Two-Step Extended Sampling Method for the Inverse Acoustic Source Problem

Recently, a newmethod, called the extended samplingmethod (ESM), was proposed for the inverse scattering problems. Similar to the classical linear sampling method (LSM), the ESM is simple to implement and fast. Compared to the LSM which uses fullaperture scattering data, the ESM only uses the scattering data of one incident wave. In this paper, we generalize the ESM for the inverse acoustic source problems. We show that the indicator function of ESM, which is defined using the approximated solutions of some linear ill-posed integral equations, is small when the support of the source is contained in the sampling disc and is large when the source is outside. *is behavior is similar to the ESM for the inverse scattering problem. Numerical examples are presented to show the effectiveness of the method.


Introduction
Recently, a new method, called the extended sampling method (ESM), was proposed for the inverse scattering problems using a single incident wave at a fixed frequency [1]. ESM is based on an idea similar to that of the linear sampling method, which has been studied extensively in the literature [2,3]. Although the ESM can only reconstruct the location and approximate the size of the target, it uses much less data than the classical linear sampling method, which needs full-aperture data (far-field pattern of all incident directions and all observation directions).
is property makes the ESM suitable for the inverse source problems.
Inverse source problems (ISP) have many applications such as pollution source detection, optical tomography, and sound source localization [4][5][6][7][8]. It is known that the inverse source problem using a single frequency far-field data does not have a unique solution [6,9]. In this paper, we consider the inverse problem to determine the location and size of an acoustic source at a fixed frequency. e ESM is generalized for the inverse acoustic source problems. We show that the indicator function of the proposed method, which is defined using the approximated solutions of some linear ill-posed integral equations, is small when the support of the source is contained in the sampling disc and is large when the source is outside. e behavior of the indicator function is similar to the ESM for the inverse scattering problem. Numerical examples are presented to validate the effectiveness of the method.
e ESM was first proposed for the inverse scattering problem to reconstruct the obstacles [1]. To treat the inverse source problem, we introduced an artificial intermediate scattering problem, whose far-field pattern is the same as the original source problem. is enables us to generalize the ESM for the inverse source problems with little change in the numerical algorithm. e rest of this paper is arranged as follows. In Section 2, we introduce the acoustic source problems and give a brief presentation of the ESM for the inverse scattering problems. Section 3 contains the generalization of the ESM for the inverse source problem and its mathematical justification. In particular, we show that the indicator function of the ESM is small when the support of the source is contained in the sampling disc and is large when the source is outside. Section 4 contains several validating numerical examples in two dimensions. We make some conclusions in Section 5.

Preliminaries
In this section, we describe the acoustic source problem and the associated inverse problem of interests. en, we briefly introduce the ESM for the inverse acoustic scattering problem.
2.1. Acoustic Source Problem. Let D ⊂ R 2 be a bounded domain and k > 0 be the wave number. Let f(x) ∈ L 2 (D) represent the source with compact support D. e acoustic source problem is to find a function u ∈ H 1 loc (R 2 ) satisfying the Helmholtz equation and the Sommerfeld radiation condition e above problem has a solution u given by where is the fundamental solution of the Helmholtz equation. Here, H (1) 0 is the Hankel function of the first kind and order 0 given by where the real part J 0 is the Bessel function of order 0 and the imaginary part Y 0 is the Neumann function of order 0 (see page 73 of [3]). Since the solution u to (1) is radiating, it has the following asymptotic behavior (see, e.g., eorem 2.6 in [3]): where x � x/|x| ∈ S ≔ x∈ R 2 | |x| � 1 and u ∞ (x) is called the far-field pattern of u. From the asymptotic behavior of the Hankel function H (1) 0 and (3), we have that e inverse source problem (ISP) considered in this paper is to find the location and size of D from the far-field pattern u ∞ (x).

