On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method

In this paper, we consider the two-dimensional Maxwell's equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of \begin{document}$O(1)$\end{document} and the other half of eigenvalues are positive with order of \begin{document}$O(10^2)$\end{document} . In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably.

1. Introduction. The transmission eigenvalue problem (TEP) has attracted a lot of attention recently in the study of direct/inverse scattering problems in inhomogeneous media [3,4,5,6,9,11,12,20,25]. The existence of transmission eigenvalues is intimately connected to the "bijectivity" of the far-field operator, which is crucial in some reconstruction methods such as the linear sampling method (LSM) [1,8] and the factorization method [21]. Conversely, the transmission eigenvalues (also eigenvalues) carry the information of the scatterer and can be estimated by the farfield data [2] or the near-field data (Cauchy data) [27]. This observation leads to the development of some reconstruction methods using the transmission eigenvalues or eigenvalues, see for example, [10,29]. In [29], an eigenvalue method using multiple frequency near-field data (EM 2 F) was proposed to detect Dirichlet or transmission eigenvalues and to reconstruct the support of the scatterer.
In this paper, we would like to propose a reliable numerical method to investigate the distribution of transmission eigenvalues for the 2d acoustic equation with an inhomogeneous index of refraction. The model is derived from the Maxwell's equations with the TM mode in pseudo-chiral media. It turns out the index of refraction of the reduced acoustic equation decreases to negative infinity when we increase the charity parameter. Precisely, we consider the TEP ∆u + λε(x)u = 0, in D, for the scattering of acoustic wave on a bounded and simply connected inhomogeneous domain D ⊆ R 2 , where ν is the outer normal to the smooth boundary ∂D, u, v ∈ L 2 (D) with u − v ∈ H 2 0 (D) = {w ∈ H 2 |w = 0, ∂w ∂ν = 0}, and ε(x) is the index of refraction. Any λ ∈ C such that (1) has nontrivial solutions u and v is called a transmission eigenvalue and u, v are called the corresponding transmission eigenfunctions for D . Equation (1a) can be considered as a reduced Maxwell's equations with transverse magnetic (TM) mode. For the standard Maxwell's equations, ε(x) is the electric permittivity corresponding to the product of the dielectric constant and the freespace dielectric constant. In practice, ε can not be taken as large as we wish. In [23] where the Maxwell's equations for Tellegen media is studied, ε(x) is the sum of the electric permittivity and the square of Tellegen parameter. By choosing large Tellegen parameter, we can enlarge the parameter ε(x) as we want. On the contrary, here ε(x) is the sum of the electric permittivity minus the square of pseudo-chiral media parameters. Therefore, by selecting the pseudo-chiral parameters sufficiently large, ε(x) becomes negative (see Section 2 below).
In recent years, several papers have been devoted to developing efficient algorithms for computing transmission eigenvalues of 2d/3d TEP [3,11,14,16,17,20,21,22,23,24,26,27]. Three finite element methods (FEMs) and a coupled boundary element method were developed to solve the 2d/3d TEP in [11,14,17] (see the book [30] for more details). Two iterative methods and the corresponding convergence analysis were given in [28]. In [17], a mixed finite element method for 2d TEP was proposed, which leads to a non-Hermitian quadratic eigenvalue problem (QEP) that was solved by an adaptive Arnoldi method. Furthermore, a multilevel correction method was used to reduce the solution of TEP into some linear boundary value problems which could be solved by the multigrid method [18]. For results related to our current work, in [23,24], the TEP for general inhomogeneous media are discretized into QEP with symmetric coefficient matrices. For such QEP, a secant-type iteration for computing several smallest positive transmission eigenvalues accurately was proposed in [24], and a quadratic Jacobi-Davidson method with nonequivalent deflation technique for computing a large number positive transmission eigenvalues was developed in [23]. Remind that the index of refraction ε(x) increases to infinity as we increase the Tellegen parameter. Numerical simulations demonstrate that the positive transmission eigenvalues are densely distributed in an interval near the origin. Similar to the method in [23], the TEP (1) is discretized into a QEP and a quadratic Jacobi-Davidson method with nonequivalence deflation is applied to compute a large number of positive eigenvalues. It turns out in the case here there exists an eigenvalue-free interval near the origin. The existence of the eigenvalue-free interval motivates us to study the reconstruction of the support of ε(x) from the near-field data in the spirit of LSM.
