1 Introduction↩︎

We consider a one-dimensional array of point sources that is unbounded, periodic along the \(x_1\)-direction with period \(L>0\), and bounded in the \(x_2\)-direction between \(-h\) and \(h\), for some \(h>0\). Each unit cell of length \(L\) contains one point source. We denote by \[\notag \Omega: = \mathbb{R}\times (-h,h), \quad \Omega_0: = \left(-\frac{L}{2}, \frac{L}{2}\right)\times (-h,h),\] and let \({\boldsymbol{x}}_0\) be the location of the source in \(\Omega_0\). The locations of all other sources in the array are then given by \({\boldsymbol{x}}_j :={\boldsymbol{x}}_0+(jL,0)^{\top}\) for \(j\in \mathbb{Z}\). These sources are represented by the delta distribution \(\delta(x;{\boldsymbol{x}}_0+(jL,0)^{\top})\) for \(x\in \Omega\) and have corresponding intensities \(\gamma_j\).

In this work, we assume that some sources are not functioning properly, in the sense that their intensities differ from \(1\). In particular, there are \(N \in \mathbb{N}^{*}\) defective sources located at \({\boldsymbol{x}}_{m_\ell}\), each having intensity \(\gamma_{m_\ell}\neq 1\), for \(\ell = 1, \ldots, N\). We denote by \({\boldsymbol{I}}\) the set of indices of all defective sources, i.e., \({\boldsymbol{I}}:=\{m: \gamma_m\neq 1\}\) and assume that \({\boldsymbol{I}}\) is a finite set.

The wave generated by all point sources in a homogeneous background with a wave number \(k > 0\) satisfies the Helmholtz equation \[\notag \Delta u + k^2 u = - \left(\sum_{m\in \mathbb{Z}\setminus {\boldsymbol{I}}} \delta(\cdot, {\boldsymbol{x}}_0+(mL,0)^{\top}) + \sum_{m \in {\boldsymbol{I}}} \gamma_m \delta(\cdot, {\boldsymbol{x}}_0+(mL,0)^{\top})\right),\] which is equivalent to \[\label{eq:direct:1s0} \Delta u + k^2 u = -\left(\delta_{\mathrm{per}}(\cdot, {\boldsymbol{x}}_0) + \sum_{m \in {\boldsymbol{I}}} \sigma_m \delta(\cdot, {\boldsymbol{x}}_m)\right),\tag{1}\] where \[\sigma_m := \gamma_m -1, \quad \delta_{\mathrm{per}}(\cdot, {\boldsymbol{x}}_{0}): = \sum\limits_{m \in \mathbb{Z}} \delta(\cdot, {\boldsymbol{x}}_0+(mL,0)^{\top}),\qquad \text{in } \Omega.\] We denote by \(G_{\mathrm{per}}(\cdot, {\boldsymbol{x}}_0)\) the outgoing periodic Green’s function with period \(L\) associated with the point source at \({\boldsymbol{x}}_0\); that is, \(G_{\mathrm{per}}(\cdot, {\boldsymbol{x}}_0)\) admits the spectral representation \[G_{\mathrm{per}}(x,{\boldsymbol{x}}_0) =\sum_{n\in\mathbb{Z}}\frac{i}{2L\theta_{n}}e^{i\frac{2\pi n}{L}\left(x_1-{\boldsymbol{x}}_{0}^{(1)}\right)+i\theta_{n}\left|x_2- {\boldsymbol{x}}_{0}^{(2)}\right|},\quad x, {\boldsymbol{x}}_0 \in \Omega_0, \; x\neq {\boldsymbol{x}}_0,\] where \(\theta_{n} = \sqrt{k^2 - \left(2\pi n/L\right)^2}, \, \theta_{n}\neq 0\), with \(\mathrm{Im}\,(\theta_n) \geq 0\). Then we impose the outgoing behavior of \(u\) by requiring that \(u - G_{\mathrm{per}}(\cdot, {\boldsymbol{x}}_0)\) satisfies the Sommerfeld radiation condition. Then the outgoing wave solution \(v\) to 1 is given by \[\label{form:v} u(x) = G_{\mathrm{per}}(x, {\boldsymbol{x}}_0) + \sum_{m \in {\boldsymbol{I}}} \sigma_m\frac{i}{4} H^{(1)}_0(k|x - {\boldsymbol{x}}_m|),\tag{2}\] where \(H^{(1)}_0\) is a Hankel function of the first kind and zero order.