Extended Sampling Method.
e ESM was first proposed in [1] to obtain the location and size of an obstacle using the scattering data of one incident plane wave. We first give a quick introduction for the scattering problem of a soundsoft obstacle. Note that the method works for other types of obstacles.
We still use D ⊂ R 2 to denote a bounded domain occupied by the sound-soft obstacle. Let u i (x) ≔ e ikx·d , x, d ∈ R 2 be the incident plane wave, where |d| � 1 is the direction. e direct exterior scattering problem is to find the scattered field u s such that Since the scattered field u s is radiating, it has an asymptotic expansion uniformly in all directions x � x/|x|. e function u ∞ (x) is the far-field pattern of u s due to the incident field u i . When the obstacle is a disc centered at the origin with radius R, denoted by B, the far-field pattern can be written down exactly (see, e.g., Chp. 3 of [3]): where J n is the Bessel function, H (1) n is the Hankel function of the first kind of order n, and θ � ∠(x, d) is the angle between x and d. If the disc is shifted to z, i.e., the far-field pattern is simply e inverse scattering problem is to find the location and size of D from the far-field pattern u ∞ (x) due to a single incident plane wave. For a point z ∈ R 2 , the ESM defines a far-field operator F z : L 2 (S) ⟶ L 2 (S) using the far-field pattern of the disc B z due to the incident plane waves of all directions d ∈ S: where s(·) indicates the surface integral. Let u s (x) and u ∞ (x) be the scattered field and far-field pattern of the scatterer D due to one incident wave, respectively. Using the far-field operator F z , we set up a farfield equation Suppose that Ω is a domain having D inside. One can generate a set of sampling points S for Ω. For z ∈ S, the following result is proved in [1]. Let B z be a sound-soft disc centered at z with radius R and assume that k 2 is not a Dirichlet eigenvalue for B z . For the far-field equation (14), if D ⊂ B z , for a given ε > 0, there exists a function g ε and the Herglotz wave function Remark 1. e value k 2 is called a Dirichlet eigenvalue for a bounded domain B if there exists at least one nontrivial solution u to the following problem: can be computed easily. e far-field pattern u ∞ (x) is the measured data due to an incident plane wave. For each sampling point z ∈ S, (14) can be solved by some regularization scheme, say Tikhonov regularization. From the above discussion, the approximate solution ‖g α z ‖ L 2 (S) should be relatively large when D is not inside B z and relatively small when D is inside B z .
en, an approximation of the location and support of D can be obtained by plotting ‖g α z ‖ L 2 (S) for all sampling points z ∈ S. (14) is similar to that of the linear sampling method by Colton and Kirsch [2]. e far-field equation of the linear sampling method is

Remark 2. Equation
where u ∞ (x; d), x, d ∈ S is the full-aperture far-field pattern of the obstacle D due to the incident plane wave . For a sampling point z ∈ Ω which contains D, using some appropriate regularization scheme, one obtains an approximate solution g α z . In general, the norm ‖g α z ‖ is smaller for z ∈ D and larger for z ∉ D. Since u ∞ (x; d) is the kernel in (19), the linear sampling method cannot directly process the far-field pattern due to a single incident plane wave.

Two-Step ESM for the ISP
Now we generalize the ESM for the inverse source problems. Let u ∞ (x) be the far-field pattern for the inverse source problem defined in (7). Since the solution of (1) is radiating, we can use the same methodology as for the inverse scattering problems. e far-field equation for the inverse source problem keeps the same as (9), i.e., e only difference is that, in (20), the far-field pattern is for the acoustic source problem defined in Section 2.1. e following theorem holds for the inverse source problem. Theorem 1. Let B z be a sound-soft disc centered at z with radius R. Assume that the radius R is large enough such that D ⊂ B z for some z ∈ R 2 . Furthermore, assume that k 2 is not a Dirichlet eigenvalue for B z . Let u ∞ (x) be the far-field pattern for the acoustic source problem defined in (7). en, the following properties hold for the far-field equation (14): and the Herglotz wave function converges to the solution w ∈ H 1 (B z ) of the Helmholtz equation with w � u on zB z as ε ⟶ 0.
Proof. It is easy to see that, under the conditions of Cases 1 and 2, one can find a domain D 0 such that D ⊂ D 0 . Define Since D ⊂ B z , there exist a positive number σ small enough such that 0 < d 2 < d 1 < σ. Consequently, we can further require that D 0 ⊂ B z for some z ∈ R 2 .
Due to the well-posedness of the acoustic source problem, there exists a radiating solution u for (1). We define an auxiliary scattering problem for a sound-soft obstacle of finding the scattered field u s such that en, there exists a radiating solution u s to the above equation. Furthermore, u s coincides with u in R 2 \D 0 . By introducing D 0 and the related scattering problem, one actually deals with an inverse scattering problem instead of the inverse source problem. e rest of the proof is then Mathematical Problems in Engineering based on the idea of eorem 3.1 of [1]. For completeness, we present the details as follows.
Recall that the Herglotz wave operator H: L 2 (S) ⟶ H 1/2 (zB z ) is defined as It is well known that H is injective and has a dense range provided k 2 is not a Dirichlet eigenvalue for the negative Laplacian for B z [3].
Define the near-to-far-field operator by which maps the boundary value of a radiating solution u ∈ H 1 loc (R 2 \B z ) to its far-field pattern u ∞ . en, the operator N is bounded and injective and has a dense range [3].
Consider the case of D ⊂ B z . For any ε, there exists g ε z such that Since D 0 ⊂ B z , we also have Subtracting (29) from (30), and using (28), we have g ε for any ε. Taking ε ⟶ 0 in (28), the Herglotz wave function converges to the unique solution w ∈ H 1 (B z ) of the following problem which proves Case 1.
for every ε > 0. Assume to the contrary that there exists a sequence v g ε n z such that ‖v g ε n z ‖ H 1 (B z ) remains bounded as ε n ⟶ 0, n ⟶ ∞. Without loss of generality, we assume that v g ε n z converges to Let v s ∈ H 1 loc (R 2 \B z ) be the unique solution of the following problem: (36) en by Rellich's lemma (see Lemma 2.12 in [3]), the scattered fields coincide in R 2 \(B z ∪ D 0 ) and We have that v s has an extension into R 2 \B z and u s has an extension into R 2 \D 0 . Since D 0 ∩ B z � ∅, U s can be extended from R 2 \(B z ∪ D 0 ) into all of R 2 ; that is, u s is an entire solution to the Helmholtz equation. Since u s also satisfies the radiation condition, it must vanish identically in all of R 2 . is leads to a contradiction since u s does not vanish identically. □ Remark 3. e case of D ∩ B z ≠ ∅ and D ⊄ B z is not covered by the above theorem. In fact, the behavior of the regularized solution g z can be analyzed if one knows D. However, this is irrelevant for the inverse problem since D is always unknown.
e above theorem provides a guidance to determine the location of the source. However, the choice of R is rather arbitrary. To find both the location and rough size of the source, we propose a two-step ESM as follows.
Two-Step ESM for the ISP (A) Find the location of D.
(i) Assume that a domain Ω containing D is known a priori. (ii) Choose a large enough radius R for B z and generate a set S of sampling points for Ω.
(a) generate S j with the distance between sampling points being roughly R j ; (b) carry out Step A using S j and a sampling disc with radius R j ; (c) find the minimum point z * j ∈ S j and set D j ≔ B