Corresponding to the TEP(1), the scattering problem is described by where D := supp(ε(x) − 1) and u i is the incident field due to a point source at x 0 , i.e., Our reconstruction method is based on the set up in [29]. We assume that the target D is inside some domain Ω which itself is inside a curve C (see Figure 1). Suppose that u s is measured on Γ = ∂Ω for all point sources x on C. We define the near-field operator N : The LSM with near-field data relies on solving the ill-posed integral equations where z ∈ T is a sampling point and T is a sampling domain inside Ω containing the target D (see Figure 1). In general, the above ill-posed equation do not have a solution. The following theorem serves as the backbone of the LSM.
• If z ∈ D, then there exists a convergent sequence v n in L 2 (D), such that lim n→∞ N v n = Φ(·, z).
• If z ∈ Ω \ D, then for every sequence v n satisfying lim n→∞ N v n = Φ(·, z), Figure 1. The target D is inside some domain Ω (Γ := ∂Ω) which itself is surrounded by a curve C. The scattered field u s is due to the scattering of the incident field u i having a point source at x 0 ∈ C.
Our reconstruction method is a direct application of Theorem 1.1. By this theorem, we can see that if k 2 is not a transmission eigenvalue then we can determine whether z ∈ D or z ∈ Ω\D from the behaviors of the solutions to (4). Since there exists an eigenvalue-free interval near the origin in TEP (1), we choose a k near the origin so that k 2 is not a transmission eigenvalue, we then discretize the integral equation (5). The strategy here is to choose m points x 1 , · · · , x m ∈ C forming a m-vector b = [Φ(x 1 , z), · · · , Φ(x m , z)] ∈ C m and n unknowns [v(y 1 ), · · · , v(y n )] := v satisfying the linear system The idea is to take m ≥ n, i.e., (6) is overdetermined. To determine whether z ∈ D or z ∈ Ω \ D, we look at the distance between the vector b and the space A = span(A), the subspace spanned by the columns of A, namely, dist(b, A) ≡ min v∈C n ||Av − b|| 2 . If dist(b, A) > 0, then, intuitively, any "solution" v satisfying (6) must contain some sufficiently large components. Mimicing the second case of Theorem 1.1, we thus assign z ∈ Ω \ D. On the other hand, if dist(b, A) = 0, i.e., b ∈ A, then the norm of v is clearly finite. We thus say that z ∈ D in view of the first case of Theorem 1.1. An easy application of truncated SVD will quickly determine the value of dist(b, A). Our criterion is very simple and easily to be implemented.
This paper is organized as follows. In Section 2, we demonstrate the derivation of 2d TEP (1) from the Maxwell's equations with TM mode in pseudo-chiral media. A corresponding discretized QEP and its spectral analysis are given in Section 3. In Section 4, a practical numerical method based on the LSM and truncated SVD for the reconstruction of the the target D is developed. Related numerical results are presented in Section 5. A concluding remark is given in Section 6. Finally, in Appendix A, we present a framework of solving the direct scattering problem using FEM. The purpose of solving the direct problem is to obtain synthetic data for numerical simulations.

2.
Maxwell's equations with the TM mode in pseudo-chiral media. Physically, the governing equations for the propagation of electromagnetic wave in biisotropic materials with complex media is modeled by the 3d source-free frequency domain Maxwell's equations where E and H are the electronic field and magnetic field respectively, k is the frequency, ε r and µ are electric permittivity and magnetic permeability respectively, ξ and ζ are 3-by-3 magnetoelectric parameter matrices in various forms for describing different types of complex media.