The objective of this work is to determine the set of indices of all defective sources \({\boldsymbol{I}}\) and the corresponding intensities \(\gamma_m\) for all \(m \in {\boldsymbol{I}}\) using measurements of \(v\) on horizontal lines \(\Gamma_{\pm t}: =\mathbb{R}\times \{\pm t\}\) for some \(t > h\). We consider the following inverse problem below.

Determine the indices \(m \in {\boldsymbol{I}}\) corresponding to the defective sources located at \({\boldsymbol{x}}_m = {\boldsymbol{x}}_0 + (mL,0)^{\top}\) and their intensities \(\gamma_m\) using measurements of the radiated waves \(v\) on \(\Gamma_{ t} \cup \Gamma_{-t}\) with \(t > h\).

We note that the results in this paper can be directly extended to the 3D setting with two-dimensional periodic source arrays. This problem arises when the source distribution deviates from its expected or designed configuration. Understanding and locating such defects is crucial for diagnosing system failures, improving performance, or reconstructing hidden inhomogeneities. Our goal is to develop a robust and computationally efficient method capable of detecting and localizing defective units in large scale periodic source systems using single frequency measurements. Unlike classical inverse source problems in free space or bounded domains, the key challenge here lies in the disruption of periodicity caused by defective sources in the infinite array. To address this, we employ the Floquet–Bloch transform to reformulate the inverse problem of interest as a quasi-periodic inverse source problem defined on a single unit cell, thereby avoiding the need to solve the inverse source problem on an unbounded domain [1], [2].

Over the past two decades, inverse source problems in free space or bounded domains have been studied extensively from both theoretical and computational perspectives. Significant progress has been made in the analysis of uniqueness and stability properties, see, e.g., [3]–[11] and the references therein. Meanwhile, various numerical algorithms have been proposed to reconstruct sources from boundary data. In particular, methods based on multi-frequency data have shown enhanced stability in recovering unknown sources, see [12]–[22].

Inverse source problems with single-frequency data can face non-uniqueness of solution unless certain a priori assumptions about the source are imposed [23]. To address this challenge, several computational strategies have been developed. Algebraic reconstruction methods have been applied to identify point sources in elliptic equations such as the Poisson and Helmholtz equations [3], [24], [25], while Newton-type iterative algorithms have been used to recover both point and extended sources [18]. In recent years, sampling-based techniques have gained popularity due to their simplicity and efficiency. Sampling-type methods have been studied in [26]–[28] for identifying point sources in the Helmholtz equation and Maxwell’s equations. The key advantages of such methods are their non-iterative nature, robustness, speed, and low computational cost.

Unlike the classical inverse source problems considered in the works mentioned above, the problem of identifying defective sources within an infinite periodic array appears, to the best of our knowledge, not to have been studied before. In this paper, we first establish uniqueness results for both the original inverse problem and its quasi-periodic counterpart. Motivated by the numerical method studied in [26] for the inverse point-source problem and the approach developed in [29] for imaging periodic scattering structures, we introduce a new numerical reconstruction scheme that combines a novel imaging function with an algebraic procedure for detecting defective units in unbounded periodic source arrays. The algebraic procedure employs linear algebra tools. The proposed method is stable with respect to noise, computationally efficient, and straightforward to implement.

It is worth emphasizing that the algebraic approach employed here identifies defective sources through the evaluation of matrix determinants and the solvability of a system of equations derived from their relationship with the imaging function. An alternative algebraic, matrix-based approach for recovering source information was investigated in [10], [11] for inverse point source problems in bounded domains. In those works, the authors employed the reciprocity gap functional together with specially designed test functions to construct Hankel matrices and associated linear systems to identify unknown sources. Although our approach also relies on matrices and linear systems, the key ingredients are derived from a new imaging function inspired by recent work of the first author and collaborators in [26], [27].

