# American Institute of Mathematical Sciences

August  2018, 12(4): 1033-1054. doi: 10.3934/ipi.2018043

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

 1 School of Mathematics, Southeast University, Shing-Tung Yau Center of Southeast University, Nanjing 211189, China 2 Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan 3 Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan 4 Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan

* Corresponding author: Jenn-Nan Wang

Received  October 2017 Revised  February 2018 Published  June 2018

Fund Project: Li is supported in parts by the NSFC 11471074. Huang was partially supported by the Ministry of Science and Technology (MOST) 105-2115-M-003-009-MY3, National Center of Theoretical Sciences (NCTS) in Taiwan. Lin was partially supported by MOST, NCTS and ST Yau Center in Taiwan. Wang was partially supported by MOST 105-2115-M-002-014-MY3

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 $O(1)$ and the other half of eigenvalues are positive with order of $O(10^2)$. 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.

Citation: Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin, Jenn-Nan Wang. On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method. Inverse Problems & Imaging, 2018, 12 (4) : 1033-1054. doi: 10.3934/ipi.2018043
##### References:
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Google Scholar [16] T.-M. Hwang, W.-W. Lin, W.-C. Wang and W. Wang, Numerical simulation of three dimensional pyramid quantum dot, J. Comput. Phys., 196 (2004), 208-232. doi: 10.1016/j.jcp.2003.10.026. Google Scholar [17] X. Ji, J. Sun and T. Turner, Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues, ACM Trans. Math. Software, 38 (2012), Art. 29, 8 pp. doi: 10.1145/2331130.2331137. Google Scholar [18] X. Ji, J. Sun and H. Xie, A multigrid method for Helmholtz transmission eigenvalue problems, J. Sci. Comput., 60 (2014), 276-294. doi: 10.1007/s10915-013-9794-9. Google Scholar [19] M. Kilmer and D. O'leary, Choosing regularization parameters in iterative methods for ill-posed problems, SIAM J. Matrix Anal. Appl., 22 (2001), 1204-1221. doi: 10.1137/S0895479899345960. Google Scholar [20] A. Kirsch, On the existence of transmission eigenvalues, Inv. Prob. Imaging, 3 (2009), 155-172. doi: 10.3934/ipi.2009.3.155. Google Scholar [21] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008. Google Scholar [22] A. Kleefeld, Numerical Methods for Acoustic and Electromagnetic Scattering: Transmission Boundary-Value Problems, Interior Transmission Eigenvalues, and the Factorization Method, Habilitation Thesis, 2015.Google Scholar [23] T. Li, T. M. Huang, W. W. Lin and J. N. Wang, An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues, Inv. Prob., 33 (2017), 035009, 21pp. doi: 10.1088/1361-6420/aa5475. Google Scholar [24] T. Li, W.-Q. Huang, W.-W. Lin and J. Liu, On spectral analysis and a novel algorithm for transmission eigenvalue problems, J. Sci. Comput., 64 (2015), 83-108. doi: 10.1007/s10915-014-9923-0. Google Scholar [25] L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753. doi: 10.1137/070697525. Google Scholar [26] A. Serdyukov, I. Semchenko, S. Tretyakov and A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Applications, Gordon and Breach Science, 2001.Google Scholar [27] J. Sun, Estimation of transmission eigenvalues and the index of refraction from cauchy data, Inv. Prob., 27 (2010), 015009, 11pp. doi: 10.1088/0266-5611/27/1/015009. Google Scholar [28] J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860-1874. doi: 10.1137/100785478. Google Scholar [29] J. Sun, An eigenvalue method using multiple frequency data for inverse scattering problems, Inv. Prob., 28(2) (2012), 025012, 15pp. doi: 10.1088/0266-5611/28/2/025012. Google Scholar [30] J. Sun and A. Zhou, Finite Element Methods for Eigenvalue Problems, Boca Raton: CRC Press, 2017. Google Scholar [31] A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems, Winston Sons, Washington, D. C., 1977. Google Scholar