We consider E and H in (7) in the transversal magnetic (TM) mode as with ξ 1 = ζ 1 = iγ 1 , ξ 2 = ζ 2 = iγ 2 , and γ 1 , γ 2 being real numbers. We assume in this paper µ = 1. Then the equation (7a) can be simplified to Substituting (10) into (7b) yields , 0] be, respectively, the electronic and magnetic plane waves with TM mode in vacuum which are governed by the free Maxwell's equations with ε 0 = µ 0 = 1. We now suppose the Maxwell's equations (7) and (11) are defined on a cylindrical set D × R satisfying the boundary conditions is the outer unit normal to the smooth boundary ∂D × R. It is easily seen from (12a) that the Dirchlet boundary condition holds. Multiplying (12b) by ik and using (11a) we get the Neumann boundary condition In view of the boundary conditions (13a), (13b), choosing u = E 3 and v = E 0,3 , we arrive at the TEP (1) with λ = k 2 and ε(x) = ε r − (γ 2 1 + γ 2 2 ). The index of refraction will become negative as long as either |γ 1 | or |γ 2 | are sufficiently large.
3. Discretization of TEP and its spectral analysis. In this section, we first briefly describe the discretized TEP and present its spectral analysis. Here we follow the notations used in [23,Section 3]. Let S h = The space of continuous piecewise linear finite elements on D, be standard nodal bases for the finite element spaces of S I h and S B h , respectively. Applying the standard piecewise linear FEM (we refer to [11] for details) to (1) and using integration by parts, we have Table 1. Stiffness and mass matrices with ε(x) < 0 for x ∈D.
with φ j , j = 1, · · · , n and ψ j , j = 1, · · · , m being the discrete test functions. By the boundary conditions (1c) and (1d), and as φ j vanishes DoF on ∂D, the boundary terms of the above equations will vanish, then it follows that Utilizing the standard FEM, we can write u and v in (1) as Hereafter, we denote 0 and I as a zero matrix/submatrix and the identity matrix, respectively, with appropriate dimensions if it is clear in the context. Now, setting u = [u 1 , · · · , u n ] , v = [v 1 , · · · , v n ] and w = [w 1 , · · · , w m ] appeared in (16), by (15), we can derive a generalized eigenvalue problem (GEP) with K, E, M 1 , M ε , F 1 , F ε , G 1 and G ε being given in Table 1. Here, A 0 means A is symmetric positive definite.
We now define and we see that On the other hand, observe that (17) can be reduced to the following QEP where p = u − v, A 2 , A 1 and A 0 are all n × n symmetric matrices given by Proof. By (18), subtracting the 1st equation by the 2nd equation in (17), and rewriting the 3rd equation in (17) we have On the other hand, from the 1st equation of (17) and the 1st equation of (22), we have From (19a) and (19c), equations (22) and (23) can be rewritten as Expressing u w in terms of p in (24a) and plugging it into (24b), we end up with a QEP Rewriting (25) as and using the fact that We can show by careful calculation that the coefficient matrices in (25) satisfy those in (21).
Let L(λ) and Q(λ) be defined in (17) and (20), respectively. Then where σ(·) denotes the spectra of the associated matrix pencil. (20) with matrices A 2 , A 1 and A 0 defined in (21). Then A 2 and A 0 are positive definite. Furthermore there are n negative and n positive eigenvalues for Q(λ).
Proof. From Lemma 3.1 and S being of full column rank, we get that the matrix A 0 in (21c) is positive definite. On the other hand, from Inverse Problems and Imaging Volume 12, No. 4 (2018), 1033-1054 which implies that A 2 0. Let (λ, p) be an eigenpair of (20), then Here and hereafter, the superscript "H" in (26) denotes the conjugate transpose. Because A 1 is symmetric and A 2 , A 0 0, we have which implies that the roots of the quadratic equation (26) are Hence, there are n negative and n positive eigenvalues for (20) and the associated eigenvectors are real vectors. and Suppose that Then the n negative and n positive eigenvalues of (20) are, respectively, in the intervals (−β * , 0) and (β * , ∞), where Proof. By the definitions of W 0 and W 1 , A 1 in (21b) can be represented as Given an orthogonal vector p, from (29) and (30a), equation (32) implies that From (27)- (29) and (33), it follows that and λ + = a 1 + a 2 1 + 4a 2 a 0 2a 2 ≥ a 1 + a 2 1 + 4a 2 a 0 2ā 2 = β * .