The structure of the paper is as follows. In Section 2, we focus on the uniqueness of the solution and the reformulation of the problem through the Floquet-Bloch transform. Section 3 is dedicated to the development of the new numerical reconstruction method for defective sources. Section 4 presents a numerical study that demonstrates the performance and effectiveness of the proposed method. Finally, in Section 5, we conclude and discuss potential future work.

2 Uniqueness of solution and Floquet-Bloch transform based formulation↩︎

We begin by establishing the uniqueness of the solution to the inverse problem. We then derive a Floquet-Bloch transform-based formulation of the problem and prove the corresponding uniqueness result.

In the following, we discuss the determination of the location and intensity of the defective sources. This is divided into two steps: the first step focuses on determining the source location within the period \(\Omega_0\), and the second step focuses on identifying the set \({\boldsymbol{I}}\) of defective indices and their corresponding intensities. The investigation of both steps will be based on the use of Floquet-Bloch transform. The one-dimensional Floquet-Bloch transform of a function \[\varphi \in \mathcal{S}(\mathbb{R}):=\left\{ \psi \in \mathcal{C}^{\infty}(\mathbb{R}): \, \sup_{x\in \mathbb{R}} |x^{j} \psi^{(\ell)}(x)| < \infty, \; \; \text{ for all} \; \; j, \ell \in \mathbb{N}\right\}\] with period \(L > 0\), is defined by \[\label{def:FBtransform} \mathcal{F}\varphi(x; \xi): = \sum_{m \in \mathbb{Z}} \varphi(x + mL)e^{- i \xi mL}, \quad \; \text{for all } \xi \in \left(-\frac{\pi}{L}, {\frac{\pi}{L}}\right), \quad x \in \mathbb{R}.\tag{3}\] Note that, for a fixed \(\textstyle\xi \in \left(-\frac{\pi}{L}, {\frac{\pi}{L}}\right)\), the function \(\phi(x):= \mathcal{F}\varphi(x; \xi)\) is \(\xi-\)quasi-periodic with period \(L\), i.e., \[\phi(x + mL) = e^{i\xi mL}\phi(x), \; \; \text{for all } m \in \mathbb{Z}, \quad x \in \mathbb{R}.\] We refer to [30], [31] for a more detailed discussion of the Floquet-Bloch transform and related properties. Now taking the Floquet-Bloch transform of 1 in the \(x_1-\)direction and denoting by \(u_{\xi} : = \mathcal{F}u (\cdot; \xi),\) \(\textstyle\xi \in \left(-\frac{\pi}{L}, {\frac{\pi}{L}}\right){\setminus \{0\}},\) we obtain from direct calculations that \(u_{\xi}\) satisfies \[\label{eq:FB:direct:1s-quasi} \Delta u_{\xi} + k^2 u_\xi = - \sum_{m \in {\boldsymbol{I}}} \sigma_m e^{-i\xi mL} \delta(\cdot, {\boldsymbol{x}}_0), \quad \text{in } \Omega_0.\tag{4}\] Note that \(\xi = 0\) is excluded since \(\mathcal{F}\delta_\mathrm{per}(\cdot;\xi) = 0\) for all \(\textstyle\xi \in \left(-\frac{\pi}{L}, {\frac{\pi}{L}}\right){\setminus \{0\}}\). In addition, the outgoing behavior of \(u\) is now described by the Rayleigh expansion condition for \(u_\xi\) as \[\label{Cond:Rayleigh:per} u_\xi(x) = \sum_{n \in \mathbb{Z}} \hat{u}_{\xi,(n)} e^{i \xi_{(n)} x_1 + i\beta_n|x_2|},\qquad \text{for }|x_2|\ge h,\tag{5}\] where \[\xi_{(n)} = \xi+\frac{2\pi}{L} n,\qquad \text{and }\qquad\beta_n = \begin{cases} \sqrt{k^2 - \xi_{(n)}^2}, k^2>\xi_{(n)}^2,\\ \notag i\sqrt{ \xi_{(n)}^2-k^2},\qquad k^2<\xi_{(n)}^2, \end{cases} \qquad n\in \mathbb{Z},\] which describes the outgoing behavior of the quasi-periodic wave \(u_\xi\). From 4 5 , we obtain the following representation of \(u_\xi,\) \[\label{sol:FB:direct:1s:quasi} u_{\xi} = \sum_{m \in {\boldsymbol{I}}} \sigma_m e^{-i\xi mL} G_{\xi}(\cdot,{\boldsymbol{x}}_0),\tag{6}\] where \(G_{\xi}\) is the \(\xi-\)quasi-periodic Green’s function defined as \[G_\xi(x,y) =\dfrac{i}{2L}\sum_{n\in\mathbb{Z}}\dfrac{1}{\beta_n}e^{i\xi_{(n)}(x_1-y_1)+i\beta_n|x_2-y_2|},\qquad x,y\in \Omega_0, \, x\neq y.\] We consider the following quasi-periodic inverse problem.

