) and use the same argument already used in the precedent proofs, then(i),(ii),(iii) is a closed subspace of ,(iv),(v)if , .

3. Integral Representation

Definition 3.1. The Dirichlet Green function for the problem (1.1) is its solution with source , , and homogeneous Dirichlet data, that is, for on .

Definition 3.2. Let be the Green function for problem (1.1) with homogeneous Dirichlet boundary conditions. Then, the solution

Remark 3.3. By using problem (1.1) linearity, we formally decompose the solution in two additive parts where is the homogeneous Dirichlet source auxiliary problem solution and is the zero source auxiliary Dirichlet problem solution. For simplicity, for a fixed or , we will denote By taking the normal trace of the solution (3.1), we obtain which is an explicit representation to the Dirichlet to Newman map with arbitrary and .
By using the same notation adopted for the additive decomposition of the solution map, fixed or , we will denote

Remark 3.4. Note that
With this decomposition, we obtain the following explicit representation to operators in Section 2:(1) with ;(2) is an explicit representation to the Dirichlet to Neumann map;(3) is an explicit representation to the Source to Neumann map.

3.1. The Inverse Source Integral Equation

Lemma 3.5. Let , , be two solutions of problem (1.1) with the same source and different Dirichlet data , , respectively. Then(i), that is, the relative Dirichlet to Newman operator is constant operator whose functional value is independent of the Dirichlet datum and depends only on the source function ;(ii) for all solutions of (1.1) with arbitrary Dirichlet data but the same source, that is, the integral is the function given by the relative Dirichlet to Newman map.

Proof. The equality in both (i) and (ii) is a trivial consequence of (3.4). Note that in this case the unique information available for source reconstruction is given by only one measurement, say that Neumann boundary measurement corresponding to some specific Dirichlet datum , which without loss of generality can be assumed zero. Note also that problem (1.1) with Dirichlet datum and source has solution . The normal trace of this regular solution is in . So we have proved that the range of is in . The domain of is since this is the set of nonzero Dirichlet data that gives the same function in the range.

Definition 3.6 (strong integral equation problem). Since in the inverse source problem the exact Cauchy data pair is given, the relative Dirichlet to Newman map value for the source problem (1.1) is known and Lemma 3.5 suggests the following integral equation formulation for the source reconstruction problem: to find such that where , for .

Remark 3.7. Note that we introduce here as a simplified notation to the extended Source to Neumann map . This notation is more usual.

The following corollary resumes all that has been discussed.

Corollary 3.8. Supposed that and . Then, (i)for a Cauchy datum , there exists a unique function solution of the inverse source problem (1.3) for the Helmholtz equation (1.1),(ii)the associated mapping defines a linear homeomorphism between these spaces, (iii)and is a right inverse of the mapping defined by the strong inverse source equation (3.12),(iv)the projection is well defined and constant in the level set (v)if the source is known to be in the class , then a single boundary measurement is sufficient to identify ,(vi)for a Cauchy datum , there are many functions , where is an observed consequence of (ii) and is an arbitrary nonobserved function.

Proof. The items are trivial consequences of the results already proved.

Remark 3.9. Given , the unique solution referred to in Section 3.1 is the unique solution of the fourth-order direct problem [1]:

Remark 3.10. The adopted Hilbert space framework for solution of the problem may be understood as an a priori information about the criteria for selecting the observable and the nonobservable part of the source. Other Sobolev spaces that induced partitions of the pivot space (in this work ) will modify this observability relation.

Remark 3.11 (relation between star-shaped and metaharmonic functions). Let us define the set The . Note that is dense in . If, for all , there exists a family of metaharmonic functions in that approach , then is dense in .

Remark 3.12. The most important classes of sources that may be reconstructed uniquely from boundary data occur when is metaharmonic or when , where is the characteristic function of an open star-shaped set with being boundary and a function. We will discuss these classes when establishing uniqueness.

Remark 3.13 (the adjoint integral equation). This equation may be used for the source reconstruction independent of solution of the direct problem (1.1). By substituting the explicit integral definition of in the duality definition of adjoint we obtain that is explicitly given by

Remark 3.14. From these formulas, we can deduce that, for a fixed , the operator for any , we will call this operator Extended Dirichlet to Neumann map.