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##### References:
 [1] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006. Google Scholar [2] F. Cakoni, D. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, C. R. Math. Acad. Sci. Pairs, 348 (2010), 379-383. doi: 10.1016/j.crma.2010.02.003. Google Scholar [3] F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inv. Prob., 26 (2010), 074004, 14pp. doi: 10.1088/0266-5611/26/7/074004. Google Scholar [4] F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255. doi: 10.1137/090769338. Google Scholar [5] F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493. doi: 10.1080/00036810802713966. Google Scholar [6] F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications: Inside Out II (ed. G. Uhlmann), vol. 60 of Math. Sci. Res. Inst. Publ., Cambridge University Press, Cambridge, 2013,527-580. Google Scholar [7] D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inv. Prob., 21 (2005), 383-398. doi: 10.1088/0266-5611/21/1/023. Google Scholar [8] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inv. Prob., 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. Google Scholar [9] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 3rd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. Google Scholar [10] D. Colton and P. Monk, A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region, SIAM J. Appl. Math., 45 (1985), 1039-1053. doi: 10.1137/0145064. Google Scholar [11] D. Colton, P. Monk and J. Sun, Analytical and computational methods for transmission eigenvalues, Inv. Prob., 26 (2010), 045011, 16pp. doi: 10.1088/0266-5611/26/4/045011. Google Scholar [12] D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inv. Prob. Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13. Google Scholar [13] P. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, Philadelphia, PA, 2010. doi: 10.1137/1.9780898718836. Google Scholar [14] G. C. Hsiao, F. Liu, J. Sun and L. Xu, A coupled BEM and FEM for the interior transmission problem in acoustics, J. Comput. Appl. Math., 235 (2011), 5213-5221. doi: 10.1016/j.cam.2011.05.011. Google Scholar [15] T.-M. Huang, W.-Q. Huang and W.-W. Lin, A robust numerical algorithm for computing Maxwell's transmission eigenvalue problems, SIAM J. Sci. Comput., 37 (2015), A2403-A2423. doi: 10.1137/15M1018927. Google Scholar [16] T.-M. Hwang, W.-W. Lin, W.-C. Wang and W. Wang, Numerical simulation of three dimensional pyramid quantum dot, J. Comput. Phys., 196 (2004), 208-232. doi: 10.1016/j.jcp.2003.10.026. Google Scholar [17] X. Ji, J. Sun and T. Turner, Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues, ACM Trans. Math. Software, 38 (2012), Art. 29, 8 pp. doi: 10.1145/2331130.2331137. Google Scholar [18] X. Ji, J. Sun and H. Xie, A multigrid method for Helmholtz transmission eigenvalue problems, J. Sci. Comput., 60 (2014), 276-294. doi: 10.1007/s10915-013-9794-9. Google Scholar [19] M. Kilmer and D. O'leary, Choosing regularization parameters in iterative methods for ill-posed problems, SIAM J. Matrix Anal. Appl., 22 (2001), 1204-1221. doi: 10.1137/S0895479899345960. Google Scholar [20] A. Kirsch, On the existence of transmission eigenvalues, Inv. Prob. Imaging, 3 (2009), 155-172. doi: 10.3934/ipi.2009.3.155. Google Scholar [21] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008. Google Scholar [22] A. Kleefeld, Numerical Methods for Acoustic and Electromagnetic Scattering: Transmission Boundary-Value Problems, Interior Transmission Eigenvalues, and the Factorization Method, Habilitation Thesis, 2015.Google Scholar [23] T. Li, T. M. Huang, W. W. Lin and J. N. Wang, An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues, Inv. Prob., 33 (2017), 035009, 21pp. doi: 10.1088/1361-6420/aa5475. Google Scholar [24] T. Li, W.-Q. Huang, W.