4.
Numerical method for reconstructing the unknown domain. In this section, we will propose a practical numerical method for the reconstruction of the support of the target D based on the LSM and the truncated SVD technique. As described in the Introduction, let z be a sampling point in the sampling domain T (see Figure 1) and k 2 is not a transmission eigenvalue for TEP (1). According to Theorem 1.1, from (4) and (5), we can determine whether z is inside D or not by solving v ∈ L 2 (Γ) which satisfies the near field integral equation where Φ(x, z) is given in (3).
To solve (34), we first discretize (34) as follows. For each point source x i ∈ C, i = 1, · · · , m, suppose we have measured the scattered fields on the discrete points y 1 , · · · , y n on Γ. With these scattered fields, the discretized equation of (34) can be formulated as the overdetermined linear system A ∈ C m×n depends on x i and y j for i = 1, · · · , m and j = 1, · · · , n, and the measured scattered fileds. v ∈ C n is the approximation to the unknown values [v(y 1 ), · · · , v(y n )] . In general, the linear system (35) is very sensitive due to the ill-posedness of (34). For the case of m < n, the problem (35) can be solved by the Tikhonov regularization method which gives (36) min v∈C n {||Av − b|| 2 + δ||v|| 2 } with δ > 0 be a regularization parameter. The proper choice of regularization parameters plays an important role in (36) in order to achieve accurate and stable numerical results. Actually, over the last four decades, many different methods for selecting regularization parameters have been proposed [13,19,31]. Even so, the selection of the optimal regularization parameter remains a difficult question in solving the discretized ill-posed linear system (35).
In this paper, we will propose a novel numerical method to reconstruct the domain D by solving the linear system (35) for the case of m ≥ n. Let A = span(A) be the subspace spanned by the columns of A. Recall from Section 1 that dist(b, A) = min v∈C n ||Av − b|| 2 . From the perspective of matrix analysis, we reinterpret Theorem 1.1 in the following way. Assume that k 2 is not an eigenvalue of (20), for instance, we choose 0 < k 2 < β * . If dist(b, A) > 0, then v satisfying (35) is in the sense that some components of v are infinity. In other words, we have v = ∞. We then assign z ∈ Ω \ D. On the other hand, if dist(b, A) = 0, then we can find a regular vector v with v < ∞ satisfying (35). In this case, we say that z ∈ D. Our reconstruction method is based on this interpretation.
We will use a truncated SVD to determine the size of v. Let where U ∈ C m×m and V ∈ C n×n are unitary, Σ = diag(σ 1 , · · · , σ n ) is diagonal matrix of singular values with σ 1 ≥ · · · ≥ σ n ≥ 0. Usually, the discretized linear system (35) is sensitive that is inherited from the ill-posedness of (34), and the coefficient matrix A in (35) is not of full column rank. Suppose rank(A) = r < n, which means that σ r > σ r+1 = · · · = σ n = 0. Denote Σ r = diag(σ 1 , · · · , σ r ) and partition U and V with respect to Σ r as U = [U 1 , U 2 ] and V = [V 1 , V 2 ] with U 1 ∈ C m×r and V 1 ∈ C n×r , respectively. Obviously, the following statements hold true.
Combing (42) and (43), the statements (39) can be represented as b ∈ span{U 1 } ⇔σ r+1 = 0, thusσ r+1 /σ r+1 = 0/0 = 1; In conclusion, in our method, to decide whether z ∈ D or not relies heavily on effectively determining the ranks of A and A. However, since the integral equation (35) is ill-posed, the matrices A and A are normally ill-conditioned and there is often no gap in the spectrums of singular values. From numerical point of view, it is difficult to compute the truncated SVD for A. Consequently, we usually set r = n.