Given \(u_\xi\) on \[\Lambda_{\pm t}:=\left(-\frac{L}{2}, \frac{L}{2}\right)\times\{\pm t\},\] for a finite number of \(\textstyle\xi \in \left( -\frac{\pi}{L},\frac{\pi}{L}\right){\setminus \{0\}}\), determine the number of defective sources \(N\), the indices \(m \in {\boldsymbol{I}}\) corresponding to their locations \({\boldsymbol{x}}_0 + (mL,0)^{\top}\), along with their intensities \(\gamma_m\).

The following theorem establishes the uniqueness of the solution to the problem using a finite number of \(\xi\). Unlike the original inverse problem, the dataset used for the quasi-periodic problem under consideration differs from the original measurements because the periodic component has been eliminated. Furthermore, using only a finite number of \(\xi\) makes the data even weaker than the dataset.

3 Numerical reconstruction method↩︎

In this section, we develop a new numerical method that combines a stable imaging function and an algebraic technique to recover the number of defects, their locations, and the corresponding intensities in the quasi-periodic inverse problem discussed in the previous section.

3.1 A stable imaging function for determining \({\boldsymbol{x}}_0\)↩︎

Recall that \(\Omega_0=\left(-\tfrac{L}{2}, \tfrac{L}{2}\right)\times (-h,h)\). Let \(\textstyle\xi \in \left(-\frac{\pi}{L}, \frac{\pi}{L}\right) { \setminus \{0\}}\). Inspired by [29], for \(z\in \Omega_0\), we define the following imaging function \[\label{eq:Iz} I_\xi(z)=\int_{\Lambda_t\cup\Lambda_{-t}} \Phi_\xi(x,z)u_\xi(x)ds(x),\tag{9}\] where \[\Phi_\xi(x,z)=\dfrac{-i}{2L}\sum_{j\in \mathbb{Z}\atop \beta_j>0} e^{-i\xi_{(j)}(x_1-z_1)-i\beta_j|x_2-z_2|},\qquad x,z\in \Omega_0.\] This imaging function aims to determine \({\boldsymbol{x}}_0\) with sampling points \(z\). The behavior of \(I_\xi(z)\) is analyzed in the following theorem.

Theorem 3 provides a complete characterization of the source position \({\boldsymbol{x}}_0\) in the period \(\Omega_0\) using \(|I_{\xi}(z,{\boldsymbol{x}}_0)|\): the maximum of \(|I_{\xi}(z,{\boldsymbol{x}}_0)|\) is attained uniquely at \(z={\boldsymbol{x}}_0\). Furthermore, this maximum forms a sharp peak, as \(|I_{\xi}(z,{\boldsymbol{x}}_0)|\) remains small for \(z \neq {\boldsymbol{x}}_0\). This property is essential for our numerical experiments, since it allows us to clearly distinguish the true source location even with oscillatory background.