-W. Lin and J. Liu, On spectral analysis and a novel algorithm for transmission eigenvalue problems, J. Sci. Comput., 64 (2015), 83-108. doi: 10.1007/s10915-014-9923-0. Google Scholar [25] L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753. doi: 10.1137/070697525. Google Scholar [26] A. Serdyukov, I. Semchenko, S. Tretyakov and A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Applications, Gordon and Breach Science, 2001.Google Scholar [27] J. Sun, Estimation of transmission eigenvalues and the index of refraction from cauchy data, Inv. Prob., 27 (2010), 015009, 11pp. doi: 10.1088/0266-5611/27/1/015009. Google Scholar [28] J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860-1874. doi: 10.1137/100785478. Google Scholar [29] J. Sun, An eigenvalue method using multiple frequency data for inverse scattering problems, Inv. Prob., 28(2) (2012), 025012, 15pp. doi: 10.1088/0266-5611/28/2/025012. Google Scholar [30] J. Sun and A. Zhou, Finite Element Methods for Eigenvalue Problems, Boca Raton: CRC Press, 2017. Google Scholar [31] A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems, Winston Sons, Washington, D. C., 1977. Google Scholar
The target $D$ is inside some domain $\Omega$ ($\Gamma: = \partial \Omega$) 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 ${\bf x}_0\in C$
Four model domains that represent the region $D$
The eigenvalues $\lambda$ of the QEP (20) in the intervel $[-80,250]$ for the four domains in Figure 2 with $\varepsilon({\bf x}) = -100$. The arrows point to the first positive eigenvalue of the QEP corresponding to each domain
Reconstruction results of four targets in 3d surface figures and in 2d contour figures with $\varepsilon({\bf x}) = -100$ and $k^*\in (0, \sqrt{\beta^*})$. The domains enclosed by red curves are the exact targets $D$. The scattered fields used have $3\%$ noise
Reconstruction results of four targets in 2d contour figures with $\varepsilon({\bf x}) = -100$ and the different $k\in (0, \sqrt{\beta^*})$. The domains enclosed by red curves are the exact targets $D$. The scattered fields have $3\%$ noise
Numerical reconstruction results of a disk when $k^2$ is a transmission eigenvalue. The scattered fields have $3\%$ noise
Stiffness and mass matrices with $\varepsilon({\bf x}) <0$ for ${\bf x} \in \bar{D }$
 stiffness matrix for interior meshes $K = [\int_D \nabla \phi_{i}\cdot\nabla \phi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{n \times n}$ stiffness matrix for interior/boundary meshes $E = [\int_D \nabla \phi_{i}\cdot\nabla \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$ mass matrices for interior meshes $M_{1} = [\int_D \phi_{i} \phi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{n \times n}$ $M_{\varepsilon} = [-\int_D\varepsilon \phi_{i}\phi_{j}d{\bf x}]\succ 0 \in \mathbb{R}^{n \times n}$ mass matrices for interior/boundary meshes $F_{1} = [\int_D \phi_{i} \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$ $F_{\varepsilon} = [-\int_D\varepsilon \phi_{i} \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$ mass matrices for boundary meshes $G_{1} = [\int_D \psi_{i} \psi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{m \times m}$ $G_{\varepsilon} = [-\int_D\varepsilon \psi_{i} \psi_{j}d{\bf x}]\succ 0 \in \mathbb{R}^{m \times m}$
 stiffness matrix for interior meshes $K = [\int_D \nabla \phi_{i}\cdot\nabla \phi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{n \times n}$ stiffness matrix for interior/boundary meshes $E = [\int_D \nabla \phi_{i}\cdot\nabla \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$ mass matrices for interior meshes $M_{1} = [\int_D \phi_{i} \phi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{n \times n}$ $M_{\varepsilon} = [-\int_D\varepsilon \phi_{i}\phi_{j}d{\bf x}]\succ 0 \in \mathbb{R}^{n \times n}$ mass matrices for interior/boundary meshes $F_{1} = [\int_D \phi_{i} \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$ $F_{\varepsilon} = [-\int_D\varepsilon \phi_{i} \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$ mass matrices for boundary meshes $G_{1} = [\int_D \psi_{i} \psi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{m \times m}$ $G_{\varepsilon} = [-\int_D\varepsilon \psi_{i} \psi_{j}d{\bf x}]\succ 0 \in \mathbb{R}^{m \times m}$
Dimensions $n$, $m$ ($K \in \mathbb{R}^{n\times n}$, $E \in \mathbb{R}^{n \times m}$) of matrices for the benchmark problems with the mesh size $h \approx 0.004$
 Domain Disk Ellipse Peanut Heart $(n, m)$ (124631, 1150) (71546,976) (149051, 1871) (168548, 1492)
 Domain Disk Ellipse Peanut Heart $(n, m)$ (124631, 1150) (71546,976) (149051, 1871) (168548, 1492)
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