We now propose a practical criterion to numerically realize (44). As above, let the singular values of A be σ 1 ≥ · · · ≥ σ n ≥ 0 and those of A beσ 1 ≥ · · · ≥σ n+1 ≥ 0. We first compute j(z) ≡ arg max 1≤j≤nσ j σ j and define I z =σ j(z) . Here I z is used as an indicator to determine whether z is in D or not. Precisely, we pick a small threshold parameter 0 < M. Then we choose the following dichotomy: The algorithm of computing the indicator I z is summarized in Algorithm 1 below.

Algorithm 1 Practical numerical reconstruction method based on the singular values
Input: Discrete point sources x i ∈ C, detection points y j ∈ Γ for i = 1, · · · , m and j = 1, · · · , n, respectively, with m > n. The wave number k ∈ R and sampling points z ∈ T . Output: The indicator I z .

Numerical experiments.
In what follows, we will demonstrate the efficiency of our numerical method for four different domains shown in Figure 2:    The arrows point to the first positive eigenvalue of the QEP corresponding to each domain.
Linux operating system, two Intel Quad-Core Xeon E5-2643 3.33 GHz CPUs and 96 GB of main memory.

Distribution of the transmission eigenvalues.
We carry out the FEM method described in Section 3 to the TEP (1) on four different domains as in Figure 2 with the regular mesh size h ≈ 0.004 for triangles of each domain D. We set ε r = 16, γ 1 = 10, γ 2 = 4, and thus ε(x) = −100. The associated dimensions n and m of matrices given in Table 1 are shown in Table 2. For each domain, we derive a corresponding QEP as given in (20). Because of the two negative signs in (20), we should modify the quadratic Jacobi-Davidson method in [23] to solve these QEPs. Of course, the partial locking and partial deflation schemes of Algorithm 2 and Algorithm 3 in [23] can be employed as well. The numerical results are shown in Figure 3.

5.2.
Reconstruction of the unknown domain from the near-field measurements. In this subsection, we apply Algorithm 1 to reconstruct the domain D from the near-field measurements for four distinct shapes as in Figure 2. As described above, our method is based on the LSM and the SVD technique.
For LSM, as shown in Figure 1, we make the following preparations for numerical experiments. Consider each domain in Figure 2 to be the target D, and let the circles with radii 3 and 6 be Γ and C, respectively. Choose the rectangle domain [−1, 1] × [−1, 1] to be the sampling domain T , which contains all possible targets D. From Figure 3, we choose a k ∈ (0, √ β * ), in other words, k 2 is not a transmission eigenvalue. Different k ∈ (0, √ β * )'s are chosen for testing in our numerical simulations.
Divid C and Γ into m and n segments uniformly, and denote the nodes as x 1 , · · · , x m and y 1 , · · · , y n , respectively. Place a uniform grid on T by drawing vertical and horizontal lines through the points with coordinates (x 1i , x 2j ), where x 1i = −1 + ih 1 for i = 1, · · · , p and x 2j = −1 + jh 2 for j = 1, · · · , q, respectively. Let each mesh point (x 1i , x 2j ) to be the sampling point z ij . In our experiment, for these parameters, we choose m = 1269, n = 693 and p = q = 201, respectively.
To construct the discretized near-field integral equation, we need to collect all the scattered fields u s on the detection points y 1 , · · · , y n for each point source x i , i = 1, · · · , m. For the purpose of numerical experiments, for a given point source x i , we can obtain the scattered fields in row [u s i1 , · · · , u s in ] by solving the direct scattering problem (2) using the FEM in Appendix A. Here, the FEM is applied to the domain enclosed by C, and the corresponding dimensions of E in Table 1 are n = 143834 and m = 1269, respectively. Adding 3% noise to the computed scattered fields u s to produce the detected waves, with which, we discretize (34) into an illposed overdetermined linear system (35) with A = [δ Γ u s ij (x i , y j )] m×n ∈ C m×n , where δ Γ = 6π/n is the arc length of the uniform segment of Γ .