This peak property can be explained by that of \(|\mathcal{K}_\xi(z,{\boldsymbol{x}}_0)|\). Figure 1 illustrates that \(|\mathcal{K}_\xi(z,{\boldsymbol{x}}_0)|\) has a significant peak at \(z = {\boldsymbol{x}}_0\) and remains significantly smaller as \(z\) is away from \({\boldsymbol{x}}_0\). The decaying behavior can be explained by considering the real and imaginary parts of \(\mathcal{K}_\xi(z,{\boldsymbol{x}}_0)\). For the real part, \[\text{Re}[\mathcal{K}_\xi(z,{\boldsymbol{x}}_0) ] = \sum_{j\in\mathbb{Z}\atop \beta_j >0} \dfrac{1}{2L\beta_j}\cos\left(\xi_{(j)}\left(z_1 -{\boldsymbol{x}}_{0}^{(1)}\right)\right)\cos\left(\beta_j\left(z_2 -{\boldsymbol{x}}_{0}^{(2)}\right)\right).\] When \(z \neq {\boldsymbol{x}}_0\), the cosine terms in the summation oscillate with different frequencies, causing cancellation in the summation. At the same time, for the imaginary part, \[\text{Im}[\mathcal{K}_\xi(z,{\boldsymbol{x}}_0) ] = \sum_{j\in\mathbb{Z}\atop \beta_j >0} \dfrac{1}{2L\beta_j}\sin\left(\xi_{(j)}\left(z_1 -{\boldsymbol{x}}_{0}^{(1)}\right)\right)\cos\left(\beta_j\left(z_2 -{\boldsymbol{x}}_{0}^{(2)}\right)\right),\] when \(z\neq {\boldsymbol{x}}_0\), the sine and cosine terms oscillate at varying frequencies, again leading to mutual cancellation. Therefore, the magnitude of \(\mathcal{K}_\xi(z,{\boldsymbol{x}}_0)\) remains significantly smaller when \(z\) moves away from \({\boldsymbol{x}}_0\).

In practice, we use \(|I_\xi|^p\), where the parameter \(p>0\), to identify \({\boldsymbol{x}}_0\) and sharpen the peak. Once \({\boldsymbol{x}}_0\) is determined, we denote \[\label{Theta} \Theta_\xi = \sum_{n\in {\boldsymbol{I}}} \sigma_n e^{-i\xi nL},\tag{13}\] which can then be computed as \[\Theta_\xi = \dfrac{I_\xi({\boldsymbol{x}}_0)}{\mathcal{K}_\xi({\boldsymbol{x}}_0,{\boldsymbol{x}}_0)}.\] This \(\Theta_\xi\) enables the detection of both locations and intensities of the defective sources, as demonstrated in the computational algorithm.

3.2 Determination of \(N\), \(m\), and \(\gamma_m\)↩︎

This section introduces a systematic approach, utilizing the imaging function \(I_\xi(z)\) and algebraic techniques, to determine the number of defective units \(N\), the indices \(m \in {\boldsymbol{I}}\), and their associated intensities \(\gamma_m\).

First, we develop the theorem to help determine the number of defective units. We employ a finite set of \(\xi_j\) as in Theorem 2, \[\xi_j = \xi_0 + j\Delta\xi,\qquad j=1,2,\ldots\] with \(0< \Delta\xi\neq \tfrac{2\pi q}{L},\, q\in \mathbb{Q}\) such that \(\textstyle\xi_j\in \left( -\frac{\pi}{L},\frac{\pi}{L}\right){ \setminus \{0\}}\) for all \(j\). For \(n=1,2,\ldots,N\), we denote further \[a_n = (\gamma_{m_n}-1) e^{-i\xi_0 m_n L}, \qquad \text{and }\qquad r_n = e^{-i\Delta\xi m_nL}.\] Recall from 13 that \(\Theta_\xi = \sum_{n\in {\boldsymbol{I}}} \sigma_n e^{-i\xi nL}\). Then \[\notag \Theta_{\xi_j}=a_1 r_1^j + a_2r_2^j+\ldots a_N r_N^j,\] where \(N=|{\boldsymbol{I}}|\) and \(m_n\in {\boldsymbol{I}}\).

We now proceed to present the method for determining the indices and intensities of the defective sources.

The following algorithm is derived directly from Theorems 4 and 5.