Remark 3.15. Once we know the integral formulation to , we can determine the integral formulation to . In fact, and from it follows that

Remark 3.16. Consider the following direct problem with . This problem has a unique solution . Let where is the associated Green function. This happens for each since . From this we deduce that the integral inherits all good properties from .

4. Integral and Variational Solutions: The Equivalence Theorem

4.1. Integral Formulation

With the integral formulas (3.8), (3.2), Definition (2.2), and supposing that the compatibility condition has been verified, we obtain that is the integral equation

4.2. Variational Formulation

Definition 4.1 (reciprocity gap functional problem). One may use the second Green theorem valid for all with a solution of problem (1.1) to formulate the reciprocity gap functional inverse problem: to find for all .
Note that the Lax-Milgram theorem assures the existence of a solution in this case.

4.3. The Equivalence Theorem

Theorem 4.2. Let one consider the two inverse source problems related with problem (1.1) with relative Dirichlet to Newman map : (i)integral equation problem given by (3.12);(ii)reciprocity gap functional problem given by (4.4). Then . Suppose additionally that the relative Dirichlet to Newman data . Then .

Proof. Let us consider the inverse source problem: to find such that where , with compatibility condition .
(i) (ii).
We start the demonstration by supposing that (i) is true; that is, there exists such that where is the Green function associated with the Helmholtz operator in .
Let be extended to the boundary of . We then have the following integral representation for : By taking the normal trace
We now multiply (4.2) by and integrate on to obtain
Now applying the Fubini theorem, we obtain which implies Since is arbitrary, we obtain the weak formulation. Note that this is almost expected, since the integral formulation is stronger than the variational, and, as usual, strong weak.
(ii) (i).
Let us now suppose that for all test functions we have the weak reciprocity integral equation (4.4) Consider the substitution of the Green function integral representation whose normal derivative is in (4.4) and apply Fubini’s theorem to obtain
Note that since(i) is the Dirichlet to Neumann map,(ii)and we obtain for all . By property of the integral, we obtain (3.12).

5. Examples on the Unitary Disk

5.1. The Green Function for the Helmholtz Equation Dirichlet Problem in Circular Domains

In this section we will consider the Green function determination when the domain in problem (1.1) is circular with respect to the polar coordinate system.

A Green function to problem (1.1) is a solution of where is the localization of the delta Dirac source. We may use the linearity of the problem for decomposing the solution in two additive parts Here is the fundamental solution for the free space Helmholtz equation and is a homogeneous source regular solution of the Helmholtz equation and . In polar coordinates with has at singular behavior solution which is a Bessel function of second kind.

Remark 5.1. When is small, this solution has a singularity that has the same behavior of the logarithmic function, that is,

Remark 5.2 (addition theorem). Let and . Then
The nonhomogeneous Dirichlet boundary condition in (5.4) is The regular solution of (5.4) is where the coefficients are determined by the Dirichlet boundary condition: Since the solutions are linearly independent, we obtain and when is not a root of the Bessel function , by noting that by addition theorem
By substituting (5.6) and (5.15) and using the addition theorem (5.9) in (5.4), we finally obtain the Green function for the circular domain Helmholtz equation problem (5.1) if we define and , which can also be rewritten as

5.2. The Integral Equation Kernel for the Unitary Disk

The kernel of the integral equation is obtain by taking the normal derivative trace of the Green function (5.18). By using the Wronskian identity for the Bessel functions we obtain which is the modified Poisson kernel for Dirichlet Helmholtz equation problem in the disk. Note that, when , it tends to the classical Poisson kernel for the disk.

Note that, when is substitute in (1.1), it becomes a modified Helmholtz equation for which extensions of the Maximum Modulo Principle and the Strong Maximum Principle for metaharmonics functions are applicable. In this case the Poisson Kernel may be rewrit with modified Bessel function of first kind

Remark 5.3. Substituting the kernel (5.23) in (3.12), we obtain the strong inverse source equation (3.12) for unitary disk

Remark 5.4. Another important class of sources to be reconstructed is the characteristic source with star-shaped boundary with parametrization given by , with . The strong inverse star-shaped characteristic source equation (3.12) for unitary disk is

Remark 5.5. Note that the kernel with finite sum converges with respect to pointwise to . Since it is bounded by the function integrable , in such a way that the limit and the integral can be transposed.

Remark 5.6. The transposed equation