Applying Algorithm 1 to each sampling point z ij ∈ T with all the above parameters to obtain the corresponding indicator I zij for i = 1, · · · , p and j = 1, · · · , q. From numerous numerical experiments by tuning k ∈ (0, √ β * ), we find that the reconstruction results will be heavily affected by the wave number k. In Figure 4, we show the 3d surface figures and the corresponding 2d contour figures by plotting I zij with respect to (x i , y j ) ∈ T , and the actual targets in Figure 2 are plotted by red lines. In fact, Figure 4 shows the best results for reconstructing targets listed in Figure 2 by choosing appropriate wave numbers k * ∈ (0, √ β * ). In Figure 5, we show the 2d contour figures for other wave numbers k ∈ (0, √ β * ). According to our experimental experience, the areas of the reconstructed domains are less than the areas of the actual domains by approximately 10% − 20%. Our numerical experiments also suggest that the reconstructions of convex domains are more stable with respect to the choice of the wave number k and noise.      In comparison, we demonstrate some numerical results of reconstructing the unknown target D for the inverse scattering problem with positive index of refraction. In [24], we considered the Maxwell's equations with the TM mode in Tellegen media and derived the acoustic equation (1a) with a positive ε(x) increasing with respect to the Tellegen parameters. When ε(x) is large enough, we have shown that the transmission eigenvalues are densely distributed near the origin. In this subsection, we show some numerical reconstruction results based on our method when k 2 is an eigenvalue of the associated QEP (presumably a transmission eigenvalue). We test two different kinds of index of refraction on the disk domain, one is a normal index of refraction ε(x) = 16 [24] and the other is ε(x) = 500 (Tellegen model) [23]. As shown in [24], when ε(x) = 16, the lowest transmission eigenvalue is approximately 1.988. In Figure 6(a), 6(b), we see that the reconstruction results are very sensitive to the choice of the wave number k. For the case of ε(x) = 500, our numerical result in [23] indicates that the transmission eigenvalues are distribute densely in the interval (0, 50). Figure 6(c) (choosing k = 5) shows that the disk is completely unrecognizable. The numerical results in this subsection verify the need of avoiding the transmission eigenvalues in the LSM. 6. Conclusion. In this paper, we propose a practical numerical method based on the LSM and the truncated SVD to reconstruct the support of the inhomogeneity in the acoustic equation with negative index of refraction, resulting from in pseudochiral media. It turns out the index of refraction is in the form ε(x) = ε r − (γ 2 1 + γ 2 2 ). We are also interested in the distribution of transmission eigenvalues for the corresponding TEP. The associated discretized TEP can be reduced into a QEP with symmetric coefficient matrices whose all the nonphysical zero eigenvalues are deflated. We prove that the corresponding QEP has half of negative eigenvalues and half of positive eigenvalues in (−β * , 0) and (β * , ∞), respectively, and there exists an eigenvalue-free interval (0, β * ). We then apply the LSM using the near-field data to reconstruct the domain D with ε(x) = 1 and propose an easily implemented truncated SVD technique to determine whether a sampling point is inside or outside of D according to the size of the indicator function. Numerical results show that the effectiveness of our method depends on the wave numbers of incident fields and the convexity of the target.
Appendix A. An FEM framework for the direct scattering problem In this appendix, we will present a numerical framework of solving the direct scattering problem (2) by the FEM given in Section 3. As described in Figure 1, let u i = Φ(x, x 0 ) be an incident field with x 0 ∈ C. Suppose C is far from the origin, since ∆u i + k 2 u i = 0, we have and (46b) ∂u s ∂r = iku s | C .
Continuing to apply the FEM and integration by parts to (46a), combining with (46b), we obtain that (∆u s , ψ j ) + k 2 (ε(x)u s , ψ j ) = − (∇u s , ∇ψ j ) + C ∇ n u s ψ j ds + k 2 (ε(x)u s , ψ j ) and where S and R are given in (47) and (48), and similarly, the mass matrices G 1 and G ε are defined as in Table 1 on domainC. Finally, we can arrange (50), (51), (52) and (53) into a linear system The approximate scattered field u s is then obtained by solving (54).