4 Numerical examples↩︎

We present several numerical examples to evaluate the performance of the proposed method. In all cases, \(\Omega_0 = (-1,1) \times (-2,2),\) with point sources arranged periodically with period \(L = 2\). The data is generated by \[u_\xi=\mathcal{F}\left( \sum_{m \in {\boldsymbol{I}}} \sigma_m\frac{i}{4} H^{(1)}_0(k|x - {\boldsymbol{x}}_m|)\right)(\cdot;\xi),\] for multiple values of \(\xi\in \left(-\tfrac{\pi}{L},\tfrac{\pi}{L} \right)\setminus\{0\}\) with a note that \(\mathcal{F}G_{\text{per}}(\cdot;\xi)=0\) for \(\xi\in \left(-\tfrac{\pi}{L},\tfrac{\pi}{L} \right)\setminus\{0\}\). Here, \(u_\xi\) is computed at wave number \(k=10.5\) on \(\Lambda_{3} \cup \Lambda_{-3}\) with added artificial random noise. The noise vector \(\mathcal{N}\) is incorporated to \(u_\xi\), the solution to 1 , as follows \[u_\xi+\delta\dfrac{\mathcal{N}}{\| \mathcal{N}\|_F}\| u_\xi\|_F,\] where \(\delta\) is the level of noise and \(\| \cdot\|_F\) is the Frobenius matrix norm. Note that \(\mathcal{N}\) consists of numbers \(a+bi\) where \(a,b\in (-1,1)\) are randomly generated with a uniform distribution. More specifically, we use \[\xi_j = -\dfrac{\pi}{L} + 0.01 + 0.295j,\qquad j=0,1,\ldots,8\] for computing \(\det D_M\) and \[\xi_j = -\dfrac{\pi}{L} + 0.03 + 0.13j,\qquad j=0,1,\ldots,22\] for reconstructing \(m\) and \(\gamma_m\).

We can clearly see a significant peak of \(|I_\xi(z)|^3\) (i.e., \(p=3\)) in Figure 2. Then we determine \({\boldsymbol{x}}_0\) as the location of this peak. Tables 3 and 4 demonstrate that the computed location of \({\boldsymbol{x}}_0\) is highly accurate.

Tables 1 and 2 summarize the computational results for determining the number of defective units by \(\det D_1,\,\det D_2,\, \det D_3,\ldots\). Following Theorem 4, if \(\det D_M \neq 0\) for \(M\le N\) and \(\det D_{M}= 0\) for \(M >N\), there are \(N\) defective sources. This pattern can be observed that in all test cases, \(\det D_M\) remain nonzero when \(M\) does not exceed the true number of defects and \(\det D_M\) become nearly zero for larger \(M\), confirming the validity and accuracy of the proposed criterion.

The reconstruction results for the indices and intensities of the defective point sources are presented in Tables 3 and 4. All indices are successfully recovered. The reconstructed intensities are also accurate, with relative errors below \(5\%\) in all cases. It is worth noting that even under \(20\%\) noise, the method maintains reasonable accuracy, with relative errors in the reconstructed intensities remaining within \(13\%\).

5 Conclusion↩︎

In this work, we investigated the problem of identifying defective sources for Helmholtz equation in unbounded periodic arrays. The main challenge is that defects break the periodic pattern, which makes classical inverse source methods ineffective. To address this, we used the Floquet–Bloch transform to reformulate the original problem into a quasi-periodic inverse source problem. We proved that both the original and the quasi-periodic problems have unique solutions, meaning the defective sources can be determined from boundary measurements. Building on this foundation, we proposed a numerical reconstruction method that combines a stable imaging function with an algebraic identification scheme. This hybrid method reliably recovers the number, locations, and intensities of defective sources. The numerical experiments show that the approach is accurate, easy to implement, computationally efficient, and highly stable even under high noise levels. Future work may extend the framework to small-volume source models and to inverse problems governed by Maxwell’s equations.

The authors would like to thank the anonymous referees for their careful reading and insightful comments, which have improved the quality of the presentation. The research of T.-P. Nguyen is partially supported by the NSF Grant DMS-2513647. The research of D.-L. Nguyen and N. H. Nguyen is partially supported by NSF Grant DMS-2243854.

References↩︎

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1. Department of Mathematics, Kansas State University, Manhattan, KS 66506 (dlnguyen@ksu.edu)↩︎

2. Department of Mathematics, Kansas State University, Manhattan, KS 66506 (nhungnh@ksu.edu)↩︎

3. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102 (thiphong.nguyen@njit.edu)↩︎