Remark 4.
e requirement that k 2 is not a Dirichlet eigenvalue of B z is not essential. One can avoid this by choosing a slightly different R.

Numerical Examples
In this section, we present some numerical examples to demonstrate the effectiveness of the proposed method. e synthetic data of the direct source problems are computed using the integral formula (7). Two wavenumbers k � 1 and k � 6 are selected for tests. We first generate a triangular mesh T for D with mesh size h ≈ 0.01 (see Figure 1). e far-field pattern is approximated by where T ∈ T is a triangle, y T is the center of T, and |T| is the area of T. e interval [0, 2π] is uniformly divided into 40 intervals, and let θ j � 2jπ/40, j � 0, . . . , 39. For all examples, assume that u ∞ (x j ), x j � (cosθ j , sinθ j ) T are the measured data. Note that the quadrature rule leads to about 3% error for the far-field pattern, which can be viewed as the noise. Let the sampling domain be given by S the set of all sampling points. For each point z ∈ S, compute using (12). Simply applying the trapezoidal rule to (20), one gets the following linear system: where en, the Tikhonov regularization is used to solve an approximated solution g α z of (43) with a fixed parameter α � 10 − 5 for all z ∈ S.
ree functions are used . We consider two domains (supports of the source functions) D 1 and D 2 (see Figure 1), the unit square with vertices and an L-shaped domain with vertices We first consider the constant function f 1 � 5. In Figure 2, we plot the contours of the indicator function I z � |g α z | in the sampling domain. It is clear that the values of I z is small in the neighborhood of the source. e boundary of the exact support is the solid line. e circle centered at the minimum value of I z (dotted line) is the output of the twostep ESM, which correctly provides the location and rough size of the source for both wave numbers k 1 � 1 and k 2 � 6. Mathematical Problems in Engineering Figure 3 is the results for f 2 � x 2 + y 2 ; that is, the source is a function of position.
Finally, in Figure 4, the results for f 3 (x, y, k) � exp(ikr) are shown. e reconstructions again correctly provide the location and rough size of the source, and the dependence of f 3 on k does not change the results significantly.

Conclusions
A two-step ESM is proposed for the inverse source problem. e ESM was originally developed for the inverse scattering problem. e new setup of the far-field equation makes it possible to treat the inverse source problem. e behavior of the solutions is theoretically justified. Numerical examples show that the two-step ESM can be used to find the location and rough size of the support of the acoustic source effectively.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.